Que Monsieur

Rafael Bravo De

L A Parra

Alain-Jacques Valleron

Ân, B àng, Cúc Cu'ò'ng

Hoàng, Hôǹg, Kathy, Kiên, Mai, Quê´, Suzzane Thu'ò'ng

Keywords: Prédation intra-guilde, migration rapide, bancs de poisson, zone de protection marine, système à trois niveaux, équation de Lotka-Volterra, système prédateur-proie, modèle SIRS, bruit télégraphique Intraguild predation, fast migration, fish school, marine protected area, three level system, Lotka-Volterra equation, predator-prey system, SIRS model, telegraph noise

successivement par Messieurs Pierre AUGER et Jean-Daniel ZUCKER, à qui j'exprime ma profonde gratitude pour la bienveillance avec laquelle ils m'ont accueilli. Je remercie aussi tout le personnel de l'équipe pour m'avoir permis de mener à bien ce travail. J'exprime ma profonde reconnaissance à mes deux directeurs de thèse, en France et au Vietnam. J'ai eu la chance de faire la connaissance il y a quelques années avec Monsieur Pierre AUGER, Directeur de Recherches à l'IRD, et de le rencontrer lors de ses nombreux voyages au Vietnam. Merci à lui d'avoir accepté d'être mon directeur de thèse en France. Je remercie chaleureusement Monsieur NGUY Ê Ñ Hu

Résumé

Résumé: Dans ce travail de thèse, nous étudions des modèles mathématiques de la dynamique des populations en environnements déterministe et stochastique. Pour les environnements déterministes, nous considérons trois modèles. Le premier est un modèle intra-guilde prenant en compte des e↵ets d'un environnement spatial hétérogène avec une migration rapide des individus entre les di↵érents sites. Le deuxième est un modèle de pêche dans une zone constituée d'une aire marine protégée où la pêche est interdite et d'une zone où la population de poissons est pêchée. Enfin le troisième est un modèle prédateur-proie considérant une proie et deux prédateurs avec des réponses fonctionnelles de Beddington-DeAngelis. Pour les environnements stochastiques, nous étudions un modèle épidémique SIRS et un modèle prédateur-proie en prenant en compte un bruit télégraphique. Nous étudions le comportement dynamique de ces modèles et nous recherchons les conditions de maintien ou de disparition des espéces modélisées. where r = b d is known as the intrinsic growth rate of species or the Malthusian parameter. If x(0) = x 0 is the size of population at the initial time, the equation (1.1.1) has solution x(t) = x 0 e rt . The growth rate r is positive only if birth rate is higher than death rate and then x(t) = x 0 e rt ! 1 as t ! 1. The model (1.1.1) is suitable for the growth of species, like bacteria in a nutrient-unlimited supplied environment.

The Logistic model

The fact that many resources are available only in limited quantities, it is impossible to have unlimited exponential growth of any population over the long run. Moreover, a large population implies fewer resources for each individual. This reason leads to logistic model which is first introduced by Verhulst [START_REF] Verhulst | Notice sur la loi que la population poursuit dans son accroissement[END_REF] and later studied further by R. Pearl and L. J. Reed [START_REF] Pearl | On the rate of growth of the population of the United States since 1790 and its mathematical representation[END_REF] dx dt = rx(1 x K ), r,K > 0.

(1. 1.3) It is observer that ẋ ⇡ rx if x small, and ẋ ⇡ 0 when x is near K. It means that for small x the population dynamics is just the exponential growth, when x is near K the size of population hardly changes. Simultaneously, if x is large, the individuals of the species compete with each other for fewer resources.

By solving equation ẋ = 0, we see that the equation (1.1.3) has two steady states (also called equilibrium or rest point) x ⇤ = 0 and x ⇤ = K. With population size at time t = 0 is x 0 , the solution of model (1.1.3) is obtained as follows

x(t) = Kx(0)e rt K + x(0)(e rt 1)

.

(1.1.4)

From the equation (1.1.3) and solution (1.1.4), we see that if 0 < x < K then ẋ > 0 and x(t) increases asymptotically to K, if x > K, ẋ < 0 and x(t) decreases asymptotically to K as t ! 1. Hence K is called carrying capacity of the environment and it is the population size of species that available resources can support. Other single species models have been used by many ecologist, for example: [START_REF] Pianka | r and K selection or b and d selection?[END_REF]) [START_REF] Pianka | r and K selection or b and d selection?[END_REF], [START_REF] Gilpin | Global models of growth and competition[END_REF] [START_REF] Gilpin | Global models of growth and competition[END_REF], ẋ = rx( I x C bx) [START_REF] Schoener | Population growth regulated by intraspecific competition for energy or time: some simple representations[END_REF]) [START_REF] Schoener | Population growth regulated by intraspecific competition for energy or time: some simple representations[END_REF], ẋ = axlog( x K ) Gompertz's model [START_REF] Swann | Optimal control analysis in the chemotherapy of IgG multiple myeloma[END_REF] [START_REF] Swann | Optimal control analysis in the chemotherapy of IgG multiple myeloma[END_REF],

ẋ = x(b + a x 1 + cx x)
ẋ = rx(1 ( x K ) ✓ )
where a, b, c, r, ✓, K, I, C are positive constants.

Biological interaction

In ecology, biological interactions can involve individuals of the same species (intraspecific interactions) or individuals of di↵erent species (interspecific interactions). Here we concentrate the interaction between species (interspecific interactions), they are categorized as either neutralism, competition, predation/parasitism, mutualism/symbiosis, commensalism or amensalism (see Table 1.1). In this section, we are interested in competition, predation and mutualism.

Competition is a negative interaction that occurs when organisms of di↵erent species use the same resource(s) at the same time and the growth rate of each species is decreased. Examples of competition are Asterionella formosa and Synedra ulna [START_REF] Tilman | Competition and nutrient kinetics along a temperature gradient: an experimental test of a mechanistic approach to niche theory[END_REF], ants and rodents near Portal, Arizona [START_REF] Townsend C R, Begon | Essentials of Ecology[END_REF].

Predation is most commonly considered to be an interaction where an organism (predator) consumes all or part of another living organism (its prey) thereby benefiting itself, but reducing the growth of the prey. For examples, lynx prey upon hares [START_REF] Murray | Mathematical Biology[END_REF], cheetahs and wild dogs kill gazelles [START_REF] Townsend C R, Begon | Essentials of Ecology[END_REF].

Mutualism is the positive interaction that species provide resources or services to each other and each species's growth rate is enhanced. One example of mutualism is the relationship between Labroides dimidiatus (cleaner fish) and Hemigymnus melapterus (client fish) [START_REF] Townsend C R, Begon | Essentials of Ecology[END_REF].

Lotka-Volterra systems

The general Lotka-Volterra equation

The classical Lotka-Volterra model is a pioneering work of Lotka and Volterra in the beginning of twentieth century [START_REF] Brauer | Mathematical Models in Population Biology and Epidemiology, Second Edition[END_REF][START_REF] Murray | Mathematical Biology[END_REF][START_REF] Takeuchi | Global Dynamical Properties of Lotka-Volterra Systems[END_REF]. The general Lotka-Volterra equation for n population is of the form ẋi = x i (r i + n X j=1 a ij x j ) i = 1, ..., n.

(1.1.5)

The x i denote densities of i-th species, the growth rate r i are intrinsic growth (or decay) rates and a ij present the e↵ect of the j-th species upon the i-th species. The matrix A = (a ij ), of which the elements are the interaction coe cients, is called the interaction matrix. Since populations can not be negative, the state space is the nonnegative orthant R n + = {x = (x 1 , ..., x n ) 2 R n : x i 0 for i = 1, ..., n}. In this model, the sign pattern of (a ij , a ji ) describes the interaction between the i-th species and the j-th species. If the sign is ( , ) or (+, +), then the relationship is competition or mutualism respectively. For predator-prey, it is (+, ) or ( , +). Some models of these relationship will be introduced in the next section.

The competition model

Let us consider a model of two competing species. They compete for the same limited resource or in some way inhibit each other's growth. This model is described by basic 2-species Lotka-Volterra competition equation with each species 1 and 2 having logistic growth in the absence of the other. It is of the form ( ẋ = x(a bx cy) ẏ = y(d ex fy), (1.1.6) in which x,y denote densities of species 1 and 2 respectively; a, b, c, d, e and f are all positive constants [START_REF] Brauer | Mathematical Models in Population Biology and Epidemiology, Second Edition[END_REF][START_REF] Edelstein-Keshet | Mathematical Models in Biology[END_REF] . The relative sizes of c, e determine the competitiveness of each species.

The system (1.1.6) has 4 equilibria (0, 0), (0, d f ), ( a b , 0) and (x ⇤ , y ⇤ ) = ( af cd bf ce , bd ae bf ce ). Figure 1.2 shows the all possibilities of solutions of (1.1.6). We summarize the result of the model (1.1.6) as follows:

• af cd > 0, bd ac > 0. In this case there is a positive steady state (x ⇤ , y ⇤ ) in the first quadrant and it is asymptotically stable. All positive solutions of (1.1.6) tend to (x ⇤ , y ⇤ ) (Figure 1.2(a)). The competition between two species is not too strong, two species can coexist.

• cd af > 0, ac bd < 0. (x ⇤ , y ⇤ ) is a positive steady state but it is saddle point.

A separatrix divides the first quadrant into two regions, one containing initial points for which orbits tend to ( a b , 0), and the other containing initial points for which orbits tend to (0, d f ). The survival of species depends on initial state (Figure 1.2(b)).

• af cd > 0, ac bd > 0. There is no positive steady state. ( a b , 0) is an asymptotically stable and all solutions tend to it (Figure 1.2(c)). The species 1 wins and the species 2 will be extinct when t ! 1.

• cd af > 0, bd ac > 0. In this case there is no interior steady state, all orbits tend to (0, d f ) (Figure 1.2(d)). Interspecific competition of species 2 dominates the species 1. Species 2 wins and drives the species 1 to extinction.

The mutualism model

We study a model in which the two species benefit from the presence of the other. We incorporate limited carrying capacities for both species. A simple two-species 1 . I n t r o d u c t i o n mutualism model is given by the next system ( ẋ = x(a bx + cy) ẏ = y(d + ex fy), (1.1.7) where a, b, c, d, e and f are all positive constants. x,y are densities of species 1 and 2 respectively [START_REF] Brauer | Mathematical Models in Population Biology and Epidemiology, Second Edition[END_REF][START_REF] Murray | Mathematical Biology[END_REF]. The constants c and e denote the coe cients interaction between the two species that increase the densities of species 1 and 2 respectively. Depending on the relationship between bf and ce, dynamical behavior of solutions have two possibilities.

• If bf > ce then (1.1.7) has one positive steady state (x ⇤ , y ⇤ ) = ( af +cd bf ce , ae+bd bf ce ) and all interior orbits converge to (x ⇤ , y ⇤ ) (Figure 1.3(a)). Moreover the inequalities x ⇤ > a b and y ⇤ > d f mean that the species converge to populations exceeding their carrying capacities.

• When bf < ce, the mutualistic e↵ects are greater than the self-limiting terms in the per capita growth rates. In this case (1.1.7) has no positive steady state, unbounded growth occurs and all interior orbits diverge to infinity (Figure 1.3(b)). The Predator-prey model

In 1926 Vito Volterra wrote a paper, entitled "Fluctuations in the abundance of a species considered mathematically" [START_REF] Bacaer | A short history of mathematical population dynamics[END_REF], to explain the change of the amount of fish in the Adriatic. He denoted by x the density of the prey fishes, also by y that of predators. He assumed that the growth rate of prey is a positive constant a in the absence of the predator and the predators will decay to zero exponentially with Malthusian parameter c if there exists no prey. This gives the equations ( ẋ = x(a by) ẏ = y( c + dx), (1.1.8) where the parameter b represents the attack rate of predators, the rate d b represents the conversion e ciency [START_REF] Edelstein-Keshet | Mathematical Models in Biology[END_REF][START_REF] Murray | Mathematical Biology[END_REF]. At the same time this model was also constructed by Lotka in the context of chemical kinetics, so it is called Lotka-Volterra model.

(1.1.8) is rewritten as ẏ(a by) y = ẋ( c + dx)

x .

This equation can be integrated directly. Then any solution (x, y) of the equation (1.1.8) satisfies the identity V (x, y) = c ln x a ln y + dx + by = h, where the constant h = V (x 0 , y 0 ) = c ln x 0 a ln y 0 + dx 0 + by 0 is determined by initial conditions (x 0 , y 0 ) and system parameters. Thus every orbit of (1.1.8) is given implicitly by an equation V (x, y) = h. They are periodic orbits with period T , in which time averages of x(t) and y(t) are constant and satisfy

1 T R T 0 x(t)dt = c d , 1 T R T 0 y(t)dt = a b .
Figure 1.4: Orbits of Lotka-Volterra predator-prey model (1.1.8).

In (1.1.8) the growth rate of prey decreases linearly as a function of the predator density. In many model the term bxy is replaced by some explicit forms for the predator functional response ↵y(1 e x ) [START_REF] Ivlev | Experimental Ecology of the Feeding of Fishes[END_REF]) [START_REF] Ivlev | Experimental Ecology of the Feeding of Fishes[END_REF],

1. Introduction ↵xy x + [START_REF] Holling | The functional response of predators to prey density and its role in mimicry and population regulation[END_REF]) [START_REF] Holling | The functional response of predators to prey density and its role in mimicry and population regulation[END_REF],

↵yx (1 > 0) [START_REF] Rosenzweig | Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time[END_REF]) [START_REF] Rosenzweig | Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time[END_REF], ✓xy ↵ + x + y [START_REF] Beddington | Mutual interference between parasites or predators and its e↵ect on searching e ciency[END_REF], [START_REF] Deangelis | A model for trophic interaction[END_REF], independently) [START_REF] Beddington | Mutual interference between parasites or predators and its e↵ect on searching e ciency[END_REF][START_REF] Deangelis | A model for trophic interaction[END_REF],

where ↵, , and ✓ are positive constants.

There are many attempts to apply the Lotka-Volterra predator-prey model to real-world oscillatory phenomena. Figure 1.5 show the data of the population size of lynx and snow hare. The data is collected by The Hudson Bay Company [START_REF] Leah | Mathematical Models In Biology[END_REF][START_REF] Murray | Mathematical Biology[END_REF]. The orbit in Figure 1.5(a) is a more or less closed curve and the oscillation of population size of two species are presented by Figure 1.5(b).

Mathematical modelling of infectious disease

Infectious diseases can be classified into two broad categories: those caused by viruses and bacteria are microparasitic diseases, and those due to worms are macroparasitic. In this section we concentrate on infectious diseases where individuals are infected by pathogen microorganisms (like, for instance, viruses, bacteria, fungi or other microparasites). Some well known examples of such infectious diseases are: Influenza, SARS, Reublla, AIDS, Ebola ...(Viral infectious diseases); Cholera, Plague, Typhus, Leprosy...(Bacterial infectious diseases); Malaria, Taeniasis... (Parasitic infectious diseases);... Mathematical modelling of infectious diseases is a tool to provides understanding of the underlying mechanisms that influence the spread of disease, explains epidemiological phenomena, and predict the future course in order to control an epidemic. Here, we summarize some of the simple classical models for microparasitic infections. In these models we assume the population has no births, deaths, migration and the total population size is a constant N . We also assume that every individual has an equal chance to meet any other member of the population and these interactions are random. The SI model is the simplest model of an infectious disease. By S + I = N , then (1.2.1) can be written as follows

İ = N I(1 I N )
This is a logistic equation with carrying capacity N . Hence, with initial value I(0) > 0, I approaches to N . It means that no matter how small the initial population of infectives, all individuals of population will become infected.

The SIR model

S SI ! I I ! R
The first paper about the SIR model was published by Kermack and McKendrick in 1927. Here, the population is divided into three classes: the susceptibles, S, who can catch the disease; the infectives, I, who have the disease and can transmit it; and the removed class, R, namely, those who have either had the disease, or are recovered, immune or isolated until recovered [START_REF] Murray | Mathematical Biology[END_REF]. Transitions between classes describe the course of transmission and recovery with rate constants and . The model is then 8 > < > :

Ṡ = SI İ = SI I Ṙ = I. (1.2.2)
Adding three equations of (1.2.2), we have Ṡ + İ + Ṙ = 0. Thus, S + I + R = N = const, where N is the total size of the population. We see that R is determined once S and I are known, and we can drop the R equation from (1.2.2), leaving the system of two equations

( Ṡ = SI İ = SI I. (1.2.3)
Let us consider an epidemic problem in a population where only a few individuals are infected at the initial time. The initial conditions are S(0) = S 0 ⇡ N, I(0

) = I 0 = N S(0) ⇡ 0, R(0) = R 0 = 0. From (1.
Ṡ = SI + ↵R İ = SI I Ṙ = I ↵R. (1.2.4)
The system (1.2.5) have two equilibria: disease free equilibrium (0, 0, 0) and endemic equilibrium (S ⇤ , I ⇤ , R ⇤ ) = ( , N

1 + ↵ , N 1 + ↵ ).
In order to exist the positive endemic equilibrium, the following condition must hold:

R 0 = N > 1
In further studying the SIRS model, we note that Ṡ + İ + Ṙ = 0, the sum S + I + R = N and it is a constant of population size. So that for convenience the removed class R can always be eliminated. The reduction of the equation (1.2.4) is then

( Ṡ = SI + ↵(N S I) İ = SI I. (1.2.5)
Depending on the value of basic reproduction number, we have two cases. If

R 0 = N < 1, (1.2.5
) has no positive endemic equilibrium, I decreases and all orbits tend to (N, 0) (Figure 1.8(a)). If R 0 = N > 1, all orbits of (1.2.5) approach to endemic equilibrium (Figure 1 In this section we assume that the models have no demography. Nowadays many kinds of epidemic models, with or without demography, have been proposed and studied. We can found some of them in [START_REF] Bastin | Lectures on Mathematical Modelling of Biological Systems[END_REF][START_REF] Brauer | Mathematical Models in Population Biology and Epidemiology, Second Edition[END_REF][START_REF] Capasso | Mathematical Structures of Epidemic Systems[END_REF][START_REF] Murray | Mathematical Biology[END_REF].

Models and obtained results

In this thesis we give out some ecosystem models and concentrate on dynamical behavior of these models. So we can see the existence or vanishing of species of ecological systems, and when they are dissipative, persistent, permanent. A species with population size x(t) with respect to time t is said to be dissipative if lim sup t!1 x(t) < where = const > 0 and to be (weak) persistent if lim sup t!1 x(t) > 0. Persistence, however, does not ensure survival of a species in a biological sense. Permanence implies the survival of all species which exist initially. A ecological system is said to be permanent if there exist a compact set K in the interior of the state space such as all orbits in the interior end up in K, i.e. there exist [START_REF] Brauer | Mathematical Models in Population Biology and Epidemiology, Second Edition[END_REF][START_REF] Hofbauer | Evolutionary Games and Population Dynamics[END_REF][START_REF] Takeuchi | Global Dynamical Properties of Lotka-Volterra Systems[END_REF]. In chapter 2 we investigate e↵ects of spatial heterogeneous environment and fast migration of individuals on coexistence of the intraguild predation (IGP) dynamics. We present a two-patch model. We assume that on one patch two species compete for a common resource, and on the other patch one species can capture the other one for the maintenance. We also assume IGP individuals are able to migrate between two patches and the migration process acts on a fast time scale in comparison with demography, predation and competition processes. This chapter show that under certain conditions the heterogeneous environment and fast migration can lead to coexistence of the two species.

1 , 2 > 0 such that 1 < lim inf t!1 x(t) < lim sup t!1 x(t) < 2

Models and obtained results

13

In chapter 3 we consider a fishery model with two sites: (1) a Marine Protected Area (MPA) where fishing is prohibited and (2) an area where the fish population is harvested. We assume that fish can migrate from MPA to fishing area at a very fast time scale and fish spatial organisation can change from small to large clusters of school at a fast time scale. The growth of the fish population and the catch are assumed to occur at a slow time scale. The complete model is a system of 5 ordinary di↵erential equations with three time scales. We take advantage of the time scales using aggregation of variables methods to derive a reduced model governing the total fish density and fishing e↵ort at the slow time scale. We analyze this aggregated model and show that under some conditions, there exists an equilibrium corresponding to a sustainable fishery. Our results suggest that in small pelagic fisheries the yield is maximum for a fish population distributed among both small and large clusters of school.

We study a predator-prey model which has one prey and two predators with Beddington-DeAngelis functional responses in chapter 4. Firstly, we establish a set of su cient conditions for the permanence and extinction of species. Secondly, the periodicity of positive solutions is studied. Thirdly, by using Liapunov functions and the continuation theorem in coincidence degree theory, we show the global asymptotic stability of such solutions. Finally, we give some numerical examples to illustrate the behavior of model. Some population models with the e↵ect of random environment were studied in chapter 5. It is structured into two sections. In the first section we present the Kernack -MacKendrick model under telegraph noise. The telegraph noise switches at random between two SIRS models. We give out conditions for the persistence of the disease and the stability of a disease free equilibrium. We show that the asymptotic behavior highly depends on the value of a threshold which is calculated from the intensities of switching between environmental states, the total size of the population as well as the parameters of both SIRS systems. According to the value of , the system can globally tend towards an endemic case or a disease free case. The aim of this chapter is also to describe completely the omega-limit set of all positive solutions to the model. Moreover, the attraction of the omega-limit set and the stationary distribution of solutions will be pointed out.

In the second section of chapter 5, we consider Lotka-Volterra predator-prey systems with the e↵ect of telegraph noise. For predator-prey system with carrying capacity of environment, we focus on persistence of the predator and thus we look for conditions that allow persistence of the predator and prey community. We also determine the subset of omega-limit set of this system. For predator-prey model with the absence of carrying capacity, it is proved that under the influence of telegraph noise, all positive trajectories of such a system always exile from any compact set of intR 2 + with probability one if two rest points of the two systems do not coincide. In case where they have the rest point in common, the trajectory either leaves from any compact set of intR 2 + or converges to the rest point. The exile of the trajectories from any compact set means that the system is neither permanent nor dissipative.

Chapter 2

Spatial heterogeneity, fast migration and coexistence of Intraguild Predation Dynamics

Introduction

In this chapter we present paper [112]. It is well-known that interactions between species are usually categorized as either competition (negative e↵ects on each other), predation/parasitism (one got positive e↵ect and the other got negative e↵ect), mutualism (positive e↵ects on each other), commensalism (one got positive e↵ect and the other got no e↵ect) or amensalism (one got negative e↵ect and the other got no e↵ect). Intraguild predation (IGP) is a combination of the first two, that is, the killing and eating species that use similar resources and are therefore potential competitors [START_REF] Polis | The ecology and evolution of intraguild predation: potential competitors that eat each other[END_REF]. Thus, broadly speaking, cannibalism is considered as a form of IGP unless there is a distinct ontogenetic niche shift that di↵erentiates the resource profile of cannibals and their victims [START_REF] Wise | Cannibalism, food limitation, intraspecific competition and the regulation of spider populations[END_REF]. For example, most spiders are generalist predators that feed on a variety of prey items such as mosquitoes and flies, making them members of the same guild. However, spiders also eat other spiders, we count this cannibalism as intraguild predation. IGP commonly involves larger individuals feeding on smaller individuals [START_REF] Polis | Exploitation competition and the evolution of interference, cannibalism, and intraguild predation in age/size-structured populations[END_REF]. We call the victim intraguild prey (IGprey) and the predator intraguild predator (IGpredator). IGP is common in nature and is found in a variety of taxa [START_REF] Polis | The evolution and dynamics of intraspecific predation[END_REF][START_REF] Polis | Exploitation competition and the evolution of interference, cannibalism, and intraguild predation in age/size-structured populations[END_REF][START_REF] Polis | Intraguild predation: The dynamics of complex trophic interactions[END_REF][START_REF] Polis | The ecology and evolution of intraguild predation: potential competitors that eat each other[END_REF][START_REF] Williams | Simple rules yield complex food webs[END_REF]. It di↵ers from classical predation because the act reduces potential exploitation competition. Thus, its impact on population dynamics is much more complex than either competition or predation alone. One characteristic of IGP is the simultaneous existence of competitive and trophic interactions between the same species.

Theoretical models predict that coexistence of IGpredator and IGprey is dicult, because IGprey experience the combined negative e↵ects of competition and predation [START_REF] Holt R D | A theoretical framework for intraguild predation[END_REF]. In systems with competition only, IGprey su↵ers no predation. In standard predator-prey interactions without competition, IGprey su↵er no exploita-2. Spatial heterogeneity, fast migration and coexistence of IGP dynamics tive competition from the IGpredator. Thus, IGP is more stressful for the intermediate consumer (IGprey) than either exploitative competition or trophic interaction alone.

The theoretical di culty in explaining IGP persistence and its observed ubiquity have identified IGP as an ecological puzzle [START_REF] Holt R D | A theoretical framework for intraguild predation[END_REF]. This led to a series of studies in order to resolve the puzzle. These studies have considered factors such as top predators (food web topology) [START_REF] Yurewicz | A growth/mortality trade-o↵ in larval salamanders and the coexistence of intraguild predator and prey[END_REF], size structure [START_REF] Borer | Intraguild predation in larval parasitoids: implications for coexistence[END_REF][START_REF] Macneil | The predatory impact of the freshwater invader Dikerogammarus villosus on native Gammarus pulex (Crustacea: Amphipoda); influences of di↵erential microdistribution and food resources[END_REF][START_REF] Mylius | Impact of intraguild predation and stage structure on simple communities along a productivity gradient[END_REF], habitat segregation [START_REF] Macneil | The predatory impact of the freshwater invader Dikerogammarus villosus on native Gammarus pulex (Crustacea: Amphipoda); influences of di↵erential microdistribution and food resources[END_REF], metacommunity dynamics [START_REF] Melían | Food web structure and habitat loss[END_REF], intraspecific predation [START_REF] Dick | Replacement of the indigenous amphipod Gammarus duebeni by the introduced G. pulex: di↵erential cannibalism and mutual predation[END_REF], and adaptive behavior [START_REF] Krivan | Optimal foraging and predator-prey dynamics[END_REF][START_REF] Krivan | Optimal intraguild foraging and population stability[END_REF][START_REF] Krivan | Trait and density mediated indirect interactions in simple food webs[END_REF].

Here, we investigate an IGP model in a two-patch environment. We assume that on one patch is pure exploitation competition, and on the other one IGpredator can capture IGprey for its maintenance. This considered scenario can potentially occur in some ecological systems. For instance, on a given patch with an abundant resource the interaction of species is more likely to be exploitation competition, while on another patch with a limit of resource one species is more likely to reduce the risk for shared resource by feeding on its competitors [START_REF] Crumrine | Partitioning components of risk reduction in a dragonfly-fish intraguild predation system[END_REF]. Aquatic invertebrates and fishes tend to prey on eggs and larvae of their resource competitors (see examples in [START_REF] Polis | The ecology and evolution of intraguild predation: potential competitors that eat each other[END_REF]). In some populations [START_REF] Polis | Exploitation competition and the evolution of interference, cannibalism, and intraguild predation in age/size-structured populations[END_REF], larger individuals feed on smaller individuals. Therefore, one can imagine complexity of environment may lead to the fact that individuals can (or cannot) encounter eggs, larvae and juveniles of their resource competitors (example includes niche and refuges), the predation can (or cannot) happen. The authors in [START_REF] Janssen | Habitat Structure A↵ects Intraguild Predation[END_REF] showed that habitat structure could reduce encounter rates between IGpredator and IGprey.

In the current contribution, we assume a non-coexistence of species locally. In the competition patch, we suppose that IGpredator is the superior exploitation competitor, i.e. without migration IGpredator out-competes IGprey. In the predation patch, we suppose that IGpredator is the inferior one so that IGpredator mainly capture IGprey in order to maintain. Moreover IGprey has a good tactics to exploit resource as well as to avoid the risk of IGpredator. This leads to the fact that without migration IGprey drives IGpredator out. Both patches are connected by density-independent migration of individuals of both IGpredator and IGprey. It is assumed that migration is fast in comparison with competition and predation in the local patches. In this work, we are going to investigate whether spatial heterogeneous environment and fast migration between patches lead to coexistence of IGP system.

We consider a fast migration in comparison with demography and interaction of species. In fact, many ecological systems highlight that migration occurs on a fast time scale relative to competition. For instance, in long lived organisms such as trees gene flow through pollination or migration can take place at a much faster time scale than selection process [START_REF] Garca-Ramos | Genetic models of adaptation and gene flow in peripheral populations[END_REF]. In host-parasite systems (in which the individual host is the patch), the interplay between within-patch and among-patch evolutionary dynamics drives the evolution of intermediate levels of virulence [START_REF] Levin | Selection of intermediate rates of increase in parasitehost systems[END_REF]. The authors in [START_REF] Morozov | Toward a correct description of zooplankton feeding in models: taking into account food-mediated unsynchronized vertical migration[END_REF] proposed a mathematical model of zooplankton moving in the water column with food-mediated fast vertical migrations. This work showed that fast vertical migration could enhance ecosystems stability and regulation of algal blooms. Another example can be found in [START_REF] Morozov | Nutrient-rich plankton communities stabilized via predator -prey interactions: revisiting the role of vertical heterogeneity[END_REF] where authors study a model of fast-moving zooplankton capable of quick adjustment of grazing load in the water column and argue that it could be a generic self-regulation process in nature. Yet the author in [START_REF] Amarasekare | Productivity, dispersal and the coexistence of intraguild predators and prey[END_REF] investigated the case where migration, demography and interaction of species act on the same time scale in an IGP model. It is shown that this migration mode can allow IGP species to coexist. We therefore consider the IGP model including the two time scales.

Taking advantage of these two time scales, we are able to use aggregation methods that allow us to reduce the dimension of the complete model and to derive a global model at the slow time scale governing the total species densities [START_REF] Auger | Aggregation of variables and applications to population dynamics[END_REF][START_REF] Auger | Aggregation methods in dynamical systems variables and applications in population and community dynamics[END_REF][START_REF] Auger | Complex ecological models with simple dynamics: From individuals to populations[END_REF]. For aggregation of variables methods, we also refer to some investigations [START_REF] Iwasa | Aggregation in model ecosystems. I. Perfect aggregation[END_REF][START_REF] Iwasa | Aggregation in model ecosystems. II. Approximate aggregation[END_REF]. Some applications of the aggregation method to population dynamics can be found in [START_REF] Auger | Emergence of population growth models: fast migration and slow growth[END_REF][START_REF] Mchich | The dynamics of a fish stock exploited between in two fishing zones[END_REF][START_REF] Nguyen | Competition and species coexistence in a metapopulation model: can fast dispersal reverse the outcome of competition?[END_REF] and [START_REF] Poggiale | From behavioral to population level: growth and competition[END_REF]. This chapter is organized as follows. Section 2.2 shows the mathematical model. In Section 2.3, we present reduction of the model. It is structured into two subsections. Subsection 2.3.1 presents the study of fast equilibrium. Subsection 2.3.2 is devoted to aggregated model. The results are discussed in Section 2.4. The last section 2.5 is about conclusion and perspectives. We consider an IGP model in a two-patch environment. We assume there is an abundant resource on patch 1 therefore IGpredator and IGprey compete with 2. Spatial heterogeneity, fast migration and coexistence of IGP dynamics each other for the common resource. A classical Lotka-Volterra competition model is used in order to represent this competition dynamics. In patch 2, it is assumed that IGpredator is the inferior exploitation competitor so that IGpredator mainly capture IGprey to maintain [START_REF] Crumrine | Partitioning components of risk reduction in a dragonfly-fish intraguild predation system[END_REF][START_REF] Polis | Exploitation competition and the evolution of interference, cannibalism, and intraguild predation in age/size-structured populations[END_REF][START_REF] Polis | The ecology and evolution of intraguild predation: potential competitors that eat each other[END_REF]. A classical predator-prey model is used to represent this predation dynamics. Both patches are connected by migration of IGP individuals (see Figure 2.1). We further assume the migration process acts on a fast time scale than the demography, the competition and predation processes in the two local patches. According to these assumptions, the complete system reads as follows: are respectively the intrinsic growth rate and the carrying capacity of IGprey in patch 2. b is predation capture rate, e is the parameter related to predator recruitment as a consequence of predator-prey interaction. d is natural mortality rate of the IGpredator on the predation patch. For the IGprey, parameter m 1 is the per capita migration rate from the predation patch to the competition patch, and m 1 , from the competition patch to the predation patch. For the IGpredator, parameter m 2 is the per capita migration rate from the predation patch to the competition patch, and m 2 , from the competition patch to the predation patch. Parameter " represents the ratio between two time scales t = "⌧ , t is the slow time scale and ⌧ is the fast one. In this chapter, we are interested in an asymmetric interaction i.e. without migration IGpredator is the superior exploitation competitor on the competition patch, but is the inferior one on the predation patch so that IGPredator mainly capture IGprey to maintain [START_REF] Crumrine | Partitioning components of risk reduction in a dragonfly-fish intraguild predation system[END_REF][START_REF] Polis | Exploitation competition and the evolution of interference, cannibalism, and intraguild predation in age/size-structured populations[END_REF][START_REF] Polis | The ecology and evolution of intraguild predation: potential competitors that eat each other[END_REF]. In the predation patch, it is further assumed that IGprey is able to avoid the risk of IGpredator leading to the fact that IGprey drives IGpredator out. Assuming the asymmetric interaction implies the next inequalities hold [START_REF] Murray | Mathematical Biology[END_REF]

Model

8 > > > > > > > > > < > > > > > > > > > : dx 1 d⌧ = (m 1 x 2 m 1 x 1 ) + "r 11 x 1 ✓ 1

Model reduction

Taking advantage of the two time scales, we now use aggregation of variables method in order to derive a reduced model [START_REF] Auger | Aggregation of variables and applications to population dynamics[END_REF][START_REF] Auger | Aggregation methods in dynamical systems variables and applications in population and community dynamics[END_REF][START_REF] Auger | Complex ecological models with simple dynamics: From individuals to populations[END_REF][START_REF] Nguyen | Coupling Equation-Based and Individual-Based Models When Studying Complex Systems: A Case Study in Theoretical Population Ecology[END_REF]. The first step is to look for existence of a stable and fast equilibrium. The fast equilibrium is the solution of the system (1) while only considering the fast part, i.e. when " = 0. The fast part corresponds to dispersal, so the fast equilibrium corresponds to the stable distribution corresponding to the dispersal process. We then consider that for the complete model, the system is always at the fast equilibrium, i.e. at any time the distribution of individuals among patches corresponds to the stable distribution. We obtain a model with two equations on which we can perform a mathematical analysis.

Fast equilibrium

Over the fast time scale ⌧ , the total IGprey population (x(t) = x 1 (t) + x 2 (t)) and IGpredator population (y(t) = y 1 (t)+y 2 (t)) are constant. After straightforward calculation, there exists a single fast and stable equilibrium that reads as follows:

-for IGprey: 8 > < > :

x ⇤ 1 = m 1 m 1 + m 1 x = ⌫ ⇤ 1 x x ⇤ 2 = m 1 m 1 + m 1 x = ⌫ ⇤ 2 x, (2.3.1)
-for IGpredator: 8 > < > :

y ⇤ 1 = m 2 m 2 + m 2 y = µ ⇤ 1 y y ⇤ 2 = m 2 m 2 + m 2 y = µ ⇤ 2 y. (2.3.2)
Therefore, the proportions of individuals of IGP in each patch rapidly tend toward to constant values which are proportional to migration rates to the patches.

Aggregated model

Coming back to the complete initial system (2.2.1), we substitute the fast equilibria (2.3.1), (2.3.2) and add the two equations of the local IGprey and IGpredator population densities, leading to the following aggregated system when using the slow time scale t:

8 > > > < > > > : dx dt = x (A Bx Cy) dy dt = y (D Ey F x) , (2.3.3)
where

A = r 11 ⌫ ⇤ 1 + r 12 ⌫ ⇤ 2 , B = r 11 K 11 ⌫ ⇤ 1 2 + r 20
2. Spatial heterogeneity, fast migration and coexistence of IGP dynamics

Results and discussions

Let's analyse the aggregated model. One can see that system (2.3.3) has four equilibria P 1 (0, 0), P 2 (0, D/E), P 3 (A/B, 0), P 4 ((CD AE)/(CF BE), (AF BD)/(CF BE)). A full stability analyses of these equilibria is given in Table 2.1.

In summary, the outcome of the dynamics of the aggregated model depends on the signs of D, F, CD AE and AF BD. These expressions depend on parameters such as the migration parameters (µ and ⌫), the competition parameters (a 12 and a 21 ), the predation parameters (b and e), the carrying capacity (K) and so on. Here, we are interested in e↵ects of the migration parameters, the competition parameters and the predation ones. To avoid dealing with complex expressions, we assume the two patches are similar for population growth r

11 = r 12 = r 21 = r and K 11 = K 12 = K 21 = K.
This yields the following simplified expressions:

D = (r + d)µ ⇤ 1 d, F = ra 21 K ⌫ ⇤ 1 µ ⇤ 1 eb(1 ⌫ ⇤ 1 )(1 µ ⇤ 1 ), AE = r 2 K (µ ⇤ 1 ) 2 , AF = ( r 2 a 21 K ebr)µ ⇤ 1 ⌫ ⇤ 1 + ebrµ ⇤ 1 + ebr⌫ ⇤ 1 ebr, BD = 2r(r + d) K µ ⇤ 1 (⌫ ⇤ 1 ) 2 2r(r + d) K µ ⇤ 1 ⌫ ⇤ 1 + r(r + d) K µ ⇤ 1 2dr K (⌫ ⇤ 1 ) 2 + 2dr K ⌫ ⇤ 1 dr K , CD = (r + d)(ra 12 + bK) K (µ ⇤ 1 ) 2 ⌫ ⇤ 1 (r + d)b(µ ⇤ 1 ) 2 (ra 12 d + brK + 2bdK) K µ ⇤ 1 ⌫ ⇤ 1 + (rb + 2bd)µ ⇤ 1 + bd⌫ ⇤ 1 bd.
Now we are going to investigate the dynamics in term of the proportion of IGprey on patch 1 (⌫ ⇤ 1 ) which is re-denote by X and of the proportion of IGpredator (µ ⇤ 1 ) which is re-denote by Y . Since the model is a combination of competition and predation models, one could expect that the outcome of the model is also a combination of the outcomes of the two. In fact, the outcome of the model can be all possibilities of the two species, i.e. coexistence and one of the two wins. Figure 2.2a shows an example where the two species coexist. Figure 2.2b illustrates the case where IGpredator wins, while Figure 2.2c illustrates the situation IGprey wins. Figure 2.2d shows the separatrix case where IGpredator or IGprey wins depending on the initial conditions. For these figures, we chose the same values of the following parameters r

11 = r 12 = r 21 = r 22 = r = 0.6, K 11 = K 12 = K 21 = K = 10, a 12 = 1.5, a 21 = 0.7, b = 0.3, d = 0.
6 and e = 0.1. Then we changed the values of X and Y which correspond to the proportions of IGprey and IGpredator, respectively, on the competition patch. We chose parameters values according to existing literatures. The growth rates are equal to 0.6, the carrying capacities are equal to 10 which are the same magnitude as those found in [START_REF] Hairston | Ecological Experiments: Purpose, Design and Execution[END_REF] (r = 0.44, K = 15) and in [START_REF] Gause | Experimental Studies on the Struggle for Existence: I. Mixed Population of Two Species of Yeast[END_REF] (r = 0.21827, K = 13). The competition coe cients, the predation coe cient and the mortality were chosen the same magnitude as those found in [START_REF] Cao | Estimating a Predator-Prey Dynamical Model with the Parameter Cascades Method[END_REF][START_REF] Gause | Experimental Studies on the Struggle for Existence: I. Mixed Population of Two Species of Yeast[END_REF][START_REF] Hairston | Ecological Experiments: Purpose, Design and Execution[END_REF][START_REF] Pearce | Some e↵ects of lates spp. on pelagic and demersal fish in Zambian waters of lake Tanganyika[END_REF] (the competition coe cients are between 0.02 to 3.15, the predation coe cients are between 0.02 to 3 and the mortalities are between 0.055 to 0.52). 2a is related to coexistence case (X = 0.2 and Y = 0.9), Figure 2b describes the case where IGpredator wins (X = 0.4 and Y = 0.7), Figure 2c shows the win of IGprey (X = 0.8 and Y = 0.2), Figure 2d is the separatrix case (X = 0.7 and Y = 0.7). enough on the predation patch in order to gain advantage over IGprey. In both situations, IGpredator wins globally. Domain III is related to the two cases. The first case is IGpredator individuals are few on the competition patch so that it decreases IGpredation's invasion on this patch. This leads to the fact IGprey is able to invade and to win globally. The second case is IGpredator has comparable distributions on the two patches, but IGprey distributes well enough on the competition patch in order to gain advantage over IGpredator. Therefore, IGprey still wins eventually. Domain I below of Figure 2.3 links to the the following situation. IGpredator has comparable distributions on the two patches and IGprey distributes mainly on the predation patch. Therefore, the e↵ect of IGprey on IGpredator on the competition patch is not strong yet the abundance of IGprey is better for maintenance of IGpredator on the predation patch. Thus, coexistence can be possible. Domain IV corresponds to the case where individuals of the two species are mainly on the competition patch. Too many individuals on the competition patch has negative e↵ects on the resource exploitation. Yet a few IGprey individuals on the predation patch not only has negative e↵ects on the maintenance of IGpredator but also decreases its invasion. Globally, this case is disadvantage for both species then species wins depending on the initial condition. Now, we study e↵ects of competition and predation parameters on the areas of the four domains. Keeping the same values of parameters as in Figure 3, we are going to change the value of one of the three parameters a 12 , a 21 and b. According to the conditions (2.2) and (2.3) we have a 12 is greater than 1, a 21 is smaller than 1 and b is smaller than d/(eK) = 0.6. of a 12 by 1.5, 5 and 8.5, respectively. According to mathematical point of view, the black dash line (AE CD = 0) changes while the grey dash line (AF BD = 0) does not change. According to ecological point of view, increase of a 12 means that the e↵ect of IGpredator on IGprey on the competition patch increases. Thus, it increases the areas of the domains which are disadvantage for IGprey. In fact, one can observe that part of domain I (both on top and below) now turns into domain II, domain I is therefore gets smaller while domain II gets bigger, and part of domain III now turns into domain IV, domain III is therefore gets smaller while domain IV gets bigger.

Figure 2.5 shows three cases from the left to the right where we changed the value of a 21 by 0.7, 0.4 and 0.1, respectively. In this case the grey dash line changes while the black dash line does not change. We decrease the value of a 21 meaning that the e↵ect of IGprey on IGpredator on the competition patch decreases too. So, it decreases the areas of the domains which are disadvantage for IGpredator. One can observe that domain IV turns into domain II, part of domain III turns into domain I below. Therefore, domain III gets smaller, domain IV disappears, domain II and domain I below get bigger.

Conclusion and perspectives

We have presented an IGP model in a two-patch environment: the interaction on a given patch is pure competition and that on the other patch is predation. We focus in particular on an asymmetric interaction i.e. without migration IGpredator is the superior exploitation competitor on the competition patch, but is the inferior one on the predation patch so that IGPredator mainly capture IGprey to maintain. We concentrated on the case where the predation is weak leading to the fact that IGprey wins on the predation patch.

The model is a coupling a classical competition model on a given patch and a classical predation model on the other patch. The two patches are connected by a fast migration of individuals. This assumption allows us to obtain the aggregated model which can be investigated analytically. As a first result, we showed that the IGP dynamics can be either competition or predation depending on the parameters. Figure 2.2 showed that all outcomes of the interaction of the two species can be achieved: coexistence, IGpredator wins, IGprey wins, species wins depending on the initial condition. Here we focus on migration, competition and predation parameters. When we fix competition and predation parameters, outcome of the dynamics depends migration parameters. We obtain four domains corresponding to the outcomes (Figure 2.3). When we respectively changed competition and predation parameters, the four domains changed (Figure 2.4, 2.5 and 2.6).

In our model, IGprey and IGpredator cannot coexist locally in the sense that each species is able to out-compete the other without migration: IGpredator wins on the competition patch and IGprey wins on the predation patch. Coexistence of the two species can be achieved under certain conditions. When each species individuals are almost there on the patch where they can invade, coexistence can be possible. The two species can also coexist when IGpredator has comparable distributions on the two patches and IGprey distributes mainly on the predation patch. In this situation, IGprey is abundant on the patch where it can invade and it survives globally. IGpredator individuals can capture more IGprey individuals on the predation patch, yet they compete with few IGprey individuals on the competition Chapter 3

E↵ect of small versus large clusters of fish school on the yield of a purse-seine small pelagic fishery including a marine protected area

Introduction

This chapter shows paper [START_REF] Nguyen | E↵ect of small versus large clusters of fish school on the yield of a purse-seine small pelagic fishery including a marine protected area[END_REF]. There was an increasing interest in modelling the dynamics of a fishery, we refer to review and classical contributions dealing with mathematical approaches [START_REF] Clark | Mathematical Bioeconomics: The Optimal Management of Renewable Resources[END_REF][START_REF] Lara M De | Sustainable Management of renewable resources: Mathematical Models and Methods[END_REF][START_REF] Smith | Economics of production from natural resources[END_REF][START_REF] Smith | On models of commercial fishing[END_REF], and more ecological ones [START_REF] Brochier | A multi-agent ecosystem model for studying changes in a tropical estuarine fish assemblage within a marine protected area[END_REF][START_REF] Fulton | Lessons in modelling and management of marine ecosystems: the Atlantis experience[END_REF][START_REF] Maury | An overview of APECOSM, a spatialized mass balanced "Apex Predators ECOSystem Model" to study physiologically structured tuna population dynamics in their ecosystem[END_REF][START_REF] Yemane | Exploring the e↵ect of Marine Protected Areas on the dynamics of fish communities in the southern Benguela: an individual-based modelling approach[END_REF]. Spatio-temporal distribution is a major factor a↵ecting fish catchability, particularly for small pelagic fish [START_REF] Arreguín-Sánchez | Catchability: a key parameter for fish stock assessment[END_REF]. Small pelagic fish species are the most exploited fish species at the world level and play a major role in world food security [START_REF] Tacon | Use of fish meal and fish oil in aquaculture: a global perspective[END_REF]. However, theses populations are threatened by both climate change [START_REF] Brochier | Climate change scenarios experiments predict a future reduction in small pelagic fish recruitment in the Humboldt Current system[END_REF][START_REF] Fréon | Conjectures on future climate e↵ects on marine ecosystems dominated by small pelagic fish[END_REF] and over-fishing [START_REF] Pinsky | Unexpected patterns of fisheries collapse in the world's oceans[END_REF]. Thus, there is a need of research to feed future management plans for these species.

Here, we present a mathematical model of a fishery targeting a small pelagic fish population distributed over two sites, a MPA and a fishing area where the fish population can be captured by purse-seine fishing boats. Following the literature [START_REF] Brehmer | Schooling behaviour of small pelagic fish: phenotypic expression of independent stimuli[END_REF][START_REF] Petitgas | Spatial organization of pelagic fish: echogram structure, spatio-temporal condition, and biomass in Senegalese waters[END_REF] we assume that small pelagic fish can either be distributed in few large clusters of fish school (hereafter referred as "cluster") or in a greater number of smaller clusters [START_REF] Petitgas | Spatial organization of pelagic fish: echogram structure, spatio-temporal condition, and biomass in Senegalese waters[END_REF]. There is evidence that large clusters are generally more easily located by fishing boats than smaller ones [START_REF] Brehmer | Omnidirectional multibeam sonar monitoring: Applications in fisheries science[END_REF]. Industrial fishing fleets use electronic devices such as sonar to detect school and the e ciency is better for large school [START_REF] Brehmer | Adaptation of fisheries sonar for monitoring large pelagic fish school: dependence of schooling behaviour on fish finding e ciency[END_REF] which may occur more often in large clusters [START_REF] Petitgas | Spatial organization of pelagic fish: echogram structure, spatio-temporal condition, and biomass in Senegalese waters[END_REF]. Fishermen of artisanal fleets can even simply detect fish school by visual observation when the school is close to the surface (upper part of the water column). Thus, once fishermen detected a school that belongs to a large cluster, they access easily the other fish schools that belong to this cluster. Furthermore, purse-seine fisheries generally operate in collaborative 3. E↵ect of Small Versus Large Clusters of Fish School fleets of several boats and join their e↵orts on large clusters. As a consequence, fish in large clusters are more exposed to fishing pressure due to increased accessibility.

The aim of the present model is to investigate the e↵ects of fish clustering on the total catch of a small pelagic purse seine fishery. What are the e↵ects of large or small clusters on the global dynamics of the fishery? Is there a proportion of small and large clusters which is optimal in terms of total catch on the long term for a given fishery and fishing e↵ort?

The complete model is a set of 5 coupled ordinary di↵erential equations (ODEs) with four variables representing fish populations divided into large or small clusters and located in MPA or in fishing area, and one variable representing a single fishing e↵ort in the fishing area whatever the cluster size. We further assume that there are three time scales: fish can migrate from MPA to the fishing area at a very fast time scale, fish can change state from small to large clusters at a fast time scale and fish growth and catch occur at a slow time scale.

To our knowledge, aggregation methods were not used to aggregate a system involving three time scales. This contribution thus shows an example of aggregation of variables in a three level system. This aggregation of a three level system requires a two-step aggregation, aggregating firstly from very fast to fast dynamics and secondly from fast to slow dynamics. Here, we simply proceed to aggregation in order to derive the slow aggregated model. We numerically show that the aggregation method is valid as soon as there exists (for the present case) an order of magnitude between two consecutive time scales (fast/very fast) or (slow/fast). Under these conditions, numerical simulations show that the dynamics of the complete and the aggregated models are very similar, i.e. the trajectories of both systems starting at the same initial conditions remain close to each other. This chapter is organized as follows. Section 3.2 presents the complete fishery model. Section 3.3 and 3.4 present the aggregation method in order to derive a global model at the slow time scale with two consecutive steps. Section 3.5 studies the e↵ects of exploited fish population structuration in small vs. large clusters on the total catch of the fishery. The chapter ends with a discussion according to our theoretical results on the yield of a given fishery and opens some perspectives.

Complete Model

We consider a population of fish that is harvested. The model takes into account fish densities and the fishing e↵ort. The model is a two sites model: a Marine Protected Area or MPA (index M ) where fishing is prohibited and a Fishing area (index F ) where the fish population is harvested. We assume that fish can migrate from MPA to fishing area F and inversely. Furthermore, fish school can belong to Small clusters (index S) or to Large clusters (index L). We assume that fish can change state from S to L and inversely. Therefore, fish school can leave large clusters to form small clusters and inversely (see figure 3.1). Fish population grows logistically with a total carrying capacity K with a fraction h in MPA and (1 h) in the fishing area. Fish are captured in the fishing area according to a Schaefer function [START_REF] Schaefer | Some considerations of population dynamics and economics in relation to the management of the commercial marine fisheries[END_REF]. As a consequence, there are 4 fish sub-populations in the model:

• n SM : density of fish in small clusters in MPA; • n LM : density of fish in large clusters in MPA; • n SF : density of fish in small clusters in fishing area; • n LF : density of fish in large clusters in fishing area. There is a single fishing e↵ort in the fishing area noted E. The model reads as follows:

8 > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > : dn SM d⌧ = (m S n SF m S n SM ) + "(kn LM kn SM ) +"µ ✓ rn SM ✓ 1 n SM + n LM hK ◆◆ dn LM d⌧ = (m L n LF m L n LM ) + "(kn SM kn LM ) +"µ ✓ rn LM ✓ 1 n SM + n LM hK ◆◆ dn SF d⌧ = (m S n SM m S n SF ) + "(kn LF kn SF ) +"µ ✓ rn SF ✓ 1 n SF + n LF (1 h)K ◆◆ "µq S n SF E dn LF d⌧ = (m L n LM m L n LF ) + "(kn SF kn LF ) +"µ ✓ rn LF ✓ 1 n SF + n LF (1 h)K ◆◆ "µq L n LF E dE d⌧ = "µ( cE + pq S n SF E + pq L n LF E), (3.2.1)
3. E↵ect of Small Versus Large Clusters of Fish School where all parameters are defined in Table 3.1.

We suppose that q S < q L , i.e. fishermen catch much better fish in large clusters than in small ones. We further assume that there exist three time scales:

• Migration (MPA/fishing area) is a very fast process;

• State change (Small clusters/Large clusters) is a fast process;

• Catch and growth are slow processes.

Therefore, we assume that there exist two dimensionless parameters " ⌧ 1 and µ ⌧ 1 being of the same order. Consequently, the model takes into account three time scales:

• a very fast time: ⌧ ;

• a fast time: t = "⌧ ;

• a slow time: T = µt = µ"⌧ ; leading to the next relation for any time dependent variable X:

dX d⌧ = " dX dt = µ" dX dT .
The MPA is assumed to be ⇠10 km diameter, roughly the maximum size for a cluster [START_REF] Petitgas | Spatial organization of pelagic fish: echogram structure, spatio-temporal condition, and biomass in Senegalese waters[END_REF], so that time scale for fish movement from MPA to fishing area (and inversely) is approximately a day. The model could be applied to any exploited aggregative small pelagic fish which forms large clusters that remain coherent at least ⇠10 days. In West Africa, one could think about the Sardinella aurita population as an example. We assume in this work that the small clusters work as a refuge, i.e. their catchability is inferior to large clusters' one, considering the case study of purse-seine fishery because of reduced accessibility as explained in the introduction. Finally, to be consistent with the mechanisms and behaviours associated to the three times scales, theses must correspond to ⇠ 1 day (very fast), ⇠ 10 days (fast) and ⇠ 100 days (slow). This respects the empirical condition for aggregation methods to work, i.e. one order of magnitude between the time scales as we show in the next section.

Building the aggregated model

Now, we shall take advantage of the three time scales to build a reduced model governing the total fish density and the total fishing e↵ort. Aggregation methods were introduced in ecology by Iwasa et al [START_REF] Iwasa | Aggregation in model ecosystems. I. Perfect aggregation[END_REF][START_REF] Iwasa | Aggregation in model ecosystems. II. Approximate aggregation[END_REF]. Here, we use time scale separation methods based on the central manifold theory and we refer to the following articles for aggregation methods [START_REF] Auger | Aggregation of variables and applications to population dynamics[END_REF][START_REF] Auger | Aggregation methods in dynamical systems variables and applications in population and community dynamics[END_REF][START_REF] Auger | A Review on Spatial Aggregation Methods Involving Several Time Scales[END_REF]. Usually, the complete system involves only two time scales. Under this condition, the aggregation is realized by calculating the fast equilibrium and the aggregated model is obtained by substituting the fast equilibrium into the complete model.

In our present case, three time scales are considered. As a consequence, the aggregation is going to require two steps. In a first step, we shall look for the existence of a very fast equilibrium and we shall substitute it into the complete model. This will lead to an "intermediate" model at the fast time scale. The second and last step will consist in looking for the existence of a fast equilibrium whose substitution in the intermediate model will lead to the aggregated and final slow model.

First step of aggregation: very fast fish movements

Let us set " = µ = 0 leading to the very fast model that describes the patch change from MPA to fishing area and inversely:

8 > > > > > > < > > > > > > : dn SM d⌧ = m S n SF m S n SM dn LM d⌧ = m L n LF m L n LM dn SF d⌧ = m S n SM m S n SF dn LF d⌧ = m L n LM m L n LF dE d⌧ = 0.
At the very fast time scale, the sub-populations small and large clusters are constant, i.e. the next variables are first integrals:

n S = n SM + n SF , n L = n LM + n LF .

E↵ect of Small Versus Large Clusters of Fish School

A simple calculation leads to the next very fast equilibrium for small clusters:

n ⇤ SM = m S m S + m S n S = ⌫ ⇤ SM n S , n ⇤ SF = m S m S + m S n S = ⌫ ⇤ SF n S , where ⌫ ⇤
SF is the proportion of small clusters in the fishing area and ⌫ ⇤ SM in MPA. Similarly for fish in large clusters we get the very fast equilibrium as follows:

n ⇤ LM = m L m L + m L n L = ⌫ ⇤ LM n L , n ⇤ LF = m L m L + m L n L = ⌫ ⇤ LF n L , where ⌫ ⇤
LF is the proportion of large fish clusters in the fishing area and ⌫ ⇤ LM in MPA. After substitution of this very fast equilibrium into the complete model, we get the "intermediate" model, i.e. the fast model (or first aggregated model) which reads:

8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : dn S dt = (kn L kn S ) + µ ✓ r⌫ ⇤ SM n S ✓ 1 ⌫ ⇤ SM n S + ⌫ ⇤ LM n L hK ◆◆ +µ ✓ r⌫ ⇤ SF n S ✓ 1 ⌫ ⇤ SF n S + ⌫ ⇤ LF n L (1 h)K ◆◆ µq S ⌫ ⇤ SF n S E dn L dt = (kn S kn L ) + µ ✓ r⌫ ⇤ LM n L ✓ 1 ⌫ ⇤ SM n S + ⌫ ⇤ LM n L hK ◆◆ +µ ✓ r⌫ ⇤ LF n L ✓ 1 ⌫ ⇤ SF n S + ⌫ ⇤ LF n L (1 h)K ◆◆ µq L ⌫ ⇤ LF n L E dE dt = µ( cE + pq S ⌫ ⇤ SF n S E + pq L ⌫ ⇤ LF n L E). (3.3.1)

Second step of aggregation: fast changes in clusters size

Let set µ = 0 in the previous first aggregated model leading to the next fast model:

8 > < > : dn S dt = kn L kn S dn L dt = kn S kn L dE dt = 0.
At the fast time scale, the total fish population is constant: n = n S + n L . A simple calculation leads to the next fast equilibrium for small clusters and large clusters:

n ⇤ S = k k + k n = ⌫ ⇤ S n, n ⇤ L = k k + k n = ⌫ ⇤ L n.
Substitution of the fast equilibrium into the "intermediate" model leads to the final aggregated model (at the slow time scale) governing the total fish density and the fishing e↵ort:

8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > : dn dT = r⌫ ⇤ SM ⌫ ⇤ S n ✓ 1 (⌫ ⇤ SM ⌫ ⇤ S + ⌫ ⇤ LM ⌫ ⇤ L )n hK ◆ +r⌫ ⇤ SF ⌫ ⇤ S n ✓ 1 (⌫ ⇤ SF ⌫ ⇤ S + ⌫ ⇤ LF ⌫ ⇤ L )n (1 h)K ◆ +r⌫ ⇤ LM ⌫ ⇤ L n ✓ 1 (⌫ ⇤ SM ⌫ ⇤ S + ⌫ ⇤ LM ⌫ ⇤ L )n hK ◆ +r⌫ ⇤ LF ⌫ ⇤ L n ✓ 1 (⌫ ⇤ SF ⌫ ⇤ S + ⌫ ⇤ LF ⌫ ⇤ L )n (1 h)K ◆ q S ⌫ ⇤ SF ⌫ ⇤ S nE q L ⌫ ⇤ LF ⌫ ⇤ L nE dE dT = ( c + pq S ⌫ ⇤ SF ⌫ ⇤ S n + pq L ⌫ ⇤ LF ⌫ ⇤ L n)E. (3.3.2) By setting = q S ⌫ ⇤ SF ⌫ ⇤ S + q L ⌫ ⇤ LF ⌫ ⇤ L , 1  = 1 K ✓ (⌫ ⇤ SF ⌫ ⇤ S + ⌫ ⇤ LF ⌫ ⇤ L + h 1) 2 h(1 h) + 1 ◆ , model (3.3.
2) can be written as:

( dn dT = rn(1 n  ) nE dE dT = ( c + p n)E. (3.3.3)
Model (3.3.3) is classic Lotka-Volterra predator-prey model with logistics growth for prey (see [START_REF] Bazykin | Nonlinear Dynamics of Interacting Populations[END_REF][START_REF] Leah | Mathematical Models In Biology[END_REF]). We see that it has two trivial equilibria: (0, 0), (, 0) and a non-trivial equilibrium point (n

⇤ , E ⇤ ) = ( c p , r (1 c p 
)). The global dynamics of model (3.3.3) depend on the sign of (n ⇤ , E ⇤ ):

• If p  > c, (n ⇤ , E ⇤
) is globally asymptotically stable;

• If p  < c, (, 0) is globally asymptotically stable. 3.3 shows a similar result in the case of fleet e↵ort extinction. This means that aggregation methods in this three level system can be successfully used when there exists at least an order of magnitude between two consecutive time scales. In the case of smaller values such as " = µ = 0.01, the approximation would be improved such that trajectories of aggregated and complete models would become extremely close and would appear confounded.

Comparison with one-step aggregation

It would have been possible to decide to perform only a one-step aggregation. The first possibility is to assume that " = 0 in order to study the very fast dynamics, and then not assuming that µ = 0. This corresponds to the first step of the previous aggregation and leads to a three equation system, which is more di cult to analyse than the previous aggregated model. The other possibility is to assume that µ = 0 in order to study the fast dynamics, without assuming at any moment that " = 0. Fast dynamics is then governed by the following set of equations:

8 > > > > > > < > > > > > > : dn SM d⌧ = (m S n SF m S n SM ) + "(kn LM kn SM ) dn LM d⌧ = (m L n LF m L n LM ) + "(kn SM kn LM ) dn SF d⌧ = (m S n SM m S n SF ) + "(kn LF kn SF ) dn LF d⌧ = (m L n LM m L n LF ) + "(kn SF kn LF ) dE d⌧ = 0. (3.4.1)
Solving dn SM /d⌧ = dn LM /d⌧ = dn SF /d⌧ = dn LF /d⌧ = 0 is equivalent to solving a linear system. We obtain:

⌫ ⇤ SM (") = k " m S k + m L k + m S (m L + m L ) k + k " m S k + m S k + m L k + m L k + (m S + m S ) (m L + m L ) , ⌫ ⇤ LM (") = k " m S k + m L k + m L (m S + m S ) k + k " m S k + m S k + m L k + m L k + (m S + m S ) (m L + m L ) , ⌫ ⇤ SF (") = k " m S k + m L k + m S (m L + m L ) k + k " m S k + m S k + m L k + m L k + (m S + m S ) (m L + m L ) , ⌫ ⇤ LF (") = k " m S k + m L k + m L (m S + m S ) k + k " m S k + m S k + m L k + m L k + (m S + m S ) (m L + m L )
.

It is easy to verify that:

lim

"!0 ⌫ ⇤ SM (") = ⌫ ⇤ SM ⌫ ⇤ S = m S k (m S + m S ) k + k , lim "!0 ⌫ ⇤ LM (") = ⌫ ⇤ LM ⌫ ⇤ L = m L k (m L + m L ) k + k , lim "!0 ⌫ ⇤ SF (") = ⌫ ⇤ SF ⌫ ⇤ S = m S k (m S + m S ) k + k , lim "!0 ⌫ ⇤ LF (") = ⌫ ⇤ LF ⌫ ⇤ L = m L k (m L + m L ) k + k .
The system obtained after a two-step aggregation appears as an approximation for " = 0 of the one-step aggregation. The dynamics obtained is a slightly better approximation of the complete dynamics than the one obtained with the two-step aggregation method. Indeed, substituting the frequencies at the fast equilibrium which are solutions of equations (3.4.1) would lead to another one-step aggregated model that could be developed as a Taylor expansion with respect to ". The zero order term of this Taylor expansion would exactly correspond to the aggregated model (3.3.3) obtained by the two-step method but, with the advantage that the first order term would give a correction term of the order of " leading to a better approximation of the complete model. However, determining the frequencies at fast equilibrium is more di cult than with the two-step aggregation methods: it requires solving a four-dimension system of equations in order to determine the fast equilibrium (first four equations of system (3.4.1)). The two-step method requires solving more (three) systems of equations, but with a lower number of equations (only two equations).

To summarize, two aggregation methods have been proposed:

• The two-step method leads to an aggregated model with less approximation but in most cases, it could be easier to handle it as it can be switched into several systems of equations, very fast and fast.

• The one-step method allows to calculate some correction terms leading to a better approximation but, we need to deal with a single system of equations to get the fast equilibrium. The later system may be more di cult to handle analytically.

Harvest Optimization

Now, we shall study the e↵ect of clusters size distribution on the total catch of the fishery at equilibrium. The catch per unit of time at equilibrium of the slow aggregated model reads as follows:

Y = n ⇤ E ⇤ = rc p (1 c p 
).

(3.5.1)

We shall study the e↵ect of the proportion of fish in small clusters on the total catch. Thus, let us write the catch Y as a function of the proportion of fish in small clusters at fast equilibrium, i.e. function of ⌫ ⇤ S . For simplicity we denote by X this proportion. According to that notation, we obtain:

Y = A 1 X 2 + A 2 X + A 3 (A 4 X + A 5 ) 2 , 0 < X < 1, (3.5.2)
in which:

A 1 = c 2 r(⌫ ⇤ SF ⌫ ⇤ LF ) 2 , A 2 = crpKh(1 h)(q S ⌫ ⇤ SF q L ⌫ ⇤ LF ) + 2c 2 r(1 h ⌫ ⇤ LF )(⌫ ⇤ SF ⌫ ⇤ LF ), A 3 = cr(1 h)(pKhq L ⌫ ⇤ LF + 2c⌫ ⇤ LF c) c 2 r⌫ ⇤2 LF , A 4 = p(q S ⌫ ⇤ SF q L ⌫ ⇤ LF ) p h(1 h)K, A 5 = pq L ⌫ ⇤ LF p h(1 h)K.
(3.5.3)

We will find out condition for the existence of a local maximum of Y with respect to X 2 (0, 1). We see that equation Y 0 (X) = 0 has unique solution:

X ⇤ = A 2 A 5 2A 4 A 3 2A 1 A 5 A 2 A 4 . (3.5.4)
The second order derivative of Y at X ⇤ is:

d 2 Y dX 2 (X ⇤ ) = (2A 1 A 5 A 2 A 4 ) 4 8( A 4 A 2 A 5 + A 2 4 A 3 + A 1 A 2 5 ) 3 . (3.5.5)
Hence, all conditions for the existence of local maximum of Y with X 2 (0, 1) can reduce as follows:

0 < A 2 A 5 2A 4 A 3 2A 1 A 5 A 2 A 4 < 1, (3.5.6) A 4 A 2 A 5 + A 2 4 A 3 + A 1 A 2 5 < 0. ( 3
Y ⇤ = 4A 3 A 1 A 2 2 4( A 4 A 2 A 5 + A 2 4 A 3 + A 1 A 2 5 )
.

(3.5.8)

In our model, fish that form small clusters act as a refuge. A fishery can be considered as a predator-prey system, the prey being the fish and the predator, the fishing fleet. Such classical prey-predator (Lotka-Volterra and Holling type II) as well as inter-specific competition models with a refuge have already been investigated [START_REF] Dao | Predator density dependent prey dispersal in a patchy environment with a refuge for the prey[END_REF][START_REF] Gonzalez | Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability[END_REF][START_REF] Krivan | On the Gause predator-prey model with a refuge: a fresh look at the history[END_REF][START_REF] Nguyen | E↵ects of refuges and density dependent dispersal on interspecific competition dynamics[END_REF].

Figure 3.4 shows that there exists a maximum of the total catch at equilibrium with respect to the proportion of small clusters. Indeed, since we consider small pelagic fish species and purse-seine fisheries, the catchability is inferior for small clusters than for large one, and captured fish mainly belong to large clusters. Starting from a fish population organized only in large clusters, increasing the proportion of small clusters firstly reduce the overall population catchability, since we assumed a lower catchability for small clusters. Such catchability reduction can be seen as a refuge e↵ect that benefit population growth, and once the equilibrium is reached it allows to increase the total catch (because the population density is higher). Globally, this is favorable to the growth of the fish population and it allows to increase the total catch at equilibrium. Besides, if a too large proportion of fish is structured in small clusters, the reduction in catchability is not anymore compensated by the growth of biomass due to the refuge e↵ect described before, and as a result the yield decrease. Consequently, there is a proportion of small fish clusters in between that maximizes the total catch at equilibrium as shown on figure 3.4.

Discussion and perspectives

Our model has shown that for small pelagic fish, there exists a maximum of the total catch with respect to the size distribution of the clusters. Another aspect regarding optimal spatial distribution of a fishing fleet in a patchy fishery was also investigated in [START_REF] Mchich | Optimal spatial distribution of the fishing e↵ort in a multi fishing zone model[END_REF].

As a result of our model, over-fishing would progressively give advantage to fish populations that are able to change rapidly from small clusters to large clusters and inversely. Being part of small clusters works as a kind of refuge for fish because the catch is less, considering that large clusters can be more easily detected by fishermen and thus exploited than small ones.

Discussion and perspectives 39

Our model does consider only two cluster sizes, large and small. In a further contribution, it would be interesting to consider a more continuous size spectrum for clusters. Does small pelagic fish species display di↵erent levels of exposure to over fishing following their specific clustering behavior?

The theoretical results obtained in this work, if they are validated on a particular fishery, could be translated in near real time management policy. Indeed, even if we do not yet understand well the determinism of small pelagic fish aggregative dynamics [START_REF] Brehmer | Schooling behaviour of small pelagic fish: phenotypic expression of independent stimuli[END_REF], we know that it is possible to control the harvesting process in near real time (e.g. as it is the case in Peru; Pers. Comm. Arnaud Bertrand). Nevertheless observation methodologies of cluster size have been already developed, particularly using acoustics devices [START_REF] Maclennan | Fisheries acoustics: Theory and Practice[END_REF], in continuous monitoring [START_REF] Brehmer | Omnidirectional multibeam sonar monitoring: Applications in fisheries science[END_REF] and near real time [START_REF] Brehmer | Field investigations and multiindicators for management and conservation of shallow water lagoons: practices and perspectives[END_REF]. Thus, a near real time management could be encouraged, on the basis of this work, to control harvesting in order to produce an optimal value for ⌫ ⇤ S . Such supervision should allow adaptive management measures, according to the variation of biotic and abiotic factors, to target the maximum sustainable yield of a fishery. However this require to improve our understanding of the e↵ect of the environment on the aggregative dynamics of exploited small pelagic fish as well as the processes a↵ecting their biomass variability and fluctuation.

The present chapter allowed us to extend aggregation of variables methods to a three level dynamical system. Aggregation in a two level system is rather usual. Here, we extended the method for a system involving three time scales and we present an aggregation method with double steps. In the present work, we simply proceed to aggregation and show by numerical simulations of a particular case that the method works quite well when there is at least an order of magnitude between two consecutive time scales.

In the future, it would be useful to present aggregation methods of three (or more) level systems in a general context and to show that the center manifold theory can be extended to the case of a system of ODEs with several time scales. The twostep methods o↵er the same benefits than well-known divide-and-conquer algorithms which aim at dividing a problem into several sub-problems that are simpler to solver. This method could prove to be of particular interest for larger dimension problems, or for problems for which fast equilibria have to be determined from non-linear systems of equations.

Chapter 4

Dynamics of species in a model with two predators and one prey

Introduction

Paper [START_REF] Ta | Dynamics of species in a model with two predators and one prey[END_REF] is presented in this chapter. The dynamical relationship between predators and prey has been studied by several authors for a long time. In those researches, to represent the average number of prey killed per individual predator per unit of time, a functional, called functional response, was introduced. One functional response can depend on only the prey's density or both the prey's and the predator's densities. However, some biologists have argued that in many situation, especially when predators have to search for food, the functional response should depend on both prey's and predator's densities [5,[START_REF] Arditi | Functional response and heterogeneities: an experimental test with cladocerans[END_REF][START_REF] Dolman | The intensity of interference varies with resource density: evidence from a field study with snow buntings, Plectrophenax nivalis[END_REF][START_REF] Jost | From pattern to process: identifying predator-prey interactions[END_REF][START_REF] Jost | Testing for predator dependence in predator-prey dynamics: a nonparametric approach[END_REF][START_REF] Skalski | Functional responses with predator interference: viable alternatives to the Holling type II model[END_REF]. One of the most popular functional responses is the fractional one as in the following prey-predator model. It is called Beddington-DeAngelis functional response.

( x 0 1 = x 1 (a 1 b 1 x 1 ) c 1 x 1 x 2 ↵+ x 1 + x 2 , x 0 2 = a 2 x 2 + c 2 x 1 x 2 ↵+ x 1 + x 2 .
In this model, x i (t) represents the population density of species X i at time t (i 1); X 1 is the prey and X 2 is the predator. At time t, a 1 (t) is the intrinsic growth rate of X 1 and a i (t) is the death rate of X 2 ; b 1 (t) measures the inhibiting e↵ect of environment on X 1 . This model was originally proposed by Beddington [START_REF] Beddington | Mutual interference between parasites or predators and its e↵ect on searching e ciency[END_REF] and DeAngelis et al. [START_REF] Deangelis | A model for trophic interaction[END_REF], independently. After the appearance of these two investigations, there are many other ones for analogous systems with di↵usion in a constant environment [START_REF] Cantrell | E↵ects of domain size on the persistence of populations in a di↵usive food chain model with DeAngelis-Beddington functional response[END_REF][START_REF] Cantrell | On the dynamics of predator-prey models with the Beddington-DeAngelis functional response[END_REF][START_REF] Cantrell | Spatial Ecology via Reaction-Di↵usion Equations[END_REF][START_REF] Cosner | E↵ects of spatial grouping on the functional response of predators[END_REF][START_REF] Hwang | Global analysis of the predator-prey system with Beddington-DeAngelis functional response[END_REF][START_REF] Hwang | Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response[END_REF]. However, the constant environment is rarely the case in real life. Most natural environments are physically highly variable, i.e., the coe cients in those models should depend on time [START_REF] Cui | extinction and periodic solution of predatorprey system with Beddington-DeAngelis functional response[END_REF][START_REF] Fan | Dynamics of a non-autonomous predator-prey system with the Beddington-DeAngelis functional response[END_REF][START_REF] Ta | Dynamics of species in a non-autonomous Lotka-Volterra system[END_REF][START_REF] Zeng | Study on a non-autonomous predator prey system with Beddington-DeAngelis functional response[END_REF]. In order to continue studying such models, in this chapter, we consider a predator-prey model of one prey and two 4. Dynamics of species in a model with two predators and one prey predators with Beddington-DeAngelis functional responses 8 > > < > > :

x 0 1 = x 1 [a 1 (t) b 1 (t)x 1 ] c 2 (t)x 1 x 2 ↵(t)+ (t)x 1 + (t)x 2 c 3 (t)x 1 x 3 ↵(t)+ (t)x 1 + (t)x 3 , x 0 2 = x 2 h a 2 (t) + d 2 (t)x 1 ↵(t)+ (t)x 1 + (t)x 2 b 2 (t)x 3 i , x 0 3 = x 3 h a 3 (t) + d 3 (t)x 1 ↵(t)+ (t)x 1 + (t)x 3 b 3 (t)x 2 i . (4.1.1)
Here x i (t) represents the population density of species X i at time t (i 1), X 1 is the prey and X 2 , X 3 are the predators. Two predators share one prey and it is assumed that there are two types of competition between two predators. The first type is direct interference where individuals of each predator species act with aggression against individuals of the other predator species. In our model, this type of competition is described by the coe cients b 2 (t) and b 3 (t). The second type of competition is interference competition that occurs during hunting because predators spend time interacting with each other rather than seeking prey. Here we assume that there is no competition of that type between individuals of the two di↵erent predator species. Therefore, the Beddington-DeAngelis functional responses are of the form

d i (t)x 1 ↵(t)+ (t)x 1 + (t)x i (i = 2, 3
). We use the same coe cients ↵, , in the functional responses of both predators, since it is assumed that both predators take the same time to handle a prey once they encounter it and that individuals of each predator species interfere with each other when hunting by exactly the same amount in both species. This assumption is somewhat restrictive from the biological viewpoint, but it could be removed without greatly changing the analysis of system (4.1.1).

Throughout this chapter, it is assumed that the functions a

i (t), b ij (t), c i (t), d i (t)
, ↵(t), (t), (t)(1  i, j  3) are continuous on R and bounded above and below by some positive constants. This chapter is organized as follows. Section 4.2 provides some definitions and notation. In Section 4.3, we state some results on invariant set, the permanence and extinction of system (4.1.1). Then, the asymptotic stability of solution is proved by using a Liapunov function. In Section 4.4, we continue using other Liapunov functions and the continuation theorem in coincidence degree theory to show the existence and global stability of a positive periodic solution. The final section 4.5 illustrates the behavior of system (4.1.1) by some computational results and gives our conclusion.

Definitions and notation

In this section we introduce some basic definitions and facts which will be used throughout this chapter. Let

R 3 + = {(x 1 , x 2 , x 3 ) 2 R 3 | x i > 0 (i 1)}. Denote by x(t) = (x 1 (t), x 2 (t), x 3 (t)
) the solution of system (4.1.1) with initial condition

x 0 = (x 0 1 , x 0 2 , x 0 3 ) = (x 1 (t 0 ), x 2 (t 0 ), x 3 (t 0 )), t 0
0. For biological reasons, through out this chapter, we only consider the solutions x(t) with positive initial values, i.e., x 0 2 R 3 + . Let g(t) be a continuous function, for a brevity, instead of writing g(t) we write g. If g is bounded on R, we denote

g u = sup t2R g(t), g l = inf t2R g(t)
,

and ĝ = 1 ! R ! 0 g(t)
dt, if g is a periodic function with period !. The global existence and uniqueness of solution of system (4.1.1) are guaranteed by the properties of the map defined by the right hand of system (4.1.1) [START_REF] Bainov | Impulsive Di↵erential Equations: Periodic Solutions and Applications[END_REF]. We have the following lemma Proof. The solution x(t) of (4.1.1) with initial value x 0 satisfies 8 > > > < > > > :

x 1 = x 0 1 exp n R t t 0 [a 1 b 1 x 1 c 2 x 2 ↵+ x 1 + x 2 c 3 x 3 ↵+ x 1 + x 3 ]du o , x 2 = x 0 2 exp n R t t 0 [ a 2 + d 2 x 1 ↵+ x 1 + x 2 b 2 x 3 ]du o , x 3 = x 0 3 exp n R t t 0 [ a 3 + d 3 x 1 ↵+ x 1 + x 3 b 3 x 2 ]du o .
The conclusion follows immediately for all t 2 [t 0 , 1). The proof is complete. Definition 4.2. System (4.1.1) is said to be permanent if there exist some positive

j (j = 1, 2) such that 1  lim inf t!1 x i (t)  lim sup t!1 x i (t)  2 (i 1)
for all solutions of (4.1.1).

Definition 4.3.

A set A ⇢ R 3 + is called an ultimately bounded region of system (4.1.1) if for any solution x(t) of (4.1.1) with positive initial values, there exists T 1 > 0 such that x(t) 2 A for all t t 0 + T 1 . Definition 4.4. A bounded non-negative solution x ⇤ (t) of (4.1.1) is said to be globally asymptotically stable (or globally attractive) if any other solution x(t) of (4.1.1) with positive initial values satisfies lim

t!1 P 3 i=1 |x i (t) x ⇤ i (t)| = 0. Remark 4.5.
It is easy to see that if a solution of (4.1.1) is globally asymptotically stable, then so are all solutions. In this case, system (4.1.1) is also said to be globally asymptotically stable.

The model with general coe cients

Let ✏ 0 be su cient small. Put

M ✏ 1 = a u 1 b l 1 + ✏, M ✏ i = d u i M ✏ 1 a l i ↵ l a l i l , m ✏ 1 = a l 1 l c u 2 c u 3 b u 1 l ✏, m ✏ i = d l i m ✏ 1 (a u i + b u i M ✏ j )( u m ✏ 1 + ↵ u ) (a u i + b u i M ✏ j ) u (i, j 2, i 6 = j) (4.3.1) then M ✏ i > m ✏ i (i 1
). We will show that max{m 0 i , 0} (i 1) are the lower bounds for the limiting bounds of species X i as time t tends to infinity. This is obvious when m 0 i  0. Therefore, it is assumed that m 0 i > 0. Hypothesis 4.6. m 0 i > 0 (i 1). Theorem 4.7. Under Hypothesis 4.6, for any su cient small ✏ > 0 such that m ✏ i > 0 (i 1), a set ✏ defined by

✏ = {(x 1 , x 2 , x 3 ) 2 R 3 | m ✏ i < x i < M ✏ i (i 1)
} is positively invariant with respect to system (4.1.1).

Proof. Through out this proof, we use the facts that the solution to the equation

X 0 (t) = A(t, X)X(t)[B X(t)] (B 6 = 0)
is given by

X(t) = BX 0 exp n R t t 0 BA(s, X(s))ds o X 0 h exp n R t t 0 BA(s, X(s))ds o 1 i + B
, where t 0 0 and X 0 = X(t 0 ). Consider the solution of system (4.1.1) with an initial value x 0 2 ✏ . From Lemma 4.1 and from the first equation of (4.1.1), we have

x 0 1 (t)  x 1 (t)[a 1 (t) b 1 (t)x 1 (t)]  x 1 (t)[a u 1 b l 1 x 1 (t)] = b l 1 x 1 (t)(M 0 1 x 1 )
. Using the comparison theorem gives

x 1 (t)  x 0 1 M 0 1 exp{a u 1 (t t 0 )} x 0 1 [exp{a u 1 (t t 0 )} 1] + M 0 1  x 0 1 M ✏ 1 exp{a u 1 (t t 0 )} x 0 1 [exp{a u 1 (t t 0 )} 1] + M ✏ 1 < M ✏ 1 , t t 0 . (4.3.2)
It follows from the third equation of (4.1.1) and from (4.3.2) that

x 0 2  a l 2 x 2 + d u 2 x 1 x 2 ↵ l + l x 1 + l x 2  a l 2 x 2 + d u 2 M ✏ 1 x 2 ↵ l + l x 2 = x 2 ⇥ (d u 2 M ✏ 1 a l 2 ↵ l ) a l 2 l x 2 ⇤ ↵ l + l x 2 = a l 2 l ↵ l + l x 2 x 2 (M ✏ 2 x 2 ). Putting C 2 (t) = a l 2 l ↵ l + l x 2 (t) , (4.3.3)
and using the comparison theorem again yield

x 2 (t)  M ✏ 2 x 0 2 exp n M ✏ 2 R t t 0 C 2 (s)ds o x 0 2 h exp n M ✏ 2 R t t 0 C 2 (s)ds o 1 i + M ✏ 2 < M ✏ 2 , t t 0 . (4.3.4)
Similarly, x 3 (t) < M ✏ 3 for every t t 0 . Now by the first equation of (4.1.1), it implies that

x 0 1 (t) x 1 ✓ a l 1 c u 2 + c u 3 l b u 1 x 1 ◆ = b u 1 x 1 (m 0 1 x 1 ). Since x 0 1 > m ✏ 1
, by the comparison theorem, we obtain x 1 (t)

x 0 1 m 0 1 exp{b u 1 m 0 1 (t t 0 )} x 0 1 [exp{b u 1 m 0 1 (t t 0 )} 1] + m 0 1 > m ✏ 1 for all t t 0 .
Similarly, for i, j 2 (i 6 = j),

x 0 i = a i x i + d i x 1 x i ↵ + x 1 + x i b i x i x j (a u i + b u i M ✏ j )x i + d l i m ✏ 1 x i ↵ u + u m ✏ 1 + u x i = ⇥ d l i m ✏ 1 (a u i + b u i M ✏ j )( u m ✏ 1 + ↵ u ) ⇤ x i (a u i + b u i M ✏ j ) u x 2 i ↵ u + u m ✏ 1 + u x i = (a u i + b u i M ✏ j ) u ↵ u + u m ✏ 1 + u x i x i (m ✏ i x i ),
from which follows that x i (t) > m ✏ i for all t t 0 . We complete the proof. In the next theorem, the permanence of system (4.1.1) is shown. A treatment called practical persistence to prove the permanence of models and its application to various types of models can be seen in [START_REF] Cantrell | Practical persistence in ecological models via comparison methods[END_REF][START_REF] Cantrell | E↵ects of domain size on the persistence of populations in a di↵usive food chain model with DeAngelis-Beddington functional response[END_REF][START_REF] Cosner | Variability, vagueness and comparison methods for ecological models[END_REF]. Theorem 4.8. Under Hypothesis 4.6, for any su cient small ✏ > 0 such that

m ✏ i > 0, m ✏ i  lim inf t!1 x i (t)  lim sup t!1 x i (t)  M ✏ i (i 1).
Consequently, system (4.1.1) is permanent.

Proof. According to the proof of Theorem 4.7 we have

x 1 (t)  x 0 1 M 0 1 exp{a u 1 (t t 0 )} x 0 1 [exp{a u 1 (t t 0 )} 1] + M 0 1 • Thus, lim sup t!1 x 1 (t)  M 0 1 , i.e.
, there exists t

1 t 0 such that x 1 (t) < M ✏ 1 for all t t 1 .
By the same arguments of making (4.3.4), it follows that x i (t) = 0, i.e., the i th -predator goes to extinction.

x 2 (t)  M ✏ 2 x 1 2 exp n M ✏ 2 R t t 1 C 2 (s)ds o x 1 2 h exp n M ✏ 2 R t t 1 C 2 (s)ds o 1 i + M ✏ 2 , ( 4
Proof. It follows from M 0 i < 0 that M ✏ i < 0 with a su ciently small ✏. Similarly to the proof of Theorem 4.7 we have

x 0 i (t)  a l i l ↵ l + l x i x i (M ✏ i x i ) < 0. (4.3.6)
Thus, there exists C 0 such that lim

t!1 x i (t) = C and C  x i (t)  x 0
i for all t t 0 . If C > 0 then from (4.3.6), there exists µ > 0 such that x 0 i (t) < µ for all t t 0 . We therefore have x i (t) < µ(t t 0 ) + x 0 i and then lim

t!1 x i (t) = 1, which contradicts x i (t) > 0 for all t t 0 . Hence, lim t!1 x i (t) = 0.
In order to consider the global asymptotic stability of system (4.1.1), we need the following result called Barbalat's lemma. Lemma 4.10. (see [START_REF] Barbȃlat | Systèmes d'équations di↵érentielles d'osillations non linéaires[END_REF].) Let h be a real number and f be a non-negative function defined on [h, +1) such that f is integrable and uniformly continuous on [h, +1). Then lim t!1 f (t) = 0.

Theorem 4.11. Suppose that Hypothesis 4.6 holds and let ✏ > 0 be su cient small such that m ✏ i > 0 (i 1). Let x ⇤ be a solution of system (4.1.1) satisfying

lim sup t!1 ⇢ b 1 + c 2 M ✏ 2 + d 2 (↵ + M ✏ 2 ) u 2 (m ✏ 1 , M ✏ 2 ) + c 3 M ✏ 3 + d 3 (↵ + M ✏ 3 ) u 3 (m ✏ 1 , M ✏ 3 ) < 0, lim sup t!1 ⇢ b j + c i (↵ + M ✏ 1 ) u i (M ✏ 1 , m ✏ i ) d i m ✏ 1 u i (m ✏ 1 , M ✏ i ) < 0, (4.3.7)
where

u i (a, b) = (↵ + x ⇤ 1 + x ⇤ i )(↵ + a + b) (i, j 2, i 6 = j)
. Then x ⇤ is globally asymptotically stable.

Proof. Let x be other solution of (4.1.1). From Corollary 4.8,

✏ is an ultimately bounded region of (4.1.1). Then there exists T 1 > 0 such that x, x ⇤ 2 ✏ for all t t 0 + T 1 . Consider a Liapunov function defined by V (t) =

P 3 i=1 | ln x i ln x ⇤ i |, t t 0 . A direct calculation of the right derivative D + V (t) of V (t)
D + V (t) = 3 X i=1 sgn(x i x ⇤ i ) ✓ x 0 i x i x ⇤ i 0 x ⇤ i ◆ =sgn(x 1 x ⇤ 1 ) h c 2 ⇣ x 2 ↵ + x 1 + x 2 x ⇤ 2 ↵ + x ⇤ 1 + x ⇤ 2 ⌘ c 3 ⇣ x 3 ↵ + x 1 + x 3 x ⇤ 3 ↵ + x ⇤ 1 + x ⇤ 3 ⌘ b 1 (x 1 x ⇤ 1 ) i + d 2 sgn(x 2 x ⇤ 2 )  x 1 ↵ + x 1 + x 2 x ⇤ 1 ↵ + x ⇤ 1 + x ⇤ 2 b 2 d 2 (x 3 x ⇤ 3 ) + d 3 sgn(x 3 x ⇤ 3 )  x 1 ↵ + x 1 + x 3 x ⇤ 1 ↵ + x ⇤ 1 + x ⇤ 3 b 3 d 3 (x 2 x ⇤ 2 )  b 1 |x 1 x ⇤ 1 | c 2 sgn(x 1 x ⇤ 1 ) ↵(x 2 x ⇤ 2 ) + (x ⇤ 1 x 2 x 1 x ⇤ 2 ) u 2 (x 1 , x 2 ) c 3 sgn(x 1 x ⇤ 1 )
↵(x

3 x ⇤ 3 ) + (x ⇤ 1 x 3 x 1 x ⇤ 3 ) u 3 (x 1 , x 3 ) + d 2 sgn(x 2 x ⇤ 2 ) ↵(x 1 x ⇤ 1 ) + (x 1 x ⇤ 2 x 2 x ⇤ 1 ) u 2 (x 1 , x 2 ) + b 2 |x 3 x ⇤ 3 | + d 3 sgn(x 3 x ⇤ 3 ) ↵(x 1 x ⇤ 1 ) + (x 1 x ⇤ 3 x 3 x ⇤ 1 ) u 3 (x 1 , x 3 ) + b 3 |x 2 x ⇤ 2 |• It follows from x, x ⇤ 2 ✏ for t t 0 +T 1 and x 1 x ⇤ i x ⇤ 1 x i = x 1 (x ⇤ i x i )+x i (x 1 x ⇤ 1 ) (i = 2, 3) that D + V (t)  b 1 |x 1 x ⇤ 1 | c 2 sgn(x 1 x ⇤ 1 ) (↵ + x 1 )(x 2 x ⇤ 2 ) x 2 (x 1 x ⇤ 1 ) u 2 (x 1 , x 2 ) c 3 sgn(x 1 x ⇤ 1 ) (↵ + x 1 )(x 3 x ⇤ 3 ) x 3 (x 1 x ⇤ 1 ) u 3 (x 1 , x 3 ) + d 2 sgn(x 2 x ⇤ 2 ) (↵ + x 2 )(x 1 x ⇤ 1 ) x 1 (x 2 x ⇤ 2 ) u 2 (x 1 , x 2 ) + b 2 |x 3 x ⇤ 3 | + d 3 sgn(x 3 x ⇤ 3 ) (↵ + x 3 )(x 1 x ⇤ 1 ) x 1 (x 3 x ⇤ 3 ) u 2 (x 1 , x 3 ) + b 3 |x 2 x ⇤ 2 |   b 1 + c 2 M ✏ 2 + d 2 (↵ + M ✏ 2 ) u 2 (m ✏ 1 , M ✏ 2 ) + c 3 M ✏ 3 + d 3 (↵ + M ✏ 3 ) u 3 (m ✏ 1 , M ✏ 3 ) |x 1 x ⇤ 1 | +  b 3 + c 2 (↵ + M ✏ 1 ) u 2 (M ✏ 1 , m ✏ 2 ) d 2 m ✏ 1 u 2 (m ✏ 1 , M ✏ 2 ) |x 2 x ⇤ 2 | +  b 2 + c 3 (↵ + M ✏ 1 ) u 3 (M ✏ 1 , m ✏ 3 ) d 3 m ✏ 1 u 3 (m ✏ 1 , M ✏ 3 ) |x 3 x ⇤ 3 | for t t 0 + T 1 . (4.3.8)
Combining (4.3.7) and (4.3.8) gives the existence of a positive number µ > 0 and of

T 2 t 0 + T 1 such that D + V (t)  µ 3 X i=1 |x i x ⇤ i | for every t T 2 . (4.3.9)
Integrating both sides of (4.3.9) from T 2 to t yields

V (t) + µ Z t T 2 ⇣ 3 X i=1 |x i x ⇤ i | ⌘ ds  V (T 2 ) < 1 for every t T 2 .
Then R

t T 2 ⇣ P 3 i=1 |x i x ⇤ i | ⌘ ds  µ 1 V (T 2 ) < 1 for every t T 2 .
Hence,

P 3 i=1 |x i x ⇤ i | 2 L 1 ([T 2 , 1

The model with periodic coe cients

In this section, we assume that the coe cients in system (4.1.1) are ! -periodic in t and bounded above and below by some positive constants. We study the existence and stability of a periodic solution of this system. To do this, we will employ the alternative approach to establish some criteria in terms of the average of the related functions over an interval of the common period. That is continuation theorem in coincidence degree theory, which has been successfully used to establish criteria for the existence of positive periodic solutions of some mathematical models of predatorprey type, we refer the reader to [START_REF] Li | Positive periodic solution for neutral delay model[END_REF][START_REF] Li | Periodic solutions of a periodic neutral delay equations[END_REF][START_REF] Li | Periodic solution of a periodic delay predator-prey system[END_REF][START_REF] Saker | Oscillation and global attractivity of hematopoiesis model with delay time[END_REF]. To this end, we shall summarize in the following a few concepts and results from [START_REF] Gaines | Coincidence Degree and Nonlinear Di↵erential Equations[END_REF] that will be basic for this section.

Let X and Y be two Banach spaces, let L : DomL ⇢ X ! Y be a linear mapping, and let N : X ! Y be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if the following conditions hold (i) ImL is closed.

(ii) dimKerL = codim ImL < 1.
If L is a Fredholm mapping of index zero and there exist continuous projections P : X ! X and Q : Y ! Y such that ImP = KerL, ImL = KerQ = Im(I Q), it follows that L p = L| DomL\KerP : (I P )X ! ImL is invertible. We denote by K p the inverse of that map. If ⌦ is an open bounded subset of X, the mapping N will be called L -compact on ⌦ if the mapping QN : ⌦ ! Y is continuous and bounded, and K p (I Q)N : ⌦ ! X is compact, i.e., it is continuous and K p (I Q)N ( ⌦) is relatively compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J : ImQ ! KerL. The following continuation theorem is from [START_REF] Gaines | Coincidence Degree and Nonlinear Di↵erential Equations[END_REF]. We now put

L 11 = ln â1 b1 , H 11 = ln â1 b1 + 2â 1 !, L 12 = ln ( â1 \ ( c 2 + c 3 ) ) , H 12 = L 12 2â 1 !, L i1 = ln â1 ( di âi l ) exp{2â 1 !} âi b1 ↵ l âi b1 l , H i1 = 2 d ( d i )! + L i1 , L i2 = ln d i exp{H 12 } (↵ u + u exp{H 12 })(â i + bi exp{H j1 }) u (â i + bi exp{H j1 }) , H i2 = L i2 2 d ( d i )! (i, j 2, i 6 = j).
The convention here is that ln x = 1 if x  0. In the next theorem, a su cient condition for existence of an ! -periodic solution of (4.1.1) is presented.

Theorem 4.13. If L i2 > 1 (i 1) then system (4.1.1) has at least one positive ! -periodic solution.

Proof. Put x i (t) = exp{u i (t)} (i 1), then system (4.1.1) becomes

> <

> :

u 0 1 = a 1 b 1 exp{u 1 } c 2 exp{u 2 } ↵+ exp{u 1 }+ exp{u 2 } c 3 exp{u 3 } ↵+ exp{u 1 }+ exp{u 3 } , u 0 2 = a 2 + d 2 exp{u 1 } ↵+ exp{u 1 }+ exp{u 2 } b 2 exp{u 3 }, u 0 3 = a 3 + d 3 exp{u 1 } ↵+ exp{u 1 }+ exp{u 3 } b 3 exp{u 2 }.
1 ! R ! 0 u 1 (t)dt 1 ! R ! 0 u 2 (t)dt 1 ! R ! 0 u 3 (t)dt 3 5 , 2 4 u 1 u 2 u 3 3 5 2 X. Then KerL = {u 2 X | u = (h 1 , h 2 , h 3 ) T 2 R 3 }, ImL = {u 2 Y | R ! 0 u
i (t)dt = 0 (i 1)}, and dimKerL = 3 = codimImL. Since ImL is closed in Y, L is a Fredholm mapping of index zero. It is easy to show that P, Q are continuous projections such that ImP = KerL, ImL = KerQ = Im(I Q). Furthermore, the generalized inverse (to L) K P : ImL ! DomL \ KerP exists and is given by

K P 2 4 u 1 u 2 u 3 3 5 = 2 4 R t 0 u 1 (s)ds 1 T R ! 0 R t 0 u 1 (s)dsdt R t 0 u 2 (s)ds 1 T R ! 0 R t 0 u 2 (s)dsdt R t 0 u 3 (s)ds 1 T R ! 0 R t 0 u 3 (s)dsdt
u 0 1 = h a 1 b 1 exp{u 1 } c 2 exp{u 2 } ↵ + exp{u 1 } + exp{u 2 } c 3 exp{u 3 } ↵ + exp{u 1 } + exp{u 3 } i , u 0 2 = h a 2 + d 2 exp{u 1 } ↵ + exp{u 1 } + exp{u 2 } b 2 exp{u 3 } i , (4.4.2) u 0 3 = h a 3 + d 3 exp{u 1 } ↵ + exp{u 1 } + exp{u 3 } b 3 exp{u 2 } i .
Suppose that (u 1 , u 2 , u 3 ) 2 X is an arbitrary solution of system (4.4.2) for a certain 2 (0, 1). Integrating both sides of (4.4.2) over the interval [0, !], we obtain

â1 ! = Z ! 0 h b 1 exp{u 1 } + c 2 exp{u 2 } ↵ + exp{u 1 } + exp{u 2 } + c 3 exp{u 3 } ↵ + exp{u 1 } + exp{u 3 } i dt, âi ! + Z ! 0 b i exp{u j }dt = Z ! 0 d i exp{u 1 }dt ↵ + exp{u 1 } + exp{u i } (4.4.3)  Z ! 0 d i dt = d ( d i )! (i, j 2, i 6 = j).
It follows from (4.4.2) and (4.4.3) that for i, j 2 (i 6 = j),

Z ! 0 |u 1 (t) 0 |dt  h Z ! 0 a 1 dt + Z ! 0 b 1 exp{u 1 }dt + Z ! 0 c 2 exp{u 2 } ↵ + exp{u 1 } + exp{u 2 } dt + Z ! 0 c 3 exp{u 3 } ↵ + exp{u 1 } + exp{u 3 } i dt <2â 1 !, Z ! 0 |u i (t) 0 |dt <2 d ( d i )!. Since u 2 X, there exist ⇠ i , ⌘ i 2 [0, !] such that u i (⇠ i ) = min t2[0,!] u i (t), u i (⌘ i ) = max t2[0,!] u i (t) (i 1). (4.4.4)
From the first equation of (4.4.3) and (4.4.4), we obtain â1 ! R

! 0 b 1 exp{u 1 (⇠ 1 )}dt = b1 ! exp{u 1 (⇠ 1 )}, from which follows u 1 (⇠ 1 ) < L 11 . Hence u 1 (t)  u 1 (⇠ 1 ) + Z ! 0 |u 0 1 (t)|dt < L 11 + 2â 1 ! = H 11 for all t 0.
On the other hand, from the first equation of (4.4.3) and (4.4.4), we also have

â1 !  Z ! 0 b 1 exp{u 1 (⌘ 1 )}dt + Z ! 0 c 2 (t) + c 3 (t) (t) dt = " b1 exp{u 1 (⌘ 1 )} + \ ( c 2 + c 3 ) # !.
Then for any t 0,

u 1 (t) u 1 (⌘ 1 ) Z ! 0 |u 0 1 (t)|dt ln ( â1 \ ( c 2 + c 3 ) ) 2â 1 ! = H 12 .
From the arguments above, we have

H 12  u 1 (t)  H 11 for all t 2 [0, !].
It then follows from the second equation of of (4.4.3) and ( 4

.4.4) that âi !  Z ! 0 d i exp{u 1 }dt ↵ + exp{u 1 } + exp{u i }  Z ! 0 d i exp{H 11 }dt ↵ l + l exp{H 11 } + l exp{u i (⇠ i )} = ! di exp{H 11 } ↵ l + l exp{H 11 } + l exp{u i (⇠ i )} , from which implies u i (⇠ i )  ln ( di âi l ) exp{H 11 } âi ↵ l âi l = ln â1 ( di âi l ) exp{2â 1 !} âi b1 ↵ l âi b1 l ,
and then

u i (t)  u i (⇠ i ) + Z ! 0 |u 0 i (t)|dt  ln â1 ( di âi l ) exp{2â 1 !} âi b1 ↵ l âi b1 l + 2 d ( d i )! = H i1 (i 2).
Similarly, for i, j 2 (i 6 = j) and t 0, we have |L ij | and letting ⌦ = {u 2 X| ||u|| < B}, then ⌦ satisfies the condition (a) of Lemma 4.12. To compute the Brouwer degree, let us consider the homotopy

âi ! = Z ! 0  d i exp{u 1 } ↵ + exp{u 1 } + exp{u i } b i exp{u j } dt Z ! 0  d i exp{H 12 } ↵ u + u exp{H 12 } + u exp{u i (⌘ i )} b i exp{H j1 } dt = " di exp{H 12 } ↵ u + u exp{H 12 } + u exp{u i (⌘ i )} bi exp{H j1 } # !, from which implies u i (⌘ i ) ln d i exp{H 12 } (↵ u + u exp{H 12 })(â i + bi exp{H j1 }) u (â i + bi exp{H j1 }) , and
u i (t) u i (⌘ i ) Z ! 0 |u 0 i (t)|dt ln d i exp{H 12 } (↵ u + u exp{H 12 })(â i + bi exp{H j1 }) u (â i + bi exp{H j1 }) 2 d ( d i )! = H i2 . Put B i = max{|H i1 |, |H i2 |} (i 1), then max t2[0,!] |u i |  B i . Thus,
H µ (u) = µQN (u) + (1 µ)G(u), µ2 [0, 1],
where

G : R 3 ! R 3 , G(u) = 2 6 4 â1 b1 exp{u 1 } â2 b2 exp{u 3 } + 1 ! R ! 0 d 2 (t) exp{u 1 }dt ↵+ exp{u 2 }+ exp{u 2 } â3 b3 exp{u 2 } + 1 ! R ! 0 d 3 (t) exp{u 1 }dt ↵+ exp{u 3 }+ exp{u 3 } 3 7 5 .
We have

H µ (u) = 2 6 6 4
â1 b1 exp{u 1 } 1 ! R ! 0 h µc 2 exp{u 2 } ↵+ exp{u 1 }+ exp{u 2 } + µc 3 exp{u 3 } ↵+ exp{u 1 }+ exp{u 3 } i dt â2 b2 exp{u 3 } + 1 ! R ! 0 d 2 exp{u 1 }dt ↵+ exp{u 1 }+ exp{u 2 } â3 b3 exp{u 2 } + 1 ! R ! 0 d 3 exp{u 1 }dt ↵+ exp{u 1 }+ exp{u 3 } 3 7 7 5
.

By carrying out similar arguments as above, one can easily show that any solution

u ⇤ of the equation H µ (u) = 0 2 R 3 with µ 2 [0, 1] satisfies L i1  u ⇤ i  L i2 (i 1). Thus, 0 / 2 H µ (@⌦ \ KerL) for µ 2 [0, 1]
, and then QN (@⌦ \ KerL) 6 = 0. Note that the isomorphism J can be the identity mapping I, since ImP = KerL, by the invariance property of homotopy, we have

deg(JQN, ⌦ \ KerL, 0) = deg(QN, ⌦ \ KerL, 0) = deg(QN, ⌦ \ R 3 , 0) = deg(G, ⌦ \ R 3 , 0) = sgn 8 > < > : det 2 6 4 b1 exp{u 1 } 0 0 @f 2 (u 1 ,u 2 ) @u 1 @f 2 (u 1 ,u 2 ) @u 2 b2 exp{u 3 } @f 3 (u 1 ,u 3 ) @u 1 b3 exp{u 2 } @f 3 (u 1 ,u 3 ) @u 3 3 7 5 9 > = > ; = sgn ⇢ b1 exp{u 1 } ✓ @f 2 (u 1 , u 2 ) @u 2 @f 3 (u 1 , u 3 ) @u 3 + b2 b3 exp{u 2 + u 3 } ◆ , (4.4.5)
where deg(•, •, •) is the Brouwer degree [START_REF] Krawcewicz | Theory of Degrees, with Applications to Bifurcations and Di↵erential Equations[END_REF] and ). Therefore, the predator X 3 vanishes and system (4.1.1) is not permanent.

f 2 (u 1 , u 2 ) = 1 ! Z ! 0 d 2 exp{u 1 }dt ↵ + exp{u 1 } + exp{u 2 } , f 3 (u 1 , u 3 ) = 1 ! Z ! 0 d 3 exp{u 1 }dt ↵ + exp{u 1 } + exp{u 3 } • It is easy to see that functions f i (u 1 , u i ) are decreasing in u i 2 R (i 2). Then @f 2 (u 1 , u 2 ) @u 2 @f 3 (u 1 , u 3 ) @u 3 > 0. ( 4
For the model with periodic coe cients, we consider the last example concerning the numerical solutions of system (4.1.1) where a

1 = 3 + 1.3 sin(2t), b 1 = 2.2 + 1.9 cos(2t), c 2 = 2.8(2.2 sin(2t)), c 3 = (3.5 2 cos(2t))/2, a 2 = 1 0.6 cos(2t), b 2 = 1.2+0.5 sin(2t), d 2 = 4+1.8 sin(2t), a 3 = (1.2 cos(2t))/3, b 3 = 1.4+1.1 sin(2t), d 3 = 3.1 2.3 sin(2t), ↵ = 0.1(3.2 2 cos(2t)), = (2.1 + 1.8 cos(2t))/5, = 3.3 sin(2t)
and initial value x 0 = (1.1, 1.9, 1.4). Under ⇡ -periodic perturbation satisfying Theorem 4.13, system (4.1.1) has positive ⇡ -periodic solution (see . Moreover, the hypothesis of Theorem 4.11 also holds then it is globally asymptotically stable.

In conclusion, this work provides some results about the asymptotic behavior The mathematical analysis presented in this model shows that according to the value of the coe cients, one can make suitable predictions about the asymptotic behavior of the overall predator -prey system including the permanence, the periodicity, the global asymptotic stability and specially the extinction of species. Those conclusions warn us to have a timely decision to protect species in our ecolog- Chapter 5

Population dynamics in random environments 5.1 Dynamical behavior of a stochastic SIRS epidemic model

Introduction

This section details paper [START_REF] Nguyen | Dynamical behavior of a stochastic SIRS epidemic model[END_REF]. The dynamics of disease spreading among a population have been investigated very widely in the frame of deterministic models e.g. [START_REF] Brauer | Mathematical Models in Population Biology and Epidemiology, Second Edition[END_REF], [START_REF] Capasso | Mathematical Structures of Epidemic Systems[END_REF], [START_REF] Leah | Mathematical Models In Biology[END_REF], [START_REF] Murray | Mathematical Biology[END_REF]. In such deterministic models, the environment is assumed to be constant. However, in most real situations, it is necessary to take into account random change of environmental conditions and their e↵ects on the spread of the disease. For instance, the disease can be more likely to spread in wet (cold) condition rather than in dry (hot) condition or any other characteristics of the environment that may change randomly. Therefore, it is important to consider the disease dynamics under the impact of randomness of environmental conditions. There are many papers about this topic in recent years e.g. [START_REF] Artalejo | Stochastic epidemic models with random environment: quasi-stationarity, extinction and final size[END_REF], [START_REF] Gray | The SIS epidemic model with Markovian switching[END_REF], [START_REF] Ji | The behavior of an SIR epidemic model with stochastic perturbation[END_REF], [START_REF] Lahrouz | Asymptotic properties of switching di↵usion epidemic model with varying population size[END_REF].

The basic simplest epidemic model that we consider is the classical SIRS model introduced by Kernack-McKendrick of the form (see [START_REF] Leah | Mathematical Models In Biology[END_REF] for details)

8 > < > : Ṡ = aSI + cR İ = aSI bI Ṙ = bI cR, (5.1.1)
where the susceptible (S), infective (I) and removed (R) classes are three compartments of the total population N . Transitions between these compartments are denoted respectively by a, b, and c. They describe the course of the transmission, recovery and loss of immunity. In further studying the SIRS model, we note that the sum S + I + R = N and it is a constant of population size. So that for convenience the removed class (R) can always be eliminated. The reduction of the equation (5.1.1) is then

( Ṡ = aSI + c(N S I) İ = aSI bI. (5.1.2)
It is easy to analyze the previous simple system (5.1.2) and to show that two situations can occur (see [START_REF] Hethcote | Qualitative analyses of communicable disease models[END_REF], [START_REF] Leah | Mathematical Models In Biology[END_REF], [START_REF] May | Parasitic infections as regulators of animal populations[END_REF]):

-If the basic reproduction number R 0 = Na b > 1 the disease spreads among the population and a positive equilibrium (s ⇤ , i ⇤ ) is globally asymptotically stable. It is therefore an endemic situation.

-If R 0 = Na b < 1 the disease is eradicated as a disease free equilibrium (N, 0), which is asymptotically stable. This situation is the eradication of the disease among the population.

In this work, we shall concentrate on the switching two classical Kernack and MacKendrick SIRS model, which will be chosen as the basic models for the epidemics. We shall assume that there are two environmental states in each of which the system evolves according to a deterministic di↵erential equation and that the system switches randomly between these two states. Thus, we can suppose there is a telegraph noise a↵ecting on the model in the form of switching between two-element set, E = {+, }. With di↵erent states, the disease dynamics are di↵erent. The stochastic displacement of environmental conditions provokes model to change from the system in state + to the system in state and vice versa.

Several questions naturally arise. For instance, in the case where the disease spreads in an environmental condition, while it is vanished in the other one, what will be the global and asymptotic behavior of the system? Using the basic reproduction number R 0 of both models and the switching intensities, can we make predictions about the asymptotic behavior of the global system, i.e., the existence of a global endemic state or a disease free state?

This section has 5 subsections. Subsection 5.1.2 details the model and gives some properties of the boundary equations. In Subsection 5.1.3, dynamic behavior of the solutions is studied and the !-limit sets are completely described for each case. It is shown that the threshold which will be given later plays an important role to determine whether the disease will vanish or be persistent. In Subsection 5. Population dynamics in random environments the interval (⌧ n , ⌧ n+1 ) and vice versa. Therefore, (S(⌧ n ), I(⌧ n )) is the switching point, that is the terminal point of one state and simultaneously the initial condition of the other. It is known that with positive initial values, solutions to both (5.1.4) and (5.1.5) remain nonnegative for all t 0. Thus, any solution to (5.1.3) starting in intR 2 + exists for all t 0 and remain nonnegative. It is easily verified that the systems (5.1.4) and (5.1.5) respectively have the equilibrium points

(s ± ⇤ , i ± ⇤ ) = ⇣ b(±) a(±) , c(±)(N b(±) a(±) ) b(±) + c(±) ⌘ , (5.1.6)
and their global dynamics depend on these equilibriums. Concretely, if i ± ⇤ > 0 then these positive equilibriums are asymptotically stable, i.e., when

N > b(±) a(±) , lim t!1 (S ± (t), I ± (t)) = (s ± ⇤ , i ± ⇤ )
. This is the endemic case, both susceptible and infective classes are together present. On the contrary, if N  b(±) a(±) then lim t!1 (S ± (t), I ± (t)) = (N, 0) and the infective class will disappear. It is called the free case

Dynamical behavior of solutions

For any (s 0 , i 0 ) 2 intR 2 + with s 0 +i 0  N , we denote by (S(t, s 0 , i 0 ), I(t, s 0 , i 0 )) the solution of (5.1.3) satisfying the initial condition (S(0, s 0 , i 0 ), I(0, s 0 , i 0 )) = (s 0 , i 0 ). For the sake of simplicity, we write (S(t), I(t)) for (S(t, s 0 , i 0 ), I(t, s 0 , i 0 )) if there is no confusion. A function f defined on [0, 1) is said to be ultimately bounded above (respectively, ultimately bounded below) by m if lim sup t!1 f (t) < m (respectively, lim inf t!1 f (t) > m).

It is easy to see that the triangle r := {(s, i) : s 0, i 0; s + i  N } is invariant for the system (5.1.3). In the future, without loss of generality, suppose that b(+) a(+)  b( ) a( ) . Define = p a(+)N b(+) + q a( )N b( ) .

( (a(⇠t)N b(⇠t))d t = p a(+)N b(+) + q a( )N b( ) = .

Denote g min = min(g(+), g( )), g max = max(g(+), g( )) for g = a, b, c. We have

lim inf t!1 1 t Z t 0 a max (N S( t))d t lim inf t!1 1 t Z t 0 a(⇠t)(N S( t))d t lim inf t!1 1 t Z t 0 a(⇠t)(N b(⇠t))d t = .
( In view of this corollary, in the following we suppose that b(+) a(+) < N. Proposition 5.1.3. S(t) is ultimately bounded below by S min > 0 and there is an invariant set for the system (5. 2a(+) I(t) < 0 for all points lying above the line BC, whereas Ṡ > m for all points that are below the line BC and on the left of AB by (5.1.10) (see the figure 5.4). Therefore, it is easy to see that the the quadrangle ABCD is invariant under system (5.1.3) and all positive solutions ultimately go there.

I min > 0 if b( ) a( ) < N.
Proof. Since b( ) a( ) < N, we can find an 0 < " Adapted from the concept in [START_REF] Brzeniak | Pathwise global attractors for stationary random dynamical systems[END_REF], we define the (random) ! limit set of the trajectory starting from an initial value (s 0 , i 0 ) by

0 < 1 such that min a(±)si + c(±)(N s i) > 0 : 0 < s  b( ) a( ) , 0 < i  " 0 > 0.
⌦(s 0 , i 0 , !) = \ T >0 [ t>T S(t, s 0 , i 0 , !), I(t, s 0 , i 0 , !) .
This concept is di↵erent from the one in [START_REF] Crauel | Attractors for random dynamical systems[END_REF] but it is closest to that of an ! limit set for a deterministic dynamical system. In the case where ⌦(s 0 , i 0 , !) is a.s constant, it is similar to the concept of weak attractor and attractor given in [START_REF] Mao | Attraction, stability and boundedness for stochastic di↵erential delay equations[END_REF][START_REF] Yuan | Attraction and Stochastic Asymptotic Stability and Boundedness of Stochastic Functional Di↵erential Equations with Respect to Semimartingales[END_REF]. Although, in general, the !-limit set in this sense does not have the invariant property, this concept is appropriate for our purpose of describing the pathwise asymptotic behavior of the solution with a given initial value.

Our task in the next part is to describe the !-limit sets of the system (5.1.3). Let ⇡ + t (s, i) = (S + (t, s, i), I + (t, s, i)), (resp. ⇡ t (s, i) = (S (t, s, i), I (t, s, i))) be the solution of (5.1.4) (resp. (5.1.5)) starting in the point (s, i) 2 R 2 + . From now on, let us fix an (s 0 , i 0 ) 2 R 2 + and suppose > 0. This implies that at least one of the systems (5.1.4), (5.1.5) has a globally asymptotically stable positive equilibrium. Without loss of generality, we assume the equilibrium point of the system (5.1.4) has this property, i.e., lim

t!1 ⇡ + t (s, i) = (s + ⇤ , i + ⇤ ) 2 int R 2 + for any (s, i) 2 int R 2
+ . Also, suppose that ⇠ 0 = + with probability 1. For " > 0 small enough, denote by U " (s, i) the "-neighborhood of (s, i) and by H " ⇢ R 2 + the compact set surrounded by AB, BC, CD and the line i = ". Set

S n = S(⌧ n , s 0 , i 0 ); I n = I(⌧ n , s 0 , i 0 ), F n 0 = (⌧ k : k  n); F 1 n = (⌧ k ⌧ n : k > n). We see that (S n , I n ) is F n 0 adapted. Moreover, given ⇠ 0 , then F n 0 is independent of F 1
n . Lemma 5.1.8. Let J r be a compact set and (s + ⇤ , i + ⇤ ) 2 J. Then, for any 2 > 0, there is a T

1 = T 1 ( 2 ) > 0 such that ⇡ + t (s, i) 2 U 2 (s + ⇤ , i + ⇤ ) for any t T 1 and (s, i) 2 J.
Proof. Consider the system (5.1.4). Since (s

+ ⇤ , i + ⇤ ) is asymptotically stable, we can find a ¯ 2 = ¯ 2 ( 2 ) > 0 such that ⇡ + t U¯ 2 (s + ⇤ , i + ⇤ ) ⇢ U 2 (s + ⇤ , i + ⇤ ) 8t 0.
On the one hand, for (s, i) 2 J, lim t!1 ⇡ + t (s, i) = (s + ⇤ , i + ⇤ ) which implies that there exists a T si satisfying ⇡ + t (s, i) 2 U¯ 2 /2 (s + ⇤ , i + ⇤ ) for all t T si . By the continuous dependence of the solutions on the initial conditions, there is a neighborhood U si of (s, i) such that for any (u, v) 2 U si we have

⇡ + T si (u, v) 2 U¯ 2 (s + ⇤ , i + ⇤ ).
As a result,

⇡ + t (u, v) 2 ⇡ + t T si U¯ 2 (s + ⇤ , i + ⇤ ) ⇢ U 2 (s + ⇤ , i + ⇤ ) 8t T si . Since J is
, i) = ⇡ + t ⇤ 2 ⇡ t ⇤ 1 (s 0 , ī0 ) 2 S.
Hence, U ⇢ ⇢ ⌦(s 0 , i 0 , !). Thus, there is a stopping time < 1 a.s. such that (S( ), I( )) 2 U . Since is a forward invariant set and U ⇢ , it follows that (S(t), I(t)) 2 8t > with probability 1. The fact (S(t), I(t)) 2 for all t > implies that ⌦(s 0 , i 0 , !) ⇢ . By combining with Theorem 5.1.12 we get = ⌦(s 0 , i 0 , !) a.s.. Moreover, we see that when condition (5.1.13) does not happen, it ensures the existence of t 0 satisfying (5.1.14). Indeed, consider the set of all (s, i) The equation (5.1.16) describes a quadratic curve (5.1.16). However, it is easy to prove that any quadratic curve is not the integral curve of the system (5.1.5). This means that we can find a t 0 such that the point (s 0 , ī0 ) = ⇡ t 0 (s + ⇤ , i + ⇤ ) satisfies the condition (5.1.14). The proof is complete.

2 int R 2 + satisfying det ✓ Ṡ+ (s, i) Ṡ (s, i) İ+ (s, i) İ (s, i) ◆ = 0, ( 5

The semigroup and the stability in distribution

It is well-known that the pair (⇠ t , S(t), I(t)) is a homogeneous Markov process with the state space V := E ⇥ intR 2 + . Let B(V) be the Borel algebra on V and be the Lebesgue measure on intR 2 + . Denote by m the product measure on (V, B(V)) defined by m(+, A) = p (A) and m( , A) = q (A).

As shown in [START_REF] Du | Dynamics of Kolmogorov systems of competitive type under the telegraph noise[END_REF]Lemma 3.1], if the distribution of (⇠ 0 , S(0), I(0)) is absolutely continuous with respect to the measure m, so is the distribution of (⇠ t , S(t), I(t)). We can therefore define P (t)f to be the density function of (⇠ t , S(t), I(t)) given that (⇠ 0 , S(0), I(0)) has the density f . If > 0 and (5.1.13) holds, all positive solutions converge almost surely to the equilibrium (s + ⇤ , i + ⇤ ) = (s ⇤ , i ⇤ ). Otherwise, we have Theorem 5.1.14. If > 0 and (5.1.13) does not hold, then (⇠ t , S(t), I(t)) has a stationary distribution ⌫ ⇤ , concentrated on E ⇥ (r \ int R 2 + ). In addition, ⌫ ⇤ is the unique stationary distribution having the density f ⇤ , and lim

t!1 kP (t)f f ⇤ k = 0 for any f 2 D.
Proof. We firstly point out the existence of a stationary distribution of the process (⇠ t , S(t, )I(t)). From the proof of Proposition 5.1.1, we have

lim inf t!1 1 t Z t 0 I( t)d t c min (a max N + c max )a max =: ⇢ > 0.
Denote by 1 A the indicator function of the set A. By using the relations 1 t

Z t 0 I( t)d t = 1 t Z t 0 I( t)1 {I( t)< ⇢ 2 } d t + 1 t Z t 0 I( t)1 {I( t) ⇢ 2 } d t  ⇢ 2 + N t Z t 0 1 {I( t) ⇢ 2 } d t,
it follows, with probability 1, that

lim inf t!1 1 t Z t 0 1 {I( t) ⇢ 2 } d t ⇢ 2N .
Applying Fatou lemma yields

lim inf t!1 1 t Z t 0 P I( t) ⇢ 2 d t ⇢ 2N .
(5.1.17)

Consider the process (⇠ t , S(t), I(t)) on a larger state space E ⇥ r\{(s, i) : s = 0, 0  i  N } . it is easy to prove that (⇠ t , S(t), I(t)) is a Feller process. Therefore, by using [102, Theorem 4.5] (or [START_REF] Stettner | On the existence and uniqueness of invariant measure for continuous time Markov processes[END_REF]) the above estimate (5.1.17) implies the existence of an invariant probability measure ⌫ for the process (⇠ t , S(t),

I(t)) on E ⇥ r \ {(s, i) : s = 0, 0  i  N } . Since {(s, i) : i = 0, 0  s  N } is invariant and lim t!1 I(t) = 0 if S(0) = 0, it follows that ⌫({(s, i) : i = 0, 0  s  N }) = 0. Thus, ⌫(E ⇥ (r \ intR 2 + )) > 0. By virtue of invariant property of E⇥intR 2 + , the measure ⌫ ⇤ defined by ⌫ ⇤ (A) = ⌫ A \ E ⇥ (r \ intR 2 + ) ⌫(E ⇥ (r \ intR 2 + )
for any measurable A 2 B(V) is a stationary distribution on E ⇥ (r \ intR 2 + ) of the process (⇠ t , S(t), I(t)). The asymptotic stability of f ⇤ can be proved by using (5.1.14); the arguments analogous to [51, Proposition 3.1] and then applying [START_REF] Pichór | Continuous Markov semigroups and stability of transport equations[END_REF]Proposition 2]. The proof is complete.

Simulation and discussion

We illustrate the above model by following numerical examples.

Example I: > 0 and the endemic is present in both states (see figure 5

Example II:

> 0 and one state is endemic, the other is disease free. The system (5.1.4) with coe cients a(+) = 1.6, b(+) = 169, c(+) = 486 has a asymptotically stable positive equilibrium and the system (5.1.5) with coe cients a( ) = 0.7, b( ) = 375, c( ) = 328 tends to the quantity of population N = 500, the number of switches n = 700, transition intensities ↵ = 8 = 15 and initial condition (S(0), I(0)) = (104, 336). Since ⇡ 402.83, the system (5.1.3) is persistent (see figure 5.9).

Example III:

< 0 and a system has positive equilibrium, the other has disease free equilibrium (figure 5.12). The parameters of the model are ↵ = 20, = 5, a(+) = 1.9, b(+) = 176, c(+) = 465, a( ) = 0.5, b( ) = 455, c( ) = 347, N = 500, (S(0), I(0)) = (64, 362), n = 100. Although the positive equilibrium of the system (5.1.4) is asymptotically stable, the system (5.1.3) is not persistent because = 9.2. The basic reproduction number R 0 is an important concept in epidemiology. R 0 is the threshold parameter for many epidemiological models, it informs whether the disease becomes extinct or whether the disease is endemic. For example, there are many recent papers about periodic epidemic models that concentrate on defining and computing R 0 (see [START_REF] Bacaer | On the biological interpretation of a definition for the parameter R0 in periodic population models[END_REF], [START_REF] Bacaer | On the final size of epidemics with seasonality[END_REF], [START_REF] Friedman | Epidemiological models with seasonality[END_REF], [144] [151]). In the classic SIRS model (5.1.1), R 0 is valued by ratio Na b , it represents the rate of increase of new infections generated by a single infectious individual in a total sane population. Based on this R 0 , we give out the key parameter for our stochastic SIRS model. reads as follows: = p a(+)N b(+) + q a( )N b( ) . This is the average of two terms associated with each system + or weighted by the switching intensities. Therefore, in the stochastic model, can be interpreted as the 5. Population dynamics in random environments Figure 5.12: Orbit of the system (5.1.3) in example III. average number of infective individuals generated by a single infectious individual in a totally sane population for the total system with random switches. We can, therefore, understand that when is positive, it signifies that asymptotically the total system will go towards an endemic state while the disease will vanish provided that it is negative. Hence, for the stochastic model, is a very important parameter that enables us to obtain important informations about the asymptotic behavior of the total system. We illustrate di↵erent situations in the following numerical simulations. Examples I and II show cases where is positive, the first one illustrates the switching between two endemic systems + and , whilst the second one depicts a system composed of an endemic case + and a case for which the disease free equilibrium is stable. In both examples, the simulations show that asymptotically the total system persists leading to an endemic situation.

The last example III considers the case of < 0, with an endemic system + and for the other one a stable disease free equilibrium. As expected, the simulation shows that after several switches, the disease is globally eradicated.

Examples II and III are interesting because they illustrate a similar case, i.e. when systems + and have opposite trends, system + being endemic and system being disease free. In those examples, it is thus questionable to predict what will be the global evolution of the complete system switching at random between these two di↵erent situations. The answer is given by looking at the sign of parameter which allow us to predict if the disease will globally invade or vanish in the long term.

Global changes may have important consequences on the spreading of emergent diseases and epidemics. Therefore, it is important to provide pertinent tools that allow us to make suitable predictions about the possibility of emergence of a disease in a changing environment undergoing climatic and environmental changes. The aim of this chapter was to provide such e cient tools.

As a perspective, the system would be extended to the case of a system switching randomly between n states, n > 2. It would also be interesting to test the model on real situations, like malaria, switching between wet and dry periods. Otherwise, for further study on the epidemic models under the e↵ect of random environmental conditions, we could add some more other stochastic factors as in [START_REF] Brauer | Mathematical Epidemiology[END_REF] to this SIRS model.

Evolution of Lotka-Volterra predator-prey

systems under telegraph noise

Model

The dynamics of predator-prey systems have been investigated very largely in the frame of deterministic models, Edelstein-Keshet (1998) [START_REF] Edelstein-Keshet | Mathematical Models in Biology[END_REF], [START_REF] Murray | Mathematical Biology[END_REF] [START_REF] Murray | Mathematical Biology[END_REF]. In [START_REF] Bazykin | Nonlinear Dynamics of Interacting Populations[END_REF] [START_REF] Bazykin | Nonlinear Dynamics of Interacting Populations[END_REF], one can find a review of most classical deterministic predatorprey models. The two variables are the prey x(t) and predator y(t) densities at time t. The classical form of a predator-prey model is the following one:

⇢ dx dt = f (x) h(x, y)y dy dt = e eh(x, y)y µy,
where the function f (x) is the natural growth function of the prey, h(x, y)y is the capture term; e e is a positive prey biomass into predator biomass conversion parameter. µ is the natural mortality rate for predators. h(x, y) is the so-called functional response, i.e., the prey density captured per unit of time and per unit of predator density. In the classical Lotka-Volterra model, it is assumed that the functional response is type I, i.e., depending only on the prey density and linear, i.e., h(x) = qx where q is a positive constant which is called the catch-ability. It is also usual to assume that prey grows logistically leading to the model:

⇢ dx dt = x r 1 x K qy = x (a bx cy) dy dt = (e eqx µ) y = ( d + ex) y.
Where r is the growth rate of the prey and K its carrying capacity. For the sake of simplification, in the next sections, we use the model under the form involving the parameter set (a, b, c, d, e) and the links with ecological parameters (r, K, q, µ, e e) is not given here because it is obvious.

This classical Lotka-Volterra model assumes that species live in a constant environment. However, it is clear that it is not the case in reality and that it is important to take into account the variability of the environment which may have important consequences on the dynamics and persistence of a predator-prey community. The variability of the environment may be expressed under the stochastic factors. For the stochastic Lotka-Volterra equation, there is not too much in mathematical literature, and almost nothing in statistical inference. Here, we mention one of the first attempts in this direction, the very interesting paper of Arnold et al. [START_REF] Arnold | The influence of external real and white noise on the Lotka-Volterra model[END_REF] in which the authors used the theory of Brownian motion processes and the related white noise models to study the sample paths of the equation. For the branching models in a varying environment, we can refer to [START_REF] Allen | A comparison of three di↵erent stochastic population models with regard to persistence time[END_REF]3,[START_REF] Sathananthan | Stability analysis of a stochastic logistic model[END_REF]. A systematic review has been given in [START_REF] Allen | An Introduction To Stochastic Processes With Applications to Biology[END_REF]. In the simplest case, one might consider that environmental conditions can switch between two states, a hot and cold one, a dry state and wet one. Thus, we can suppose there is a telegraph noise a↵ecting on the model in the form of switching between two-element set, E = {1, 2}. With di↵erent states, the coe cients of model are di↵erent. The stochastic displacement of environmental conditions provokes model to change from the system in state one to the system in state two and vice versa.

Let's consider again process (⇠ t ) mentioned in section 5.

P{⇠ t = 1} = ↵ + ; q = lim t!1 P{⇠ t = 2} = ↵ ↵ + .
Lotka-Volterra predator-prey system with telegraph noise (⇠ t ) is described by equation

( ẋ = x (a(⇠ t ) b(⇠ t )x c(⇠ t )y) , ẏ = y ( d(⇠ t ) + e(⇠ t )x) , (5.2.1)
where g : E ! (0, 1) for g = a, b, c, d, e. This section studies the dynamical behavior of system under telegraph noise. It is structured into four subsections. In subsection 5.2.2, we summarize our paper [START_REF] Auger | Evolution of Lotka-Volterra predator-prey systems under telegraph noise[END_REF]. Paper [START_REF] Auger | Evolution of Lotka-Volterra predator-prey systems under telegraph noise[END_REF] presents this model with carrying capacity of environment (K < 1 i.e. b 6 = 0) where the dynamics of the system is quite di↵erent. The predator may be extinct when one deterministic system has only non positive rest point. However, if two deterministic systems have positive rest points, it is proved that the model will be permanent with probability 1. Moreover, we show the existence of stationary distribution of solution in this case. When the carrying capacity of environment is absent (K = 1 i.e. b = 0), the telegraph noise does the model chaotically. It can not be permanent if the positive rest points of two deterministic systems do not coincide [START_REF] Takeuchi | Evolution of predator-prey systems described by a Lotka -Volterra equation under random environment[END_REF]. This case is shown in our paper [START_REF] Takeuchi | Evolution of predator-prey systems described by a Lotka -Volterra equation under random environment[END_REF] and summarized in subsection 5.2.3. Last subsection is discussion and conclusion. Figure 5.17: Orbit of system in case y ⇤ 1 > 0, y ⇤ 2 < 0. enough rectangle, containing the rest point, such that every trajectory, passing through a point H 1 , is contained in H 1 = H 1 (ε). We see that the number of the switching points in H \ H 1 must be finite. Otherwise, the orbit of z(t) has to leave from H by Theorem 4.3. Therefore, there is k > 0 such that z n ∈ H 1 for any n > k which implies that lim t→∞ (x(t, x, y), y(t, x, y)) = (p, q). Thus, we have (4.5). ✷

Conclusion

Up to now various dynamical models in ecology have been proposed under the environmental fluctuations corresponding to seasonality. The deterministic switching between two different predator-prey systems exhibits more complex dynamics including stable equilibrium point, limit cycle, and also chaos [START_REF] Arnold | The influence of external real and white noise on the Lotka-Volterra model[END_REF]. The possibilities or the conditions for the coexistence of temporally segregated competitors in a cyclic environment [START_REF] Auger | Aggregation of variables and applications to population dynamics[END_REF] or two competing species following enough rectangle, containing the rest point, such that every trajectory, passing through a point H 1 , is contained in H 1 = H 1 (ε). We see that the number of the switching points in H \ H 1 must be finite. Otherwise, the orbit of z(t) has to leave from H by Theorem 4.3. Therefore, there is k > 0 such that z n ∈ H 1 for any n > k which implies that lim t→∞ (x(t, x, y), y(t, x, y)) = (p, q). Thus, we have (4.5). ✷

Conclusion

Up to now various dynamical models in ecology have been proposed under the environmental fluctuations corresponding to seasonality. The deterministic switching between two different predator-prey systems exhibits more complex dynamics including stable equilibrium point, limit cycle, and also chaos [START_REF] Arnold | The influence of external real and white noise on the Lotka-Volterra model[END_REF]. The possibilities or the conditions for the coexistence of temporally segregated competitors in a cyclic environment [START_REF] Auger | Aggregation of variables and applications to population dynamics[END_REF] or two competing species following In both cases, the green line shows the trajectory of (5.2.8) subjecting to System (5.2.9) and the red line shows the trajectory of (5.2.8) subjecting to System (5.2.10). eco-system. This conclusion warns us to have a timely decision to protect species in our ecosystem.

Remark 5.1. By simulation results, we observe that (4.5) does not occur. However, so far we are unable to prove this conjecture. This is still an open problem.

We illustrate our result by simulations. Figure 5 shows the behavior of the trajectory of the systems ẋ = x(a(ξ t )b(ξ t )y), ẏ = y(-c(ξ t ) + d(ξ t )x). eco-system. This conclusion warns us to have a timely decision to protect species in our ecosystem.

Remark 5.1. By simulation results, we observe that (4.5) does not occur. However, so far we are unable to prove this conjecture. This is still an open problem.

We illustrate our result by simulations. Figure 5 shows the behavior of the trajectory of the systems ẋ = x(a(ξ t )b(ξ t )y), ẏ = y(-c(ξ t ) + d(ξ t )x).

(5.

Discussion and conclusion

Up to now there have been proposed various dynamical models in ecology under the environmental fluctuations corresponding to seasonality. The deterministic switching between two di↵erent predator-prey systems exhibits more complex dynamics including stable equilibrium point, limit cycle, and also chaos [START_REF] Hanski | Population oscillations of boreal rodents: regulation by mustelid predators leads to chaos[END_REF]. The possibilities or the conditions for the coexistence of temporally segregated competitors in a cyclic environment [START_REF] Levin | Dispersion and population interactions[END_REF] or two competing species following Lotka-Volterra competition system in a seasonally fluctuating environment [START_REF] Mao | Asymptotic Behavior of the Stochastic Lotka-Volterra model[END_REF] are studied. The stochastic change of environment gives the similar e↵ect on population dynamics with seasonality [START_REF] Chesson | Environmental variability promotes coexistence in lottery competitive systems[END_REF]. Most of such models are formulated by deterministic dynamics, but some kind of stochasticity reflecting complexity of biological or environmental factors should be introduced in population dynamics. Slatkin [START_REF] Slatkin | The dynamics of a population in a Markovian environment[END_REF] introduced 5. Population dynamics in random environments and analyzed a class of general models of single population which grows under the telegraph noise exactly same in this paper. In section 5.2 we obtained the results for two-species Lotka-Volterra predator-prey systems under telegraph noises. We believe the necessity to collect the results for various types of two-species dynamical systems, and to proceed to obtain the general conditions for the behaviors of the systems.

In subsection 5.2.2 we consider an ecology system where there are two species related by predator-prey relation and environment has a finite carrying capacity. The mathematical analysis presented in this model shows that according to the value of some number , one can make suitable predictions about the asymptotic behavior of the overall predator-prey system. Suppose that the evolution of every species depends on the quantity of rainfall for every period. If the rainfall is su cient (good state), the catch ability of the predator is good and the quantity of every species asymptotes to the positive values (the prey and predator co-exist). Whenever the rainfall is small (bad state), the hunting potential of the predator becomes very weak and the amount of predator gets smaller with increasing of time (the predator vanishes). Suppose that the rainfall is in a stationary regime (switching stationarily between dry season and rainfall one). If the two states are good, i.e., both y ⇤ 1 > 0 and y ⇤ 2 > 0, although the quantity of two species is chaotic, but the system is still permanent. Consequently, none of species is extinct. When there is at least a system having the bad state, i.e., either y ⇤ 1 < 0 or y ⇤ 2 < 0 we see that lim inf t!1 y(t) = 0. Depending on the sign of the value , the quantity of the predator y(t) can be recovered or not. In case > 0 we have lim sup t!1 y(t) > 0, i.e., the amount of the predator is recovered (of course in the rainfall season). If < 0 we have lim t!1 y(t) = 0, i.e., the predator vanishes. The model of predator-prey with the absence of carrying capacity is studied in subsection 5.2.3. We obtain an interesting result. In the deterministic environment, this model has periodic solutions and two species are permanent. However, with the e↵ect of the random switching of environment, the orbits of predator-prey model are very chaos and could be not permanent.

In view of practice, when the amount of a species is smaller than a threshold, we consider this species disappears in our system. The obtained results tell us that although the environment condition changes constantly (since the Markov switching process is stationary), species may vanish from ecology system. This warns us to have a timely decision to protect species in our ecology system.

Chapter 6 Conclusion

In this thesis we consider some ecosystem models. We try to show the dynamical behavior and the condition for existence of these models. The obtained results could be useful to explain some phenomena and give us some recommendations for the ecosystems.

Intraguild predation (IGP) occurs when one species prey upon another species with which it competes for a shared resource. IGP is common in nature, for examples lion and lynx, wolf and coyote, insects and arachnids, shark and other predator in marine ecosystems... In chapter 2 we study the e↵ects of spatial heterogeneous environment and fast migration of individuals on coexistence of the IGP dynamics. We concentrate on a two-patch model including two species with IGP interaction. The interaction of two species is competition in one patch and in the other patch it is predation. We also assume two species are not coexistence in each patch. However, we have a interesting results that by the heterogeneous environment and fast migration between two patch, two species can be coexistence in some situation. In addition we are also interested in the e↵ects of the migration parameters, the competition parameters and the predation ones on this coexistence. In fact, the shared resource of two species is hidden in this model. Hence, to describe IGP interaction in more detail, in a further work we will investigate a model with three species where two species have IGP interaction and they prey upon the third species. In the other hand, we can change functional response of predation to other type. We hope the model will have limit cycle if functional response is Holling type 2.

Chapter 3 presents a mathematical model of a fishery targeting a small pelagic fish population distributed over two sites, a marine protected area (MPA) and a fishing area where the fish population can be captured by purse-seine fishing boats. An aggregated model is obtained from complete model by using aggregation methods. Hence we describe the condition for the stableness of aggregation model and show that there is a proportion of small clusters which is maximum in terms of total catch on the long term for a given fishery and fishing e↵ort. These results should allow adaptive management measures not only for the maintenance of model but also for the maximum of total catch. In this model we only investigate the e↵ects of fish clustering on the total catch of a small pelagic purse seine fishery. In future 6. Conclusion work, we will study the e↵ects of both proportion of small cluster and proportion of MPA on this catch, then it can propose a suitable MPA.

In chapter 4 we study the dynamical relationship between three species in a nonautonomous model of one prey and two predators, in which the functional responses are of the form Beddington-DeAngelis functional response. We consider the model with general coe cients and periodic coe cients. The results show us when the model has globally asymptotically stable solution and the condition for the coexistence or extinction of species.

Randomness or stochasticity play a vital role in the dynamics of an ecological system and the variation of random factors can cause sharp changes in it. Chapter 5 shows some ecosystem models under random environment. This random factor is described by telegraph noise and it can do the models chaotically. We try to show the dynamics, persistence and permanence of the models. In some conditions, the models is persistent, permanent or not. The persistence is "not good" for epidemic model, it means the disease will globally invade in the long term. However, the permanence is "good" for predator-prey models, it ensures the species coexistence. We only investigate the models with the e↵ect of telegraph noise in this chapter, then, for future work we can mention seasonality of environment characteristics to describe the models in more detail. How will the dynamical behavior of the models be if they have more than the two states of environment or if they have the fast migration of individuals as in chapter 2 and 3? They are the problems to work on after.

1

 1 Single species growth The exponential growth model In 1798 Malthus published a book entitled An Essay on the Principle of Population [94]. In this book he investigated the simplest population model of single species dx dt = bx dx, (1.1.1) in which b and d are per capita birth and death rate, x(t) is the population size of the species at time t. The equation (1.1.1) can be written more succinctly as dx dt = rx, (1.1.2)

Figure 1 . 1 :

 11 Figure 1.1: Solutions of logistic model

1. 1 dynamics 3 Table 1 . 1 :

 1311 Population Organizing ecological interactions. E↵ect on species 1 E↵ect on species 2

Figure 1 . 2 :

 12 Figure 1.2: Orbits of model (1.1.6) in four cases of coe cients

Figure 1 . 3 :

 13 Figure 1.3: Orbits of model (1.1.7)

Figure 1 . 5 :

 15 Figure 1.5: Realistic data of lynx-hare population. (a) Phase plane plot of the data. (b) Oscillation of population size.

  We consider the spread of a disease in a population. We assume that the size of population is N and fixed. The population has only two class of individuals: susceptible S and infected class I. is the disease transmission (infection) rate. So we have the model İ = SI.(1.2.1)

  It follows I + S ln S = const = I 0 + S 0 ln S 0 . Since equation İ = ( S )I, it shows that İ > 0 if and only if S > . Hence I increases so long as S > , but since S always decreases, I ultimately decreases and tend to 0. I gets I max = N + ln ( S 0 ) (Figure 1.6). If S 0 < , I decreases to 0 and this is no epidemic case. While if S 0 > , I firstly increases to I max and then decreases to 0, this is epidemic case.

Figure 1 . 6 :Figure 1 . 7 :

 1617 Figure 1.6: An epidemic trajectory of model (1.2.3).

Figure 1 . 8 :

 18 Figure 1.8: Orbits of the SIRS model (1.2.5)

Figure 2 . 1 :

 21 Figure 2.1: IGpredation on two patches. IGpredator compete with IGprey on patch 1 or else competition patch. The system is predation on patch 2 or else predation patch.

Figure 2 .Figure 2 . 2 :

 222 Figure 2.2: Phase portraits of the outcome.Figure2ais related to coexistence case (X = 0.2 and Y = 0.9), Figure2bdescribes the case where IGpredator wins (X = 0.4 and Y = 0.7), Figure2cshows the win of IGprey (X = 0.8 and Y = 0.2), Figure2dis the separatrix case (X = 0.7 and Y = 0.7).

Figure

  Figure 2.2: Phase portraits of the outcome.Figure2ais related to coexistence case (X = 0.2 and Y = 0.9), Figure2bdescribes the case where IGpredator wins (X = 0.4 and Y = 0.7), Figure2cshows the win of IGprey (X = 0.8 and Y = 0.2), Figure2dis the separatrix case (X = 0.7 and Y = 0.7).

Figure 2 . 3 :

 23 Figure 2.3: Outcomes of the dynamics in term of migration parameters X and Y . The black dash line is about AE CD = 0, the grey dash line is about AF BD = 0. Domain I: coexistence; domain II: IGpredator wins; domain III: IGprey wins; domain IV: separatrix case. Parameters values are chosen as follows: r 11 = r 12 = r 21 = r 22 = r = 0.6, K 11 = K 12 = K 21 = K = 10, a 12 = 1.5, a 21 = 0.7, b = 0.3, d = 0.6 and e = 0.1.

Figure 2 . 4 : 2 .Figure 2 . 5 :

 24225 Figure 2.4: The change of the four domains in term of a 12 . Figure 2.4 shows three cases from the left to the right where we changed the value

Figure 2 . 6 :

 26 Figure 2.6: The change of the four domains in term of b.

Figure 2 .

 2 Figure 2.6 shows three cases from the left to the right where we changed the value of b by 0.5, 0.3 and 0.1, respectively. In this case, both the black dash and grey dash lines change. For instance, when the value of b increases from 0.3 to 0.5 it implies that IGpredator's predation ability increases. So, it increases the areas of

Figure 3 . 1 :

 31 Figure 3.1: Diagram of the model used in this study showing the interactions between aggregative dynamics (small to large clusters and vis versa) and the migration between fishing and marine protected areas (MPA). See table 3.1 for parameters description.

Figure 3 .

 3 Figure 3.2 shows comparison of the trajectories of complete and aggregated models in the same case and initial conditions for di↵erent values of the small parameters, (a) " = µ = 1, (b) " = 1 and µ = 0.1, (c) " = 0.1 and µ = 1, (d) " = µ = 0.1. Grey trajectory corresponds to the complete model and the black one the aggregated model. The solutions of both models (3.2.1) and (3.3.3) have the same dynamical behaviour. However, to have trajectories close enough of each other we need to chose " and µ at least smaller than 0.1 as shown on figure 3.2(d).Figure3.3 shows a similar result in the case of fleet e↵ort extinction. This means that aggregation methods in this three level system can be successfully used when there exists at least an order of magnitude between two consecutive time scales. In the case of smaller values such as " = µ = 0.01, the approximation would be improved such that trajectories of aggregated and complete models would become extremely close and would appear confounded.

Figure

  Figure 3.2 shows comparison of the trajectories of complete and aggregated models in the same case and initial conditions for di↵erent values of the small parameters, (a) " = µ = 1, (b) " = 1 and µ = 0.1, (c) " = 0.1 and µ = 1, (d) " = µ = 0.1. Grey trajectory corresponds to the complete model and the black one the aggregated model. The solutions of both models (3.2.1) and (3.3.3) have the same dynamical behaviour. However, to have trajectories close enough of each other we need to chose " and µ at least smaller than 0.1 as shown on figure 3.2(d).Figure3.3 shows a similar result in the case of fleet e↵ort extinction. This means that aggregation methods in this three level system can be successfully used when there exists at least an order of magnitude between two consecutive time scales. In the case of smaller values such as " = µ = 0.01, the approximation would be improved such that trajectories of aggregated and complete models would become extremely close and would appear confounded.

3 .Figure 3 . 2 :

 332 Figure 3.2: Orbit of complete (grey) and aggregated (black) models in case of a sustainable fishery. E ⇤ > 0 when: (a) " = µ = 1, (b) " = 1, µ = 0.1, (c) " = 0.1, µ = 1, (d) " = µ = 0.1 and m S = 0.8, m S = 0.3, m L = 0.6, m L = 0.5, k = 0.7, k = 0.4, r = 0.9, h = 0.4, K = 100, c = 0.6, p = 1, q S = 0.07, q L = 0.1, with initial values n SM (0) = 20, n LM (0) = 15, n SF (0) = 10, n LF (0) = 20, E(0) = 35.

3. 4 35 Figure 3 . 3 :

 43533 Figure 3.3: Orbit of complete (grey) and aggregated (black) models in case of a stable fishery free equilibrium. E ⇤ < 0 when " = µ = 0.1, m S = 0.4, m S = 0.7, m L = 0.5, m L = 0.5, k = 0.3, k = 0.6, r = 0.7, h = 0.3, K = 50, c = 0.9, p = 1, q S = 0.02, q L = 0.04, with initial values n SM (0) = 15, n LM (0) = 10, n SF (0) = 12, n LF (0) = 8, E(0) = 30.

.5. 7 )

 7 Under condition (3.5.6) and (3.5.7), we obtain the maximum of Y :

Figure 3 . 4 :

 34 Figure 3.4: Total catch as a function of the proportion of small clusters showing a maximum corresponding to maximum sustainable yield. The coe cients are m S = 0.8, m S = 0.2, m L = 0.7, m L = 0.3, r = 0.9, h = 0.4, K = 100, c = 0.6, p = 1, q S = 0.07, q L = 0.1.

Lemma 4 . 1 .

 41 Both the non-negative and positive cones of R 3 are positively invariant for (4.1.1).

  along the solution of 4.3 The model with general coe cients 47 (4.1.1) gives

Lemma 4 .

 4 12. (Continuation theorem) Let X and Y be two Banach spaces and L a Fredholm mapping of index zero. Assume that N : ⌦ ! Y is L -compact on ⌦ with ⌦ is open and bounded in X. Furthermore, assume that (a) for each 2 (0, 1), x 2 @⌦ \ DomL, Lx 6 = N x; (b) for each x 2 @⌦ \ KerL, QN x 6 = 0; (c) deg{QN x, ⌦ \ KerL, 0} 6 = 0; then the operator equation Lx = Nx has at least one solution in DomL \ ⌦.

3 5 .

 35 Obviously, QN and K P (I Q)N are continuous. It is easy to see that N is Lcompact on ⌦ with any open bounded set ⌦ ⇢ X. Now we will find an appropriate open, bounded subset ⌦ for application of the continuation theorem. Corresponding to the operator equation Lu = N u, 2 (0, 1), we have

Figure 4 . 4 :

 44 Figure 4.4: Population sizes X 3 with respect to time

Figure 4 . 5 :Figure 4 . 6 :x3Figure 4 . 7 :

 454647 Figure 4.5: Orbit of non permanent system

Figure 4 . 8 :

 48 Figure 4.8: Orbit of periodic system with respect to time

Figure 5 . 1 :

 51 Figure 5.1: SIRS diagram.

Figure 5 . 2 :Figure 5 . 3 :

 5253 Figure 5.2: An example of endemic case

  1.3), which absorbs all positive solutions.Proof. Let Smin be chosen such thatNa(±)S min + c(±) let A = (S min , 0), B = (S min , N b(+) 2a(+) ), C = ( b(+) 2a(+) , N b(+)2a(+) ), D = (N, 0). In the interior of the triangle r we have İ(t) = a(⇠ t )(S(t) b(⇠t) a(⇠t) )I(t)  a(⇠ t )(S(t) b(+) a(+) )I(t)  a(⇠ t )( b(+) 2a(+) b(+) a(+) )I(t) = a(⇠ t ) b(+)

Figure 5 . 4 :

 54 Figure 5.4: An example of invariant set. The invariant set is defined by 4 dash dot lines.

Figure 5 . 5 :

 55 Figure 5.5: An example of existence of I min when b(+) a(+) < b( ) a( ) < N.

1 I

 1 Then, while I(t)  " 0 and S(t)  b( ) a( ) we have Ṡ > 0 andİ Ṡ = (a(⇠ t )S b(⇠ t ))I a(⇠ t )SI + c(⇠ t )(N S I) > kIwhere k is some positive number. Denote by the piece of the solution curve to the equation dI dS = kI starting at (S min , " 0 ) and ending at the intersection point ( b( ) a( ) , "1 ) of this solution curve with the line s = b( ) a( ) (see the figure 5.5). Let G be the subdomain of quadrangle ABCD consisting of all (s, i) 2 ABCD lying above the curve if s  b( ) a( ) and lying above the line i = "1 if b( ) a( )  s  N .Obviously, G is invariant domain because İ Ṡ > kI, Ṡ > 0 on and İ > 0 on the segment I = "1 , b( ) a( )  S  N . Since lim sup t!(t) > 1 > " 0 and (S(t), I(t)) must eventually enter the quadrangle ABCD, (S(t), I(t)) also eventually enters G which implies that I(t) ultimately bounded below by I min = " 1 . Corollary 5.1.7. If b( ) a( ) < N then the system (5.1.3) is permanent.

  .1.15) or⇥ a(+)si+c(+)(N s i) ⇤⇥ a( )s b( ) ⇤ [ a( )si+c( )(N s i) ⇤⇥ a(+)s b(+) ⇤ = 0. (5.1.16)

  .6). It corresponds to ↵ = 15, = 18, a(+) = 1.2, b(+) = 432, c(+) = 265, a( ) = 1.5, b( ) = 139, c( ) = 428, N = 500, the initial condition (S(0), I(0)) = (250, 10) and number of switches n = 500. In this example, ⇡ 369.36, the solution of (5.1.3) switches between two asymptotically stable positive equilibriums of the systems (5.1.4) and (5.1.5).

Figure 5 . 6 :

 56 Figure 5.6: Orbit of the system (5.1.3) in example I.

Figure 5 . 7 :Figure 5 . 8 :

 5758 Figure 5.7: Trajectory S(t) in example I

Figure 5 . 9 :

 59 Figure 5.9: Orbit of the system (5.1.3) in example II.

Figure 5 . 10 :Figure 5 . 11 :

 510511 Figure 5.10: Trajectory S(t) in example II

Figure 5 . 13 :Figure 5 . 14 :

 513514 Figure 5.13: Trajectory S(t) in example III

Figure 5 . 18 :

 518 Figure 5.18: The oscillations of x(t) and y(t) in case A.

Figure 5 . 19 :

 519 Figure 5.19: The oscillations of x(t) and y(t) in case B.

  Fig. 5.

Figure 5 . 20 :

 520 Figure 5.20: Orbits of system (5.2.8) for Case A

  Fig. 5.

Figure 5 . 21 :

 521 Figure 5.21: Orbits of system (5.2.8) for Case B

Figure 5 .

 5 22 (or 5.23) shows the time evolution of x(t) and y(t) corresponding to Case A (or B), respectively.

Figure 5 .Fig. 7 .

 57 Figure 5.22: (a) Time evolution of x(t) for Case A. (b) Time evolution of y(t) for Case A 956 Y. Takeuchi et al. / J. Math. Anal. Appl. 323 (2006) 938-957

(5. 1 )Fig. 7 .

 17 Fig. 7. (a) Time evolution of x(t) for Case B. (b) Time evolution of y(t) for Case B.

1 )

 1 Case A corresponds to a(1) = 2, b(1) = 2, c(1) = 3, d(1) = 2 and a(2) = 3, b(2) = 3, c(2) = 6, d(2) = 4 with the initial condition (2, 1.5). In this case, two systems have the rest

Figure 5 .

 5 Figure 5.23: (a) Time evolution of x(t) for Case B. (b) Time evolution of y(t) for Case B

  x

			dx d⌧ 2	= (m	1 x	K	1 11	a 12	y K 1 11	◆
	and r	21 represent the growth rates of IGprey and IGpredator in patch 1. K	11 and
	K 21 are the carrying capacities in the competition patch of IGprey and IGpredator,
	respectively. a	12 and a 21 represent the competition coe cients showing the e↵ect of
	IGpredator on IGprey and of IGprey on IGpredator. r	12 and K 12

Table 2 .

 2 1: Global outcome of the aggregated model.

	D F CD AE AF BD				Equilibria and Stability
	+ +		+	P 1 , P				4 < 0
	+ +				P	1 , P	2 , P	3 : unstable, P	4 : stable
	+ +	+	+		P	1 , P	4 : unstable, P	2 , P	3 : stable
	+ +	+		P	1 , P	3 : unstable, P	2 : stable and P	4 < 0
	+				P	1 , P	2 , P	3 : unstable, P	4 : stable
	+	+		P	1 , P

2 : unstable., P 3 : stable and P 3 : unstable, P 2 : stable and P 4 < 0 P 1 , P 2 , P 3 : unstable, P 4 : stable + P 1 : unstable, P 3 : stable, P 2 , P 4 < 0 + + P 3 : stable, P 1 , P 2 , P 4 < 0 Next we study in detail the dynamics of the aggregated model in term of the migration parameters (i.e. X and Y ).

Table 3 .

 3 1: Description of all parameters for the complete model.

	K		total carrying capacity of MPA and fishing area
	h		proportion of MPA, 0 < h < 1
	k		rate of change of fish state from large clusters to small clusters
	k		rate of change of fish state from small clusters to large clusters
	m	L	rate of migration from fishing area to MPA for fish in large clusters
	m	L	rate of migration from MPA to fishing area for fish in large clusters
	m	S	rate of migration from fishing area to MPA for fish in small clusters
	m	S	rate of migration from MPA to fishing area for fish in small clusters
	r		growth rate of fish
	q	S	catchability for fish in small clusters
	q	L	catchability for fish in large clusters
	c			average cost per unit of fishing e↵ort
	p		constant market price

  )). On the other hand, it follows from x, x ⇤ 2

	✏ for all t t	0 +T	1 and from equations
	of (4.1.1) that the derivatives of x consequence P 3 i=1 |x i x ⇤ i | is uniformly continuous on [T i (t), x ⇤ i (t)(i 1) are bounded on [T 2 , 1). By Lemma 4.10 we 2 , 1). As a have lim t!1 P 3 i=1 |x

i

x ⇤ i | = 0, which completes the proof.

  for any solution u 2 X of (4.4.2), we have ||u||  P

	of . Taking B = P P	P 4 i=1 B	3 i=1 B i where B 4 is taken su ciently large such that B i , and clearly, B i (i 1) are independent 4
	3	2
	i=1	j=1

  .4.6)

	1.5		t-x1
	1.26
	1.02
	x1
	0.78
	0.54
	0.3
	0	6	12	18	24	30
			t
	Figure 4.2: Population sizes X	1 with respect to time
	1.6		t-x2
	1.28
	0.96
	x2
	0.64
	0.32
	0
	0	6	12	18	24	30
			t
	Figure 4.3: Population sizes X	2 with respect to time
	1.56		t-x3
	1.32
	1.08
	x3
	0.84
	0.6
	0.36
	0	6	12	18	24	30
			t

  The proof is complete. By definition of in (5.1.7) we have the following corollary.

	5. Population dynamics in random environments
	it follows
	.1.8) t )(N S(t)), t ))I(t)+c(⇠ t )N +c(⇠ Z t 0 c(⇠t)(N S( t))d t. c(⇠t)(N S( t))d (a(⇠ (a(⇠t)N + c(⇠t))I( t)d t + t ))I(t)+c(⇠ On the other hand, from Ṡ(t) = (a(⇠ t )S(t)+c(⇠ t )(N S(t)) S(t) S(0) t 1 t Z t 0 1 t Since lim t!1 S(t) S(0) t = 0, lim sup t!1 ⇣ 1 t Z t 0 (a(⇠t)N + c(⇠t))I( t)d t + 1 t Z t 0 t⌘  0. Hence, lim inf t!1 1 t Z t 0 (a max N + c max )I( t)d t lim inf t!1 1 t Z t 0 (a(⇠t)N + c(⇠t))I( t)d t lim inf t!1 1 t Z t 0 c(⇠t)(N S( t))d t lim inf t!1 1 t Z t 0 c min (N S( t))d t. (5.1.9) Combining (5.1.8) and (5.1.9), we obtain lim inf t!1 1 t Z t 0 I( t)d t lim inf t!1 1 t Z t 0 c min a max N + c max (N S( t))d t c min (a max N + c max )a max > 0. This inequality implies that there exists 1 > 0 such that lim sup t!1 I(t) > 1 . b) From the inequality İ(t) I(t) = a(⇠ t )S(t) b(⇠ t )  a(⇠ t )N b(⇠ t ), we have lim sup t!1 ln I(t) ln I(0) t  lim sup t!1 1 t Z t 0 (a(⇠t)S( t) b(⇠t))d t  < 0, which implies that lim t!1 I(t) = 0. On the other hand, Ṡ(t) = a(⇠ t )S(t)I(t)+c(⇠ t )(N S(t) I(t)) a max NI(t)+c min (N S(t) I(t)). Thus, S(t) Z t 0 e c min (t t) ( a max N + c min )I( t)d t + S(0)e c min t + c min N Z t 0 e c min (t t) d t. We see that lim t!1 R t 0 e c min (t t) d t = 1 c min . Further, by paying attention that lim t!1 I(t) = 0 we also obtain lim t!1 Z t 0 e c min (t t) ( a max N + c min )I( t)d t = 0. Hence, lim inf t!1 S(t) N . Combining S(t)  N for all t > 0 gets lim t!1 S(t) = a(+) N. Corollary 5.1.2. If b(+) N then lim t!1 I(t) = 0 and lim t!1 S(t) = N .

  compact and the family {U si : (s, i) 2 J} is an open covering of J, by Heine-Borel lemma, there is a finite subfamily, namely {U s i i i , i = 1, 2, ..., n}, which covers J. Let T

	consequence, U is an open set. Moreover, for every (s, i) 2 U , there exists a (t ⇤ 1 , t ⇤ 2 ) 2 (0, d 1 ) ⇥ (0, e 1 ) such that (s
	U	1 = max ⇤ ) for any t T ⇤ , i + 2 (s + 1 .	1in {T	s i i i }. We see that if (s, i) 2 J then ⇡ + t (s, i) 2
	Lemma 5.1.9. There is a compact set K 2 int R 2 + such that, with probability 1, there are infinitely many k = k(!) 2 N satisfying (S 2k+1 , I 2k+1 ) 2 K.

Spatial heterogeneity, fast migration and coexistence of IGP dynamics patch, hence IGpredator can also survive globally.The current contribution is the first attempt to investigate the e↵ects of heterogeneous environment on IGP dynamics. In this study, we do not take into account density-dependent migration, di↵erent predation types such as Holling type II, III and so on. It would be also interesting to consider these factors in the near future.

E↵ect of Small Versus Large Clusters of Fish School

Remerciements

Ce travail a été e↵ectué au sein de 'ũ Du', Professeur à l

Combining (4.4.5) and (4.4.6) gives deg(JQN, ⌦ \ KerL, 0) = 1 6 = 0. By now we have proved that ⌦ verifies all requirements of Lemma 4.12, then Lu = Nu has at least one solution in DomL \ ⌦, i.e., (4.4.1) has at least one ! -periodic solution u ⇤ in DomL \ ⌦. Set x ⇤ i = exp{u ⇤ i }(i 1), then x ⇤ is an ! -periodic solution of system (4.1.1) with strictly positive components. We complete the proof.

Corollary 4.14. If the ! -periodic solution x ⇤ in Theorems 4.13 satisfies the assumptions in Theorems 4.11, then x ⇤ is globally asymptotically stable.

Proof. The proof of this corollary are derived directly from Theorems 4.11 and 4.13.

Numerical examples and conclusion

In this section, we present some numerical examples. At the first example, we consider the case a i > 0, m 0 i > 0 (i 1) then Hypothesis 4.6 holds and system (4.1.1) has invariant set. Figure 4.1 is the orbit of solution with initial value x 0 = (1.3, 1.4, 1.1), it seems to be very chaotic but it is permanent. According to theorem 4.11, it is not only permanent but also globally asymptotically stable. In spite of di↵erent initial value, In next example, Theorem 4.9 will be illustrated by system (4.1.1) with coecients a

3 sin(0.4t), ↵ = (1.8 cos(5.7t))/3, = 1 0.2 cos(0.2⇡t), = 3.4 1.6 sin(1.8t) and initial condition x 0 = (1.3, 2.1, 2.4). Since M 0 3 < 0 then the density of species X 3 goes extinct (see 5.1.4, we study stability in distribution of the system. In the last subsection 5.1.5, some simulation results illustrate the behavior of the SIRS model under telegraph noise. The conclusion presents a summary of the results and some perspectives of the work.

Preliminary

Let us consider a continuous-time Markov process ⇠ t , t 2 R + , defined on the probability space (⌦, F, P), with values in the set of two elements, say E = {+, }.

Suppose that (⇠ t ) has the transition intensities +

↵ ! and ! + with ↵ > 0, > 0. The process (⇠ t ), say "telegraph noise", has a unique stationary distribution

The trajectory of (⇠ t ) is piecewise constant, cadlag functions. Suppose that [0,1) = 1 for t 0 (= 0 for t < 0). In this section, we consider the Kernack-MacKendrick model under the telegraph noise ⇠ t of the form:

Population dynamics in random environments

Proof. In the case b( ) a( ) < N, we can choose K ⌘ G, that was established in Proposition 5.1.6 and see that (S 2k+1 , I 2k+1 ) 2 K for every k > k 0 . Suppose that b(+) a(+) < N < b( ) a( ) . With 1 is shown in Proposition 5.1.1, by a similar way to the construction of the set G in the proof of Proposition 5.1.6, we construct a curve + for the system (5.1.4), with the initial point (S min , 1 ) and the end point ( b(+) a(+) , I + min ); then define the subdomain K of quadrangle ABCD consisting of every (s, i) 2 ABCD lying above the curve + if s  b(+) a(+) and lying above the line i = I + min if b(+) a(+)  s  N . It is seen that K is an invariant set for the system (5.1.4)

Proof. This result is a consequence of Lemma 5.1.8 and the continuous dependence of the solutions on the initial values.

Lemma 5.1.11. For any 6 > 0 and (s, i) 2 H " , there exists a 5 > 0 such that

Proof. The conclusion follows straightforward from the continuous dependence of the solutions on the initial values.

To describe the pathwise dynamic behavior of the solutions of the system (5.1.

where %(k) = ( 1) k .

Theorem 5.1.12. If > 0 then for almost all !, the closure of is a subset of ⌦(s 0 , i 0 , !).

Proof. With K is mentioned in Lemma 5.1.9, we construct a sequence

Similarly,

Continuing this way we obtain

Hence,

(5.1.12)

. By virtue of Lemma 5.1.10, for any 4 > 0, there are

We now construct the sequence of stopping times

.. By the same argument as above we obtain P{! :

s, ī) for infinitely many times and (s, ī) 2 ⌦(s 0 , i 0 , !) a.s. Thus, {⇡ t (s + ⇤ , i + ⇤ ) : t 0} ⇢ ⌦(s 0 , i 0 , !). Based on Lemmas 5.1.10, 5.1.11 and using a similar argument, we see that b) If (5.1.13) is not satisfied then, with probability 1, the = ⌦(s 0 , i 0 , !). Moreover, absorbs all positive solutions in the sense that for any initial value (s

Proof. a) It is easy to see that the systems (5.1.4) and (5.1.5) have the same equilibrium, (s

if and only if the condition (5.1.13) is satisfied. Let " > 0 be arbitrary. Since (s ⇤ , i ⇤ ) is globally asymptotically stable, there is a neighborhood V " ⇢ U ✏ (s ⇤ , i ⇤ ), invariant under the system (5.1.4) (see The Stable Manifold Theorem, [115, pp 107]). Under the condition (5.1.13), the vector fields of both systems (5.1.4) and (5.1.5) have the same direction at every point (s, i). As a result, V " is also invariant under the system (5.1.5), which implies that V " is invariant under the system (5.1.3). By Theorem 5.1.12, (s ⇤ , i ⇤ ) 2 ⌦(s 0 , i 0 , !) for almost all !. Therefore, T

V" = inf t > 0 : (S(t),

. This property says that (S(t), I(t)) converges to (s ⇤ , i ⇤ ) with probability 1 if S(0) > 0, I(0) > 0.

b) We will show that if there exists a t 0 > 0 such that the point (s 0 , ī0 ) = ⇡ t 0 (s + ⇤ , i + ⇤ ) satisfies the following condition det ✓ Ṡ+ (s 0 , ī0 ) Ṡ (s 0 , ī0 ) İ+ (s 0 , ī0 ) İ (s 0 , ī0 )

then, with probability 1, the closure of coincides ⌦(s 0 , i 0 , !) and absorbs all positive solutions. Indeed, let (s 0 , ī0 ) = ⇡ t 0 (s + ⇤ , i + ⇤ ) be a point in intR 2 + satisfying the condition (5. 1.14). By the existence and continuous dependence on the initial values of the solutions, there exist two numbers d > 0 and e > 0 such that the function '(t

) is defined and continuously di↵erentiable in ( d, d) ⇥ ( e, e). We note that

Therefore, by the Inverse Function Theorem, there exist 0 < d 1 < d, 0 < e 1 < e such that '(t 1 , t 2 ) is a di↵eomorphism between V = (0, d 1 ) ⇥ (0, e 1 ) and U = '(V ). As a

Predator-prey system with carrying capacity

The following subsection presents main results of paper [START_REF] Auger | Evolution of Lotka-Volterra predator-prey systems under telegraph noise[END_REF]. We consider Lotka-Volterra predator-prey system (5.2.1) in case b(⇠ t ) 6 = 0

where g : E ! (0, 1) for g = a, b, c, d, e. The noise (⇠ t ) intervenes virtually into the equation (5.2.2), it makes a switching between the deterministic system

and the deterministic one

(5.2.4)

where g i = g(i) for i = 1, 2 and g = a, b, c, d, e. It is well-known that the systems (5.2.3) and (5.2.4) respectively have the rest points

and their global dynamics depend on these rest points (see [START_REF] Bazykin | Nonlinear Dynamics of Interacting Populations[END_REF]). Concretely, if y ⇤ i > 0 then the i th -rest point is asymptotically stable, i.e., lim

The behavior of two boundary equations is easy to be studied. In the case where the prey is absent, the quantity v(t) of predator at the time t satisfies the equation v = d(⇠ t )v. Thus, v(t) decreases exponentially to 0. Similarly, without the predator, the quantity u(t) of the prey at the time t satisfies the logistic equation

Denote by (x(t, x 0 , y 0 ), y(t, x 0 , y 0 )) the solution of (5.2.2) satisfying the initial condition (x(0, x 0 , y 0 ), y(0, x 0 , y 0 )) = (x 0 , y 0 ). For the sake of simplification, we write (x(t), y(t)) for (x(t, x 0 , y 0 ), y(t, x 0 , y 0 )) The following propositions will give the conditions for the persistence of system (5.2.2) Let !(x 0 , y 0 ) be the !-limit set of the solution (x(t, x 0 , y 0 ), y(t, x 0 , y 0 )) of the system (5.2.2). We have Theorem 5.2.4.

a) Suppose that > 0

)) of the system (5.2.3) and 2 of the solution (x 2 (t, x ⇤ 1 , y ⇤ 1 ), y 2 (t, x ⇤ 1 , y ⇤ 1 )) of system (5.2.4) are subsets of !(x 0 , y 0 ). Moreover, 2. Any positive orbit ¯ 2 of the solution (x 2 (t, x, ȳ), y 2 (t, x, ȳ)) of the system (5.2.4), starting in a point (x, ȳ) 2

1 at t = 0, is a subset of !(x 0 , y 0 ). Similarly, any positive orbit e 1 of the solution (x 1 (t, e

x, e y), y 1 (t, e

x, e y)) of the system (5.2.3), starting in a point (e x, e y) 2 2 at t = 0, is a subset of !(x 0 , y 0 ).

If y ⇤

1 > 0 and y ⇤ 2 < 0, then we have a similar result as in 1.; provided that

1 is replaced by closure of e 1 -says b 1 . Concurrently, u ⇢ !(x 0 , y 0 ) with u to be the !-limit set of (u(t), 0), here u(t) is the solution of the system (5.2.6). b) If < 0, y ⇤ 1 < 0 and y ⇤ 2 < 0 then u ⌘ !(x 0 , y 0 ). Theorem 5.2.4 describes the subsets of !-limit set of solution for each case. It is shown that the !-limit sets include every orbit starting at a point on the curves linking two rest points of the subsystems.

The last theorem proves the existence of invariant measures Theorem 5.2.5. If y ⇤ 1 > 0, y ⇤ 2 > 0, > 0, the system (5.2.2) is permanent. Moreover, for the Markov process (x(t), y(t), ⇠ t ) t 0 , there exists a stationary distribution. ). Thus the !-limit set of all solutions starting in int R 2 + is the segment [(4.7, 0); (2.9, 0)] (see the theorem 5.2.4). We sketch the oscillations of x(t) and y(t) in these cases in the figure 5.18 and 5.19.

Predator-prey system with the absence of carrying capacity

This subsection is summarization of paper [START_REF] Takeuchi | Evolution of predator-prey systems described by a Lotka -Volterra equation under random environment[END_REF]. We study Lotka-Volterra predator-prey system (5.2.1) in case b(⇠ t ) = 0

where g : E ! R + \ {0} for g = a, c, d, e. In the case where the noise (⇠ t ) intervenes virtually into Equation (5.2.8), it makes a switching between the deterministic system ( ẋ1 (t) = x 1 (t)(a

(5.2.9) and a deterministic one

(5.2.10) [START_REF] Murray | Mathematical Biology[END_REF] and [START_REF] Takeuchi | Global Dynamical Properties of Lotka-Volterra Systems[END_REF] show that system (5.2.9) (resp. (5.2.10)) has a rest point

2 )) and its solutions are periodic orbits. Theorem 5.2.6. Suppose that (5.2.9) and (5.

or,

x(t, x 0 , y 0 ) = 0, (5.2.12)

(5.2.13) Theorem 5.2.6 and 5.2.7 mean that if the rest points of two systems do not coincide, all trajectories of the system perturbed by telegraph noise always leave from any compact set in intR 2 + . In case two systems have the rest point in common, either the trajectory of a random predator-prey system converges to the common rest point or it leaves from any rectangle in intR 2 + . These properties imply that such a system is neither permanent nor dissipative. Lotka-Volterra competition system in a seasonally fluctuating environment [START_REF] Auger | Evolution of Lotka-Volterra predator-prey systems under telegraph noise[END_REF] are studied.

The stochastic change of environment gives the similar effect on population dynamics with seasonality [START_REF] Allen | A comparison of three di↵erent stochastic population models with regard to persistence time[END_REF]. Most of such models are formulated by deterministic dynamics, but some kind of stochasticity reflecting complexity of biological or environmental factors should be introduced in population dynamics. Slatkin [START_REF] Auger | Complex ecological models with simple dynamics: From individuals to populations[END_REF] introduced and analyzed a class of general models of single population which grows under the telegraph noise exactly same in this paper. Du et al. [3,[START_REF] Amarasekare | Productivity, dispersal and the coexistence of intraguild predators and prey[END_REF] analyzed two-species Lotka-Volterra competition systems under telegraph noises. In this paper we obtained the results, stated in Theorem 4.5, for two-species Lotka-Volterra predator-prey systems under telegraph noises. We believe the necessity to collect the results for various types of two-species dynamical systems including the asymptotically stable Lotka-Volterra predatorprey systems, and to proceed to obtain the general conditions for the behaviors of the systems. In view of practice, when the amount of a species is smaller than a threshold, we consider this species disappears in our system. Theorem 4.5 tells us that although the environment condition changes constantly (since the Markov switching process is stationary), species may vanish from Lotka-Volterra competition system in a seasonally fluctuating environment [START_REF] Auger | Evolution of Lotka-Volterra predator-prey systems under telegraph noise[END_REF] are studied. The stochastic change of environment gives the similar effect on population dynamics with seasonality [START_REF] Allen | A comparison of three di↵erent stochastic population models with regard to persistence time[END_REF]. Most of such models are formulated by deterministic dynamics, but some kind of stochasticity reflecting complexity of biological or environmental factors should be introduced in population dynamics. Slatkin [START_REF] Auger | Complex ecological models with simple dynamics: From individuals to populations[END_REF] introduced and analyzed a class of general models of single population which grows under the telegraph noise exactly same in this paper. Du et al. [3,[START_REF] Amarasekare | Productivity, dispersal and the coexistence of intraguild predators and prey[END_REF] analyzed two-species Lotka-Volterra competition systems under telegraph noises. In this paper we obtained the results, stated in Theorem 4.5, for two-species Lotka-Volterra predator-prey systems under telegraph noises. We believe the necessity to collect the results for various types of two-species dynamical systems including the asymptotically stable Lotka-Volterra predatorprey systems, and to proceed to obtain the general conditions for the behaviors of the systems.

In view of practice, when the amount of a species is smaller than a threshold, we consider this species disappears in our system. Theorem 4.5 tells us that although the environment condition changes constantly (since the Markov switching process is stationary), species may vanish from