Jérôme Coville
email: jerome.coville@avignon.inra.fr

Frédéric Fabre
email: frederic.fabre@avignon.inra.fr

CONVERGENCE TO THE EQUILIBRIUM IN A LOTKA-VOLTERRA ODE COMPETITION SYSTEM WITH MUTATIONS

Keywords: Demo-genetic dynamics, Lokta-Volterra competition system with mutation, equilibria, Relative entropy, Global stability. 2010 Mathematics Subject Classification: 34A34, 34A40, 34D05, 34D23, 92D15, 92D25

In this paper we are investigating the long time behaviour of the solution of a mutation competition model of Lotka-Volterra's type. Our main motivation comes from the analysis of the Lotka-Volterra's competition system with mutation which simulates the demo-genetic dynamics of diverse virus in their host :

In a first part we analyse the case where the competition terms Ψ i are independent of the virus type i. In this situation and under some rather general assumptions on the functions Ψ i , the coefficients r i and the mutation matrix µ ij we prove the existence of a unique positive globally stable stationary solution i.e. the solution attracts the trajectory initiated from any nonnegative initial datum. Moreover the unique steady state v is strictly positive in the sense that vi > 0 for all i. These results are in sharp contrast with the behaviour of Lotka-Volterra without mutation term where it is known that multiple non negative stationary solutions exist and an exclusion principle occurs (i.e For all i = i 0 , vi = 0 and vi 0 > 0 ). Then we explore a typical example that has been proposed to explain some experimental data. For such particular models we characterise the speed of convergence to the equilibrium. In a second part, under some additional assumption, we prove the existence of a positive steady state for the full system and we analyse the long term dynamics. The proofs mainly rely on the construction of a relative entropy which plays the role of a Lyapunov functional.

INTRODUCTION

In this paper we are investigating the long time behaviour of the solution of some models that have been recently used in epidemiology. Our analysis focuses on a Lotka-Volterra competition system with mutation which basically simulates the demo-genetic dynamic of a genetically diverse virus population in its hosts, highliting the numerous links existing between ecological and within-host infection dyanmics. Such type of model has been proposed to explain some experimental data e.g. [START_REF] Cuevas | Evolution of rna virus in spatially structured heterogeneous environments[END_REF][START_REF] Fabre | Modelling the evolutionary dynamics of viruses within their hosts: A case study using high-throughput sequencing[END_REF][START_REF] Miralles | Multiple infection dynamics has pronounced effects on the fitness of rna viruses[END_REF]. To be more specific the demo-genetic dynamic is modelled by N ordinary differential equations which simulates at host scale the dynamics of v i (t) the number of virus particles of genotype i at time t :

(1.1)

dv i (t) dt = v i r i - 1 K Ψ i (v) + N j=1 µ ij (v j -v i )
where r i , K and µ ij represent respectively the growth rate for each genotype, the total carrying capacity of the host and a nonnegative matrix characterising the rate of mutation from a virus of genotype i to a virus of genotype j. For each i, Ψ i (v) : R N → R is a locally Lipschitz application describing the intensity of the interaction between a virus of genotype i with all its competitors. The mutation term of the system can also be interpreted as a dispersal term. Indeed, mutation naturally corresponds to dispersal into the discrete space of genotype. But the mutation term can also handle dispersal between physical patches. With this in mind, the above system of equation can also be used to model the demo-genetic dynamic of a diverse virus population in structured hosts, each host tissue types being virus "habitats" connected to each others by dispersal via fluid flow (e.g. [START_REF] Orive | Viral infection in internally structured hosts. i. conditions for persistent infection[END_REF]). Ways of derive results for this interpretation are discussed in the biological comments subsection.

In what follows we will always make the following assumptions on r i , Ψ i and

µ ij           
For all i, r i > 0,

The matrix (µ ij ) is nonnegative symmetric and irreducible Ψ i (v) ∈ C 0,1 loc (R N , R), Ψ i (0) = 0 Ψ i is monotone increasing with respect to the natural order of R N (1.2) Furthermore we will assume that for all i there exists positive constants R i , k i , c i with k i > 0 so that the function Ψ i satisfies for all v ∈ R N,+ \ Q Ri (0),

c i   N j=1 v j   ki ≤ Ψ i (v), (1.3)
where Q Ri (0) denotes the ball of radius R i and centred at 0 associated to the l 1 norm.

A typical example of model satisfying our assumption is given by

dv i (t) dt = v i   r i - 1 K 4 j=1 β ij v j   + M v with M :=     (1 -µ) 2 -1 µ(1 -µ) µ(1 -µ) µ 2 µ(1 -µ) (1 -µ) 2 -1 µ 2 µ(1 -µ) µ(1 -µ) µ 2 (1 -µ) 2 -1 µ(1 -µ) µ 2 µ(1 -µ) µ(1 -µ) (1 -µ) 2 -1     ,
where µ is a parameter giving the point mutation rate per replication cycle and per nucleotide. This mutation matrix corresponds to a viral population composed of 4 variants differing only by one or two substitutions involved in adaptative proprieties (e.g. pathogenicity). The interactions terms Ψ i := 4 j=1 β ij v j can handle a wide range of possible inter-specific (inter-variants) competition rates between any pairs of virus variants.

This particular structure has been used recently to model the adaptation of plant virus to resistance genes [START_REF] Fabre | Key determinants of resistance durability to plant viruses: Insights from a model linking within-and between-host dynamics[END_REF][START_REF] Fabre | Modelling the evolutionary dynamics of viruses within their hosts: A case study using high-throughput sequencing[END_REF]. This particular form of competition is commonly used to model virus evolution, [START_REF] Cuevas | Evolution of rna virus in spatially structured heterogeneous environments[END_REF][START_REF] Lafforgue | Differences in accumulation and virulence determine the outcome of competition during tobacco etch virus coinfection[END_REF][START_REF] Miralles | Multiple infection dynamics has pronounced effects on the fitness of rna viruses[END_REF]. Without the mutation's matrix (µ ij ), the system (1.1) reduces to a classical competition system in the sense of Hirsch [START_REF] Hirsch | Systems of differential equations which are competitive or cooperative: I. limit sets[END_REF][START_REF] Hirsch | Systems of differential equations that are competitive or cooperative ii: Convergence almost everywhere[END_REF][START_REF] Hirsch | Systems of differential equations which are competitive or cooperative: Iii. competitive species[END_REF] (1.4)

dv i (t) dt = v i r i - 1 K Ψ i (v) .
Such systems has been intensively studied and many aspects are now well understood see for example [START_REF] Akin | The geometry of population genetics[END_REF][START_REF] Bürger | The mathematical theory of selection, recombination, and mutation[END_REF][START_REF] Bürger | Mutation load and mutation -selection -balance in quantitative genetic traits[END_REF][START_REF] Champagnat | Convergence to equilibrium in competitive lotka-volterra and chemostat systems[END_REF][START_REF] Crow | An introduction to population genetics theory[END_REF][START_REF] O; Diekmann | The dynamics of adaptation: An illuminating example and a hamilton-jacobi approach[END_REF][START_REF] Hadeler | Stable polymorphisms in a selection model with mutation[END_REF][START_REF] Hirsch | Systems of differential equations which are competitive or cooperative: I. limit sets[END_REF][START_REF] Hirsch | Systems of differential equations that are competitive or cooperative ii: Convergence almost everywhere[END_REF][START_REF] Hirsch | Systems of differential equations which are competitive or cooperative: Iii. competitive species[END_REF][START_REF] Hofbauer | The selection mutation equation[END_REF][START_REF] Hofbauer | Coexistence for systems governed by difference equations of lotka-volterra type[END_REF][START_REF] Jabin | On selection dynamics for competitive interactions[END_REF][START_REF] Li | The criteria for globally stable equilibrium in n-dimensional lotka-volterra systems[END_REF][START_REF] Perthame | Transport equations in biology[END_REF] and references therein. In particular, the existence of stationary solution and the asymptotic behaviour of the solution has been obtained in [START_REF] Champagnat | The evolutionary limit for models of populations interacting competitively via several resources[END_REF][START_REF] Champagnat | Convergence to equilibrium in competitive lotka-volterra and chemostat systems[END_REF][START_REF] Jabin | On selection dynamics for competitive interactions[END_REF][START_REF] Perthame | Transport equations in biology[END_REF]. Those systems are characterised by the existence of at least as many equilibrium states that the number of competiting species (or genotypes) involved. In addition, the dynamics exhibit a competitive exclusion principle which state that the fittest species initially present will overcome all the other ones.

When the mutations's matrix (µ ij ) is non trivial, the system (1.1) does not fall into Hirsch's definition of competitive system and less results are known. If for reasonably smooth interaction functions, the existence of solution of (1.1) defined for all times is not an issue, the existence of a non trivial stationary solution and the analysis of the asymptotic behaviour are challenging questions.

Most of the known results concerns either particular interaction functions Ψ i for which the existence of steady states and their local stability are investigated [START_REF] Bates | Structure of principal eigenvectors and genetic diversity[END_REF][START_REF] Crow | An introduction to population genetics theory[END_REF][START_REF] Hadeler | Stable polymorphisms in a selection model with mutation[END_REF][START_REF] Hofbauer | The selection mutation equation[END_REF] or for some ODE's systems (1.1) where the mutation matrix µ ij is considered as a small parameter. In the latter the system (1.1) is seen as a perturbation of (1.4) and analysed using perturbative techniques [START_REF] Bürger | Mutation load and mutation -selection -balance in quantitative genetic traits[END_REF][START_REF] Calsina | Stationary solutions of a selection mutation model: The pure mutation case[END_REF][START_REF] Calsina | Asymptotic stability of equilibria of selection-mutation equations[END_REF]. Recently, there has been an intense activity on continuous trait version of (1.1) where some of the techniques can be used to obtain the existence of locally stable steady states for (1.1) see for example [START_REF] Barles | Dirac concentrations in lotka-volterra parabolic pdes[END_REF][START_REF] Calsina | Stationary solutions of a selection mutation model: The pure mutation case[END_REF][START_REF] Calsina | Asymptotic stability of equilibria of selection-mutation equations[END_REF][START_REF] Calsina | Asymptotics of steady states of a selection mutation equation for small mutation rate[END_REF][START_REF] Canizo | Measure solutions for some models in population dynamics[END_REF][START_REF] Carrillo | Adaptive dynamics via hamilton-jacobi approach and entropy methods for a juvenile-adult model[END_REF][START_REF] Champagnat | The evolutionary limit for models of populations interacting competitively via several resources[END_REF][START_REF] Champagnat | Convergence to equilibrium in competitive lotka-volterra and chemostat systems[END_REF][START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF][START_REF] O; Diekmann | The dynamics of adaptation: An illuminating example and a hamilton-jacobi approach[END_REF][START_REF] Jabin | On selection dynamics for competitive interactions[END_REF][START_REF] Perthame | Transport equations in biology[END_REF][START_REF] Raoul | Long time evolution of populations under selection and vanishing mutations[END_REF][START_REF] Raoul | Local stability of evolutionary attractors for continuous structured populations[END_REF]. However, to our knowledge there is no results on global stability of the steady states for systems like(1.1) neither for its continuous trait version.

In this direction, our first results concern the systems (1.1) where the competition terms Ψ i are independent of i. A typical case is Ψ i (v) = N j=1 v j which corresponds to a situation where inter-and intra-species (genotypes) competitions are equals (i.e. blind and uniform competition between variants, see Lafforgue et al. [START_REF] Lafforgue | Differences in accumulation and virulence determine the outcome of competition during tobacco etch virus coinfection[END_REF]). In this situation one can show that there exists a unique positive stationary solution of (1.1) v ∈ R N , which attracts all the trajectories initiated from any nonnegative and non zero initial data. Namely we show that Theorem 1.1. Assume that the interaction Ψ i is independent of i and is satisfying the assumptions (1.2)-(1.3), then there exists a unique positive stationary solution v to (1.1). Moreover for any nonnegative initial datum v(0) not identically zero, the corresponding solution v(t) of (1.1) converges to v.

A case of particular interest is when the interactions Ψ i take the following form Ψ i (v) = N j=1 r j v j . This particular structure of interaction was initially introduced on a theoretical ground by Sole et al. [START_REF] Sole | Red queen dynamics, competition and critical points in a model of rna virus quasispecies[END_REF] to model the competition between viruses. Recently, this form of interactions has been used to explain experimental results of virus evolution [START_REF] Cuevas | Evolution of rna virus in spatially structured heterogeneous environments[END_REF][START_REF] Fabre | Modelling the evolutionary dynamics of viruses within their hosts: A case study using high-throughput sequencing[END_REF][START_REF] Miralles | Multiple infection dynamics has pronounced effects on the fitness of rna viruses[END_REF]. Sole et al. showed that Eigen's model of molecular quasi-species [START_REF] Eigen | Molecular quasi-species[END_REF] was to a large extent equivalent to the Lotka-Volterra competition equations under this assumption.

For this type of interaction, we can rewrite the system (1.1) as follows

(1.5) dv i (t) dt = v i   r i - 1 K N j=1 r j v j   + N j=1 µ ij (v j -v i ).
For (1.5) besides the asymptotic behaviour of the solution obtained as an application of Theorem 1.1 we can precise the speed of convergence to the equilibrium. Furthermore we can give an estimate of the time to reach near the equilibrium. In epidemiology, this type of informations is of practical use for building tractable nested models. Nested models are a class of model which explicitly links the relationships between processes at different levels of biological organization.They are often used to study the pathogen evolution by linking the disease dynamic of within-and between-host, see Mideo et al. [START_REF] Mideo | Linking within-and between-host dynamics in the evolutionary epidemiology of infectious diseases[END_REF]. Their formalisation becomes more simple when the within-host pathogen dynamic is faster than the between-hosts dynamic. Indeed, in such cases, using for example slow-fast reduction techniques commonly used in ecology [START_REF] Auger | Aggregation methods in dynamical systems and applications in population and community dynamics[END_REF], the within-host dynamic can be approximated by its equilibrium, see for example [START_REF] Fabre | Durable strategies to deploy plant resistance in agricultural landscapes[END_REF].

More precisely, we show that Theorem 1.2. Assume that the interactions Ψ i take the form Ψ i (v) = N j=1 r j v j then for any nonnegative initial datum v(0) not identically zero, the solution v(t) of (1.5) converges exponentially fast to its unique equilibrium. That is to say there exists two positive constants

C 1 and C 2 so that v -v ∞ ≤ C 1 e -C2t .
Next, we analyse the situation where the functions Ψ i are not reduced to a single function. From a biological point of view, this situation could appears as, often, genotypes (species) exhibit particular association patterns (see for example [START_REF] Takahashi | Analysis of the spatial distribution of identical and two distinct virus populations differently labeled with cyan and yellow fluorescent proteins in coinfected plants[END_REF] in the case of plant viruses). A way to model such phenomena is to take Ψ i of the form Ψ i (v) = N j=1 α i,j r j v j where α i,j are crowing index [START_REF] Lloyd | Mean crowding[END_REF][START_REF] Hartley | A general framework for the aggregation model of coexistence[END_REF]. This index is equal to 1 when the 2 species are distributed independently and ranges from 0 (complete avoidance) to a large constant (near overlap) according to patterns of species association.

In this general context, our first result concerns the existence of a positive stationary solution for the system (1.1) assuming we have the following extra condition

∀ i N j=1 µ ij ≤ r i 2 . (1.6)
This condition makes senses in our application framework since mutation rates (expressed as substitutions per nucleotide per cell infection) range from 10 -8 to 10 -6 for DNA viruses and from 10 -6 to 10 -3 for RNA viruses [START_REF] Sanjuan | Viral mutation rates[END_REF] whereas growth rates r i are of the order of the unit or higher. For example, the overall growth rate of the RNA virus VSV was estimated to 0.6 virus/hour [START_REF] Cuevas | Following the very initial growth of biological rna viral clones[END_REF]. Each cell infected by a single VSV particle produces from 50 to 8000 progeny virus particle/ cell infection [START_REF] Zhu | Growth of an rna virus in single cells reveals a broad fitness distribution[END_REF].

Under the extra assumption (1.6) we prove Theorem 1.3. Assume that r i , Ψ i and µ ij satisfy the assumptions (1.2)- (1.3). Assume further that (1.6) holds . Then there exists a positive stationary solution v to the system (1.1).

Lastly, we obtain the convergence to the equilibria for some general interactions Ψ i . Namely, we show that Theorem 1.4. Assume that r i , Ψ i and µ ij satisfy (1.2)- (1.3). Assume further that (1.6) holds and

Ψ i (v) = α(v) + ǫψ i (v) with ψ i ∈ C 1 loc uniformly bounded and α ∈ C 1 loc satisfying (1.2)-(1.3) is so that ∀ x ∈ R N , ∇α(x) > 0.
Then there exists ǫ 0 so that, for all ǫ ≤ ǫ 0 , the positive stationary solution vǫ of (1.1) attracts all the possible trajectories initiated from any non zero and nonnegative initial data.

Note that C 1 loc perturbation of the particular interaction function α(v) = N j=1 r j v j satisfied the assumptions of the above Theorem.

General remarks.

Before going to the proofs, we want to make some general remarks and comments.

Mathematical comments : First as remarked in [START_REF] Champagnat | Convergence to equilibrium in competitive lotka-volterra and chemostat systems[END_REF], we can interpret the system of equations (1.5) as a discrete version of the continuous model

(1.7) ∂v(t, x) ∂t = v(t, x) r(x) - 1 K R n r(y)v(t, dy) + R n µ(x, y)v(t, dy) -µ(x)v(t, x)
by posing for each sub population corresponding to a typical trait

x i ∈ R d , v(t, x) = N i v i (t)δ xi , r(x i ) = r i , µ(x i , x j ) = µ ij and µ(x) = R n µ(x, y)dy.
Recently there have been a lot of works dealing with (1.7) and generalisation of it, see for example [START_REF] Barles | Dirac concentrations in lotka-volterra parabolic pdes[END_REF][START_REF] Bürger | The mathematical theory of selection, recombination, and mutation[END_REF][START_REF] Calsina | Stationary solutions of a selection mutation model: The pure mutation case[END_REF][START_REF] Calsina | Asymptotic stability of equilibria of selection-mutation equations[END_REF][START_REF] Calsina | Asymptotics of steady states of a selection mutation equation for small mutation rate[END_REF][START_REF] Canizo | Measure solutions for some models in population dynamics[END_REF][START_REF] Carrillo | Adaptive dynamics via hamilton-jacobi approach and entropy methods for a juvenile-adult model[END_REF][START_REF] Champagnat | Convergence to equilibrium in competitive lotka-volterra and chemostat systems[END_REF][START_REF] Champagnat | The evolutionary limit for models of populations interacting competitively via several resources[END_REF][START_REF] Desvillettes | On selection dynamics for continuous structured populations[END_REF][START_REF] O; Diekmann | The dynamics of adaptation: An illuminating example and a hamilton-jacobi approach[END_REF][START_REF] Jabin | On selection dynamics for competitive interactions[END_REF][START_REF] Perthame | Transport equations in biology[END_REF][START_REF] Raoul | Long time evolution of populations under selection and vanishing mutations[END_REF][START_REF] Raoul | Local stability of evolutionary attractors for continuous structured populations[END_REF] and references therein. A large part of the analysis are concerning (1.7) in absence of mutation (ie µ ≡ 0) or in the limit µ → 0. In the latter case, much have been done in developing a constrained Hamilton-Jacobi approach to analyse the long time behaviour of positive solution of this type of models see for instance [START_REF] Barles | Dirac concentrations in lotka-volterra parabolic pdes[END_REF][START_REF] O; Diekmann | The dynamics of adaptation: An illuminating example and a hamilton-jacobi approach[END_REF]. Other approaches based on semigroup theory have also been developed to analyse the asymptotic behaviour and local stability of the stationary solution of (1.7) see [START_REF] Calsina | Stationary solutions of a selection mutation model: The pure mutation case[END_REF][START_REF] Calsina | Asymptotic stability of equilibria of selection-mutation equations[END_REF]. Although some of the techniques developed in this two frameworks may be adapted to analyse the system (1.1) most of them fail when we try to prove the global stability of the stationary states. To tackle this difficulty we construct a set of Lyapunov functionals for the solution of (1.1) when the interactions Ψ i are independent of i. Properly used these Lyapunov functionals enable us to analyse the asymptotic behaviour of the solution of (1.1) in this particular situation. From this analysis, we derive some consideration on the asymptotic behaviour of the solution of (1.1) in a general situation. The Lyapunov functionals are constructed in the spirit of the relative entropy introduced for linear parabolic operators in [START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF]. Even if our problem is nonlinear such type of relative entropy can still be constructed. These is an interesting new feature of nonlinear dissipative system.

It is worth noticing that a similar construction can be made for (1.7) giving us access to a simple way of analysing the asymptotic behaviour and the global stability of steady solution of (1.7), see [START_REF] Coville | Convergence to the equilibria on some mutation selection model[END_REF].

Along some of the proofs we notice that the existence of a steady state in Theorem 1.1 and Theorem 1.3 can be generalised to situation where the mutation matrix (µ ij ) is not symmetric. Indeed, when the interactions Ψ i are independent of i (Theorem 1.1), the construction of a unique stationary solution relies only on the Perron-Frobenius Theorem which holds true for non symmetric matrices. However, for the general case (Theorem 1.3) to obtain the existence of a steady state, we do require that the condition (1.6) is replaced by

∀ i N j=1 µ ij + µ ji 2 ≤ r i 3 . (1.8)
Biological comments : First, we emphasize the biological interpretation of our result and particularly the role of the mutation term. Indeed, under biological compatible assumptions concerning the competition ψ i , the mutation matrix (µ ij ) and the growth rate r i we have shown that, in sharp contrast with the classical results known for the Lotka-Volterra system without mutation, the mutation term deeply changes the dynamics of the system. On one hand, the mutation term stabilizes the dynamic of system by reducing the number of equilibrium up to a single equilibrium and on the other hand, the mutation term precludes the competitive exclusion principle to occur.

Second, we emphasize the biological relevance of relaxing some hypothesis regarding (i) the monotony of Ψ i and (ii) the symmetry of the mutation matrix µ ij to study viral demo-genetics dynamics in structured hosts. To illustrate this point, let consider 2 patches p 1 and p 2 , and 2 virus genotypes v 1 and v 2 . The demo-genetics dynamics of the viral population in this system can be modelled by

dw dt = M w + Rw -Ψ(w)w
where w is the vector (v 1,p1 , v 2,p1 , v 1,p2 , v 2,p2 ) and R, M and Ψ(w) are the following matrices R := (r i δ ij ),

M :=     µ 1 -µ 0 0 1 -µ µ 0 0 0 0 µ 1 -µ 0 0 1 -µ µ     +     -d 1 0 d 2 0 0 -d 1 0 d 2 d 1 0 -d 2 0 0 d 1 0 -d 2     , Ψ(w) :=     r 1 v 1,p1 + r 2 v 2,p1 0 0 0 0 r 1 v 1,p1 + r 2 v 2,p1 0 0 0 0 r 1 v 1,p2 + r 2 v 2,p2 0 0 0 0 r 1 v 1,p2 + r 2 v 2,p2     .
In structured hosts some tissue types often act as virus "sources" and others are "sinks", creating an asymmetry in the exchange between patches. In the above example this implies that d 1 > d 2 if the patch p 1 is a "source" and the patch p 2 is a "sink", making the mutation matrix M nonsymmetric. Moreover, in this example since the competition takes place only inside a given patch, we can check that the two monotone interaction functionals involved Ψ 1 (v) := r 1 v 1,p1 + r 2 v 2,p1 and Ψ 2 (v) := r 1 v 1,p2 + r 2 v 2,p2 do not satisfy the monotone properties (1.2).

Such type of structure, inducing asymmetries in the exchanges and some weak interaction functionals, are expected to play a role in constraining or facilitating adaptive evolution of viruses in heterogeneous host environment [START_REF] Orive | Viral infection in internally structured hosts. i. conditions for persistent infection[END_REF]. It is then relevant to study extension of our results in a context of more general assumptions on µ ij and Ψ i .

Organization of the Paper :

In Section 2 we start by establishing some preliminaries results about the system (1.1). We derive the relative entropy identities in 3 Then in Section 4 we analyse in details the system (1.1) for a particular type of interactions and we prove the Theorem 1.1. We show Theorem 1.2 in Section 5. Finally, we are proving the Theorems 1.3 and 1.4 in Sections 6 and 7.

2. GLOBAL FACTS ON THE SYSTEM (1.1) In this section we establish some useful properties of solution of (1.1) and prove the existence of a positive global in time solution of the system (1.1), that for convenience we rewrite (2.1)

dv dt = A(Ψ(t))v
where A(Ψ(t)) is the following matrix:

A :=    (r 1 -1 K Ψ 1 (t)) -µ 1 + µ 11 µ ij . . . µ ij (r N -1 K Ψ N (t)) -µ N + µ N N    , with Ψ i (t) := Ψ i (v(t)) and µ i = N j=1 µ ij .
Since the function Ψ i are locally Lipschitz the local existence of a solution of (2.1) is a straightforward application of the Cauchy-Lipschitz Theorem. To obtain a global solution starting with non-negative initial data, we need more a priori estimates on the solutions (v i ) i∈{1,...,N } . Let us first show that the system (2.1) preserve the positivity.

Positivity.

Lemma 2.1 (Positivity). The system (2.1) preserves the positivity. That is to says that if v(0) is a nonnegative initial value, then the solution v of (2.1) stays nonnegative. Moreover, for all i v i (t) > 0 for all times t ∈ (0, T ].

Proof:

We argue by contradiction and assume that v changes its sign. Let t 1 ∈ (0, T ] be the first time where v i (t 1 ) = 0 for some i, v(t) > 0 for all t ∈ (0, t 1 ) and there exists t > t 1 so that v(t) < 0. t 1 > 0 is well defined since v changes its sign and v i (0) ≥ 0 for all i, v i (0) > 0 for some i and lim t→0 + dvi(t) dt > 0. Now at the time t 1 , we have

0 ≥ dv i (t 1 ) dt = j µ ij v j (t 1 ) ≥ 0.
Therefore for all i v i (t 1 ) = 0 = dvi(t1) dt and by the Cauchy-Lipschitz Theorem we deduce that for all i and all t ∈ [t 1 , T ] v i (t) = 0. Thus v ≥ 0 in (0, T ] which contradicts that v changes its sign.

Next we show that v i (t) > 0 for all times t ∈ (0, T ]. From the above argumentation, v is nonnegative for all times. To show that v stays positive for all time, we can see that it is enough to show that

N (t) = N i=1 v j (t) > 0 for all times t ∈ [0, T ]. Indeed, if there is t 1 ∈ (0, T ] so that t 1
is the first time where for some i, v i (t 1 ) = 0 then arguing as above we see that for all i and

T ≥ t 1 v i (t) = 0 so N (t) = 0 for all t ≥ t 1 .
Now, let us denote Q(0, 1) the following set

Q(0, 1) := x ∈ R N | N i=1 |x i | ≤ 1 .
Since Ψ i is locally Lipschitz, we have on Q(0, 1),

(2.2) ∀ i, ∃κ i so that Ψ i (x) ≤ κ i N i=1 |x i |.
Let κ 0 := N i=1 κ i and δ a positive constant so that δ < min 1, N (0) 2 , rmin 2κ0 . We will prove that N ≥ δ for all times in (0, T ]. We argue by contradiction and assume there exists t 1 < t 2 so that N (t 1 ) = δ and N (t) ≤ δ in (t 1 , t 2 ]. Using that the v i are non negative functions and (2.2), from our assumption we deduce that on [t 1 , t 2 ] and for all i,

dv i dt ≥ v i (r min -κ i N ) + N j=1 µ ij (v j -v i ).
By summing over all the possible i we end up with the following equation

dN dt ≥ N (r min -N κ 0 ), (2.3) N (t 1 ) = δ. (2.4)
Using the comparison principle on the latter equation, we achieve on [t 1 , t 2 ], N ≥ N where N is the solution of the logistic type equation (2.3). By construction we have N > δ for all times t > t 1 . Thus we get the contradiction δ < N ≤ δ. Hence N > δ for all t ∈ (0, T ].

Remark 2.2. Of the above proof, we also have a bound from below for N . Namely, we have for all t

N (t) ≥ min 1, N (0) 2 , r min 2κ 0 .

Existence of a global solution.

Next we show the following estimate an Lemma 2.3. Let v(t) be a solution of the Cauchy problem (2.1), then there exists two constants C 0 and C 1 and a positive vector v p so that v(t) ≤ C 0 e C1t v p

Proof:

From Lemma 2.1 we know that the v i are non negative functions. Therefore by (2.1) we can see that the v i satisfy (2.5)

dv i dt ≤ r i v i + N j=1 µ ij (v j -v i ).
Let us denote M and R the two following matrices

M :=    -µ 1 + µ 11 µ ij . . . µ ij -µ N + µ N N    , R :=    r 1 0 . . . 0 r N    .
We can then rewrite the inequalities (2.5) as follows

(2.6) dv dt ≤ (R + M )v.
By choosing μ = sup i∈{1,...,N } µ i , we can see that R + M + μ is nonnegative matrix. Since R + M + μId is also irreducible, by the Perron-Frobenius Theorem R + M + μId posses a unique principal eigenpair (ν p , v p ) so that v p is a positive vector, i.e. there exists (ν p , v p ) so that v p > 0 and

(R + M + μId)v p = ν p v p .
Note that e (νp-μ)t v(0) ∞ v p satisfies

du dt = (R + M )u.
So, by the Cauchy-Lipschitz Theorem, from (2.6) we deduce that v ≤ e (νp-μ)t v(0) ∞ v p for all times t.

The existence of a global in time solution for the system (2.1) is then a consequence of the Cauchy-Lipschitz Theorem and the above a priori estimates.

Remark 2.4. By adapting some ideas in [START_REF] Perthame | Transport equations in biology[END_REF] and for a particular type of Ψ i , we can derive an explicit formula for the solution of (2.1). Indeed when the interaction terms take the form Ψ i (v) = α(v) = N j=1 α j v j for all i then the solution v i (t) can be expressed by the formula

(2.7) v i (t) = e (R+M)t v(0) i 1 + N j=1 α j t 0 e (R+M)s v(0) j ds .
To obtain this formula we argue as in [START_REF] Perthame | Transport equations in biology[END_REF] (Chapter 2 Section 2.1 ) and we start by introducing the functions

α(t) := N j=1 α j v j and V i (t) := e t 0 α(s) ds v i (t).
We remark that the V i satisfy the linear equation

(2.8) dV i (t) dt = r i V i + N j=1 µ ij (V j -V i ).
Thus V i (t) := e (R+M)t v(0) i and v i (t) is implicitly given by the formula

v i (t) = e -t 0 α(s) ds e (R+M)t v(0) i .
Now let us evaluate the term e -t 0 α(s) ds . By differentiating e -t 0 α(s) ds one has

d dt (e t 0 α(s) ds ) = α(t)e t 0 α(s) ds = N j=1 α j V j (t).
Therefore we have

e t 0 α(s) ds = 1 + t 0 N j=1 α j V j (s) ds = 1 + N j=1 α j t 0 e (R+M)s v(0) j ds.
Hence

v i (t) = e (R+M)t v(0) i 1 + N j=1 α j t 0 e (R+M)s v(0) j ds .

RELATIVE ENTROPY IDENTITIES.

Here, we prove the following general principle which give us access to some useful identifies that we constantly use along this paper.

Theorem 3.1. Assume that r i , (µ ij ), Ψ i satisfies (1.2)-(1.
3). Let v and v be respectively a positive solution and a positive stationary solution of (1.1), and let H be a smooth (at least C 1 ) function. Then the function

H(v) := N i=1 v2 i H vi vi satisfies (3.1) dH(t) dt = -D(v) + 1 K N i=1 vi H ′ v i vi Γ i v i where D(v) := N i,j=1 µ ij vi vj H v j vj -H v i vi + N i,j=1 µ ij vi vj H ′ v i vi v i vi - v j vj Γ i := Ψ i (v) -Ψ i (v)

Proof:

By (1.1), for all i we have

(3.2) dv i dt =   r i v i - 1 K Ψ i (v)v i + N j=1 µ ij (v j -v i )   + 1 K Γ i v i
Using that v is a stationary solution, we have for all i

(r i - 1 K Ψ i (v))v i = - N j=1 µ ij (v j -vi ),
and we can rewrite the above equation as follows

dv i dt = N j=1 µ ij v j - vj v i vi + 1 K Γ i v i .
By multiplying the above equality by vi H ′ vi vi and by summing over all i we achieve

N i=1 vi H ′ v i vi dv i dt = 1 K N i=1 vi H ′ v i vi Γ i v i + N i=1 vi H ′ v i vi N j=1 µ ij v j - vj v i vi .
Thus we have

dH(t) dt = 1 K N i=1 vi H ′ v i vi Γ i v i - N i,j=1 µ ij vi vj H ′ v i vi v i vi - v j vj .
Hence we have

dH(t) dt = 1 K N i=1 vi H ′ v i vi Γ i v i -D(v).
since by symmetry of µ ij , N i,j=1

µ ij vi vj H v j vj -H v i vi = 0.
As immediate corollary of the Theorem 3.1, we have the following identities that we constantly used along this paper,

Lemma 3.2. Assume that r i , (µ ij ), Ψ i satisfies (1.2)-(1.
3). Let v and v be respectively a positive solution and a positive stationary solution of (1.1), then we have (i)

d dt N i=1 v i vi = 1 K N i=1 (Ψ i (v) -Ψ i (v))v i vi (ii) d dt N i=1 v 2 i = - N i,j=1 µ ij vi vj v j vj - v j vj 2 + 2 K N i=1 v 2 i (Ψ i (v) -Ψ i (v))
.

Proof:

(i) and (ii) can be obtained straightforwardly from the Theorem 3.1. Indeed, by using H(s) = s in Theorem 3.1, and by observing that from the symmetry of µ ij , N i,j=1

µ ij vi vj v i vi - v j vj = 0,
we easily get

dH(t) dt = 1 K N i=1 Γ i vi v i ,
which proves (i) since for the function H(s) = s, we have

H(t) = N i=1 vi v i .
We can obtain (ii) in a similar way by using the function H(s) = s 2 in the Theorem 3.1 and by observing that

N i,j=1 µ ij vi vj 2 v i vi v i vi - v j vj = N i,j=1 µ ij vi vj v i vi - v j vj 2 .
4. THE SPECIAL CASE Ψ i (.) = α(.):

In this section we analyse in details the asymptotic behaviour of a positive solution of (1.1) when the competition functional Ψ i is independent of i, i.e for all i,

Ψ i (v) = α(v) where α is a function from R N → R which satisfies (1.2) -(1.
3). As we expressed in Theorem 1.1 that we recall below, in this situation the system (1.1) has a unique positive stationary solution which attracts all the trajectories initiated from any nonnegative and non zero initial data. More precisely, we prove Theorem 4.1. Assume that for all i the function Ψ i (.) = α(.), then there exists a unique stationary solution v of (2.1). Moreover, for all nonnegative and non zero initial datum v(0), the corresponding solution v(t) converges to v.

To prove the Theorem, we first analyse the existence of stationary solution 4.1. Study of the existence of equilibria. Recall that we look for a stationary solution of (4.1)

dv dt = A(α(t))v.
Therefore if there exists a stationary equilibria for the system (4.1) the v i must satisfies the following equations:

A(ᾱ)v = 0 (4.2) ᾱ = α(v) (4.3)
where A(ᾱ) is the matrix

A(ᾱ) :=    (r 1 -ᾱ K ) -µ 1 + µ 11 µ ij . . . µ ij (r N -ᾱ K ) -µ N + µ N N    .
Note that we can rewrite the matrix A(ᾱ) := R -ᾱ K Id + M where R and M are matrices defined in Section 2.

Therefore, a solution v to (4.2) is a solution to

(4.4) (M + R)v = ᾱ K v.
Let us now establish some important property of the equilibrium.

Lemma 4.2. If v is a nonnegative stationary solution of (4.1), then either v ≡ 0 or v > 0 (i.e ∀ i, v i > 0).

Proof:

First, we observe that 0 is a solution of the problem (4.2). Now let v be nonnegative stationary solution of (4.1) so that v i = 0 for some i. From (4.4) we see that

(M + Rv) i = ᾱ K v i = 0. Therefore 0 = r i v i + N j=1 µ ij (v j -v i ) = N j=1 µ ij v j .
Thus v j = 0 for all j since by assumption v j ≥ 0 and (µ ij ) is irreducible. Hence any nonnegative stationary solution is either positive or the zero solution.

Observe that from the above Lemma and (4.4), one can see that v is a positive eigenvector of the matrix M + R associated to the eigenvalue ᾱ K . Now we are in position to prove that there exists a unique positive stationary solution to (4.4).

Lemma 4.3.

There exists a unique (ᾱ, v) solution of the equations (4.2) and (4.3). Moreover v satisfies α(v) = ν p .

Proof:

By choosing μ = sup i∈{1,...,N } µ i , we can see that R + M + μ is nonnegative matrix. Since R + M + μId is also irreducible, by the Perron-Frobenius Theorem R + M + μId posses a unique principal eigenpair (ν p , v p ) so that v p is a positive vector, i.e. there exists (ν p , v p ) so that v p > 0 and

(4.5) (R + M + μId)v p = ν p v p .
Moreover, the linear subspace associated to the eigenvalue ν p is one dimensional [START_REF] Zeidler | Nonlinear functional analysis and its applications[END_REF]. So without any loss of generality we can assume that

N i=1 (v p ) 2 i = 1.
From the equation (4.5) we deduce that the vector v p is a positive eigenvector of the matrix M + R associated with the eigenvalue λ p := (ν pμ). By construction one can see that λ p is the unique eigenvalue of the matrix M + R associated with a positive eigenvector. A quick computation shows that λ p = (ν pμ) > 0. Indeed, if not we have (R + M )v p ≤ 0. Thus for all i ∈ {1, . . . , N } we have

r i (v p ) i + N j=1 µ ij (v p ) j -(v p ) i ≤ 0. Let (v p ) i0 := min i∈{1,...,N } (v p ) i then for (v p ) i0 we have N j=1 µ ij (v p ) j -(v p ) i0 ≥ 0.
Since R is a positive matrix we achieve the contradiction

0 < r i0 (v p ) i0 + N j=1 µ ij (v p ) j -(v p ) i0 ≤ 0.
Now from (4.4) we deduce that there exists an unique positive ᾱ so that ᾱ K = λ p . Let us now construct our solution. Note that for any µ ∈ R, the vector µv p is also a solution to (4.4) with the eigenvalue λ p . So to obtain a solution v to (4.2) and (4.3) we only have to adjust µ in such a way that α(µv p ) = ᾱ. This is always possible for a unique µ since α(0) = 0, lim µ→∞ α(µv p ) = +∞ and α is an increasing function.

Convergence to the unique equilibrium.

Let us look at the convergence of v(t) toward its equilibrium. Let us first establish some useful identities.

Lemma 4.4. Let us denote (ᾱ, v) the stationary solution constructed above. Let v be a solution of (4.1) then v satisfies the following identities (i)

d dt N i=1 v i vi = 1 K (ᾱ -α(t)) N i=1 v i vi .
(ii)

d dt N i=1 (v i ) 2 = - N i,j=1 µ ij vi vj v j vj - v j vj 2 + 2 K (ᾱ -α(t)) N i=1 (v i ) 2 .

Proof:

Since here for all i, Ψ i = α, from Lemma 3.2 we deduce that

d dt N i=1 v i vi = 1 K N i=1 (α(v) -α(v(t)))v i vi , d dt N i=1 (v i ) 2 = - N i,j=1 µ ij vi vj v j vj - v j vj 2 + 2 K (ᾱ -α(t)) N i=1 (α(v) -α(v)) (v i ) 2 .
Thus (i) and (ii) hold true.

From the above Lemma we can derive a useful Lyapunov functional.

Lemma 4.5. Let v be the positive stationary solution of (4.1). For any positive solution v of (4.1), let us denote E, β and F the following quantities

E(v) := N i=1 (v i ) 2 , β(v) := N i=1 v i vi F (v) := log E(v) (β(v)) 2 .
Then F ≥ log( 1 sup i vi ) and for any positive solution v of (4.1) we have

d dt F (v) = - 1 E(v) N i,j=1 µ ij vj vi v j vj - v i vi 2 < 0.

Proof:

First let us show that F is bounded from below. By construction and using a standard convexity argument, we see that β 2 (v) ≤ (sup i vi )E(v). So from the monotonicity of the log, we conclude that F (v) ≥ log( 1 sup i vi ). Now let us show that F is non increasing. To do so let us compute d dt F (v). From the definition of F we see that

d dt F (v) = d t E(v) E(v) -2 d t β(v) β(v) ,
where d t denotes d dt . By Lemmas 4.4 we have

d t E(v) E(v) = - 1 E(v) N i,j=1 µ ij vj vi v j vj - v i vi 2 + 2 K (ᾱ -α(t)) d t β(v) β(v) = 1 K (ᾱ -α(t)).
Thus,

d dt F (v) = - 1 E(v) N i,j=1 µ ij vj vi v j vj - v i vi 2 ≤ 0.
Next we derive some a priori estimates on the solution v of (4.1) from the previous Lemmas. Namely, we show that Lemma 4.6. Let v be a positive solution of (4.1), then there exists C 1 so that

E(v) + β(v) ≤ C 1 .

Proof:

Observe that from Lemma 4.5, to obtain the bound it is sufficient to have a uniform bound on β. Indeed, since F is decreasing in time, we have for all times

E(v) ≤ β 2 (t) E(v(0)) β 2 (v(0)) .
Now recall that by Lemma 4.4, β(v) satisfies the following equation

dβ(t) dt = 1 K (ᾱ -α(t))β(t).
Since α satisfies the assumptions (1.2)-(1.3), there exists R α , c α , k α so that for all

x ∈ R N,+ \ Q Rα (0), (4.6) c α N i=1 x i kα ≤ α(x).
Assume that for some t > 0, v(t) ∈ R N,+ \ QRα (0) otherwise we are done since

β(v) ≤ max i∈{1,...,N } vi N j=1 |v j |.
Let Σ be

Σ := {t ∈ R + | N (t) > R α }.
From (4.6) and using that the v i are non negative, we see that for all t ∈ Σ

α(t) ≥ c α N (t) kα ≥ c α 1 vmax kα β(t) kα ,
where vmax := max i∈{1,...,N } vi . Therefore, on Σ we have

dβ(t) dt ≤ 1 K ᾱ -c 0 β kα (t) β(t).
Using the logistic character of the above equation, we can check that

β(v(t)) ≤ sup β(v(0)), max x∈ QRα (0) β(x), ᾱ c 0 1 kα .
We are now in position to prove the convergence of v toward its equilibrium.

Lemma 4.7. Let (v i ) i=1...N be the unique stationary solution of (4.1). Then for any non negative initial datum (v i (0)) i=1...N not identically zero, the corresponding solution (v i (t)) i=1...N of (4.1) converges to (v i ) i=1...N as t goes to infinity.

Proof:

For simplicity we denote <, > the standard scalar product in R N . Now, since v = 0 and for all times t, v(t) = (v 1 , . . . , v N ) is a vector of R N , we can write v(t) := λ(t)v + h(t) with for all t, < h(t), v >= 0. Substituting v by this decomposition in (4.1), it follows that

λ ′ (t)v + dh(t) dt = λ(t)A(α(t))v + A(α(t))h, (4.7) = 1 K (ᾱ -α(t))λ(t)v + A(α(t))h. (4.8)
Therefore, we have

< λ ′ (t)v + dh(t) dt , h >=< 1 K (ᾱ -α(t))v + A(α(t))h, h > . Thus < dh(t) dt , h >= 1 2 dE(h) dt =< A(α(t))h, h > .
By following the computation developed for the proof of (ii) in Lemma 4.4, we see that

dE(h) dt = - N i,j=1 µ ij vi vj h j vj - h j vj 2 + 2 K (ᾱ -α(t))E(h).
Since E(h) ≥ 0 for all times, we will analyse separately two situations: Either E(h(t)) > 0 for all times t or there exists t 0 ∈ R so that E(h(t 0 )) = 0. In the latter case, from the above equation we see that we must have E(h(t)) = 0 for all t ≥ t 0 and so for all t ≥ t 0 , we must have v(t) = λ(t)v. Hence from (4.8) we are reduced to analyse the following ODE equation

λ ′ (t) = λ(t) K (ᾱ -α(λ(t)))
where α is the increasing locally Lipschitz function defined by α(s) := α(sv). Note that since λ(t) < v, v >= β(v) > 0, we have λ(t) ≥ 0 for all times t. The above ODE is of logistic type with non negative initial datum therefore by a standard argumentation we see that λ(t) converges to λ > 0 where λ is the unique solution of α( λ) = ᾱ. By construction we have α(1) = ᾱ, so we deduce that λ = 1. Hence, in this situation, v converges to v as time goes to infinity.

In the other situation, E(h(t)) > 0 for all t and we claim that Claim 4.8. E(h(t)) → 0 as t → +∞.

Assume the Claim holds true then we can conclude the proof by arguing as follows. From the decomposition v(t) = λ(t)v + h(t), we can express the function

β(v(t)) by β(v(t)) =< v, v >= λ(t) < v, v >. Therefore from Lemma 4.4 we deduce that (4.9) λ ′ (t) = 1 K (ᾱ -α(λ(t)v + h(t)))λ(t).
Now by using E(h) → 0, we deduce that h → 0 as t → ∞ and from (4.9) we are reduced to analyse the ODE

λ ′ (t) = 1 K (ᾱ -α(λ(t)))λ(t) + λ(t)( α(λ(t)v) -α(λ(t)v + h(t))) = 1 K (ᾱ -α(λ(t)))λ(t) + λ(t)o(1)
where |o(1

)| = | α(λ(t)v) -α(λ(t)v + h(t))| ≤ C E(h) → 0 as t → ∞.
As before we can conclude that λ(t) → 1 and v converges to v.

Proof of Claim 4.8:

Since E(h(t)) > 0 for all t, as in Lemma 4.5 we have (4.10)

d dt log E(h) (β(v)) 2 = - 1 E(h) N i,j=1 µ ij vi vj h j vj - h j vj 2 .
Thus the function F := log E(h) (β(v)) 2 is a decreasing smooth function. First we observe that the claim is proved if there exists a sequence (t n ) n∈N converging to infinity so that E(h(t n )) → 0. Indeed, assume such sequence exists and let (s k ) k∈N be a sequence converging to +∞. Then there exists k 0 and a subsequence (t n k ) k∈N of (t n ) n∈N so that for all k ≥ k 0 , we have s k ≥ t n k . Therefore from the monotonicity of F we have for all

k ≥ k 0 log E(h(s k )) (β(v(s k ))) 2 ≤ log E(h(t n k )) (β(v(t n k ))) 2 .
By letting k to infinity in the above inequality, we deduce that

lim k→∞ log E(h(s k )) (β(v(s k ))) 2 = -∞,
which implies that E(h(s k )) → 0, since by Lemma 4.6 (β(v(t k ))) k∈N is uniformly bounded. The sequence (s k ) k∈N being chosen arbitrarily this implies that E(h(t)) → 0 as t → +∞.

Let us now prove that such sequence (t n ) n∈N exists. We argue by contradiction and assume that inf t∈R + E(h(t)) > 0. Therefore from the monotonicity and the smoothness of F we deduce that there is c 0 ∈ R so that

F (h(t)) → c
µ ij vj vi h j vj - h i vi 2 = 0.
From the a priori estimates of Lemma 4.6, there exists a sequence t n → ∞ so that for all i h i (t n ) → h i . Passing to the limit along this sequence in the equation (4.11) it yields

0 = N i,j=1 µ ij vj vi h j vj - h i vi 2 .
By using the irreducibility assumption on the nonnegative matrix µ ij and the positivity of the quantities vi , one can deduce from the above equality that we must have for all i and j h j vj = h i vi .

Thus if we set λ := h1 v1 we have h = λv. So by using that < h, v >= 0 for all time it follows that λ = 0. Hence, we get the contradiction

0 < inf t∈R E(h(t)) ≤ E(h(t n )) → 0.
From the above differentials inequalities, by comparing N , N min and N max via the Cauchy Lipschitz Theorem, we obtain N min ≤ N ≤ N max for all t ∈ [0, t 1 ). Since N max (t 1 ) < K, we can bootstrap the above argument and show that (5.4) holds true for all t.

A useful functional inequality.

Next we establish a useful functional inequality satisfied by vectors h ∈ v⊥ where v⊥ denotes the linear subspace of R N orthogonal to v. Lemma 5.3. There exists C 1 > 0 so that for all h ∈ v⊥

C 1 E(h) ≤ N i,j=1 µ ij vi vj h j vj - h j vj 2 .
Moreover C 1 = λ 2 where λ 2 is the minimal eigenvalue strictly positive of the linear eigenvalue problem

h i N j=1 µ ij vj vi - N j=1
µ ij h j = λh i .

Proof :

Let I be the following Rayleigh quotient

I(h) := 1 E(h) N i,j=1 µ ij vi vj h j vj - h j vj 2 .
Observe that the first part of the Lemma is proved if we show that (5.7) inf

h∈v ⊥ I(h) > 0, or equivalently (5.8) inf h∈v ⊥ ,E(h)=1 I(h) = inf h∈v ⊥ I(h) > 0,
since for all real µ, I(h) = I(µh).

To obtain (5.7), we argue by contradiction and assume that inf h∈v ⊥ I(h) = 0. By (5.8) we can take (h n ) n∈N a minimising sequence so that h n ∈ v⊥ , E(h n ) = 1 for all n. Since {x ∈ R N |x ∈ v⊥ , E(x) = 1} is a closed bounded set, (h n ) n∈N is uniformly bounded and we can extract a subsequence (h n k ) k∈N which converges to h ∈ v⊥ with E( h) = 1. Passing to the limit along this subsequence, we see that I( h) = 0 which combined with h ∈ v⊥ implies that h = 0. Thus we get the contradiction 0 = E( h) = 1. Hence (5.7) holds true.

Let C 1 be defined by C 1 = inf h∈v ⊥ ,E(h)=1 I(h), let us try to compute C 1 . By construction C 1 is the result of a constrained minimisation problem. Therefore, by the standard optimization Theory [START_REF] Rockafellar | Convex Analysis[END_REF], the minimizers must satisfy the following Euler-Lagrange equations (D -M )h = λh, (5.9)

E(h) = 1, (5.10) < h, v >= 0, (5.11)
where λ ∈ R is a Lagrange Multiplier to be determined and M and D are the following matrices

M :=    µ 11 µ ij . . . µ ij µ N N    , D :=     1 v1 N j=1 µ 1j vj 0 . . . 0 1 vN N j=1 µ N j vj     .
By taking k > 0 large enough, the irreducible matrix M -D + kId becomes non-negative and so M -D + kId is a symmetric positive definite matrix. As a consequence, the eigenvectors associated to the eigenvalue ν i of M -D + kId form an a orthogonal basis of R N , i.e. R N = ⊥ E νi , where E νi denotes the eigenspace associated to the eigenvalue ν i .

Since M -D + kId is non negative and irreducible by the Perron-Frobenius Theorem there exists a unique eigenvalue, says ν 1 associated with a positive eigenvector φ 1 . Furthermore ν 1 is the largest eigenvalue and is algebraically simple. By a direct computation one can see that

( M -D + kId)v = kv, thus ν 1 = k and φ 1 = γv. From the properties of ν 1 we have E ν1 = lin(v), for all i = 1, ν i < ν 1 = k and v⊥ = ⊥ i =1 E νi .
By construction, the λ i := ν ik ≤ 0 are the eigenvalues of M -D and we can see that for all h ∈ v⊥ , E(h) = 1 we have

I(h) = t h( M -D)h ≥ min i =1 {-λ i }.
Let λ 2 < 0 be second largest eigenvalue of M -D and φ 2 an associated eigenvector. By normalising φ 2 properly and since φ 2 ∈ v⊥ a straightforward computation shows that

I(φ 2 ) = t φ 2 (D -M )φ 2 = -λ 2 = min i =1 {-λ i }.
Hence, min h∈v ⊥ I(h) = -λ 2 .

Asymptotic behaviour of the solution.

We are now in position to obtain the exponential rate of convergence for any solution v of (1.5).

Proof of Theorem 1.2:

First we claim that (5.12)

N i=1 vi = K.
Indeed from Lemma 4.3 we deduce that v = Kλp N i=1 ri(vp)i v p where v p is the positive normalised eigenvector associated to the principal eigenvalue λ p of the matrix R + M . A straightforward computation shows that

N i=1 r i (v p ) i = N i=1 ((R + M )v p ) i = λ p N i=1 (v p ) i . Since λ p > 0 it follows that N i=1 (v p ) i N i r i (v p ) i = 1 λ p . Thus Kλ p N i r i (v p ) i N i=1 (v p ) i = K.
Now recall that in the proof of Lemma 4.7 from the orthogonal decomposition v(t) = λ(t)v + h(t) we have

dE(h) dt = - N i,j=1 µ ij vi vj h j vj - h j vj 2 + 2 K (ᾱ -α(t))E(h).

By Lemma 5.3 we obtain

dE(h) dt ≤ -C 1 E(h) + 2 K (ᾱ -α(t))E(h).
Now arguing as in the proof of Lemma 4.5, we end up with

d dt log E(h) β(v) 2 ≤ -C 1 .
Therefore, thanks to Lemma 4.6 we deduce that (5.13)

E(h) ≤ E(h(0)) β(v(0)) 2 e -C1t β 2 (v) ≤ C 2 e -C1t .
Finally we get the exponential rate of convergence, by observing that

N (t) =< v, 1 >= λ(t) < v, 1 > + < h, 1 >,
which thanks to (5.3) implies that

λ(t) = K < v, 1 > + < h, 1 > < v, 1 > + |N (0) -K|e -1 K t 0 α(s) ds < v, 1 > .
By Lemma 5.2 and the estimates (5.13), (5.12), ,using standard norm estimates, we have

v -v ∞ ≤ < h, 1 > K + |N (0) -K|e -1 K t 0 α(s) ds K v ∞ + N E(h) ≤ |N (0) -K| v ∞ K e -1 K t 0 α(s) ds + 1 + v ∞ K C 2 e -C1t .

THE GENERAL CASE:THE STATIONARY SOLUTION

In this section we investigate the existence of a positive stationary solution of (2.1) under the additional condition (1.6) on the matrices R and M that we recall below,

∀ i N j=1 µ ij ≤ r i 2 .
This assumption has for consequence that the matrix R + M is positive definite. Indeed, we have

t h(R + M )h = N i=1 r i h 2 i - 1 2 N i,j=1 µ ij (h i -h j ) 2 ≥ 2 N i,j=1 µ ij h 2 i - 1 2 N i,j=1 µ ij (h i -h j ) 2 ≥ N i,j=1 (µ ij h 2 i + h 2 j - 1 2 (h i -h j ) 2 ≥ 1 2   N i,j=1 µ ij (h i + h j ) 2   .
Thus kern(R + M ) = {0} and the matrix R + M is invertible. Moreover from the last inequality we see that there exists positive constants c 0 and C 0 so that (6.1)

c 0 < u, u > R+M ≤ E(u) ≤ C 0 < u, u > R+M ,
where < u, u > R+M := t u(R + M )u.

Let Ξ(v) be the diagonal matrix defined by

(Ξ(v)) ij = δ ij Ψ i (v).
With this notation, a positive stationary solution of (2.1) is then a non negative solution of the following problem:

(6.2) (R + M )v = Ξ(v)v
Note that when Ξ(v) can be written as Ξ(v) = α(v)Id, the construction of a positive solution has already been made in Section 4. So in the later, we will assume that Ξ(v) cannot be written as Ξ(v) = α(v)Id. It is worth mentioning that in this situation the method used in Section 4 does not work and we have to use another strategy.

Let T be the following map

T : R N → R N v → T v := (R + M ) -1 [Ξ(v)v]
.

Since R + M is invertible, T is well defined and one can easily check that a positive solution of (6.2) is a positive fixed point of the map T . To check that T has a positive fixed point we use a degree argument. Let Ψ i (v) s and Ξ(v) s defined by

Ψ s i := sΨ i + (1 -s)Ψ 1 , (Ξ s (v)) ij := δ ij Ψ s i (v), we consider the homotopy H ∈ C([0, 1] × R N , R N ) defined by H : [0, 1] × R N → R N (s, v) → H(s, v) := (R + M ) -1 [Ψ s (v)v].
One can see that H(1, .) = T and H(0, .) = T 0 where T 0 corresponds to the map

T 0 : R N → R N v → T 0 v := Ψ 1 (v)(R + M ) -1 v.
Note that there exists a unique positive fixed point to T 0 which can be constructed by arguing as in Section 4.

The next step in this degree argument is to obtain for all s, a good a priori estimates on the fixed point of the map H(s, v), i.e. a good estimate on the positive solutions of the following problem:

(6.3) (R + M )V = Ψ s (V )V.
In this direction we show the following: Lemma 6.1. Let V be a non negative solution of (6.3). Then either V ≡ 0 or V > 0 and there exists c1 and C1 independent of s so that c1

≤ i V i ≤ C1 .

Proof:

To obtain that V is either positive or V = 0 we can argue as in the proof of Lemma 2.1. So assume that there exists a i 0 so that V i0 = 0. By construction V i0 is a minimum of the V i and from the equation satisfied by V i0 we get

0 ≤ N j=1 µ i0j (V j -V i0 ) = 0.
Therefore V j = V i0 for all j where µ i0j = 0. Since M is irreducible there exists j = i 0 so that µ i0j = 0. Let ρ := {k | V k = 0} then i 0 and all j so that µ i0j = 0 belongs to ρ. In the previous argument, by replacing i 0 by any k ∈ ρ, we see that all j so that µ kj = 0 belongs to the set ρ. By iterating enough times the above argument and using the irreducibility of the matrix M we can see that ρ = {1, . . . , N } so V i = V i0 = 0 for all i. Therefore a nonnegative solution of (6.3) is either

V ≡ 0 or V > 0.
Now let us assume that V > 0. Recall that by (6.1) there exists positive constants c 0 and C 0 so that for all u ∈ R N c 0 < u, u > R+M ≤ E(u) ≤ C 0 < u, u > R+M . So for a solution V of (6.3) one has

< V, V > R+M = i Ψ s i (V )V 2 i ≥ s min i∈{2,...,N } Ψ i (V ) + Ψ 1 (V ) < V, V > ≥c 0 Ψ 1 (V ) < V, V > R+M ,
and we also get

< V, V > R+M ≤ C 0 s max i∈{2,...,N } Ψ i (V ) + Ψ 1 (V ) < V, V > R+M .
Therefore we have

Ψ 1 (V ) ≤ 1 c 0 , (6.4) 1 C 0 ≤ s max i∈{2,...,N } Ψ i (V ) + Ψ 1 (V ). (6.5)
Now thanks to the assumptions (1.2)-(1.3) made on the functions Ψ i , there exists R 1 , c 1 , k 1 and N positive constants κ i so that :

For all x ∈ R N,+ \ Q R1 (0), c 1 N i=1 x i k1 ≤ Ψ 1 (x), (6.6)
For all i, and for all

x ∈ Q R1 (0), Ψ i (x) ≤ κ i N j=1 |x j |. (6.7)
By combining (6.4),(6.6), (6.5) and (6.7) we deduce that

N = N j=1 V j ≤ sup 1 c 0 c 1 1 k 1 , R 1 , N = N j=1 V j ≥ min R 1 , 1 C 0 (κ 1 + sup i κ i ) .

Computation of the degree.

We are now in position to prove the existence of a positive solution to the equation (6.2) by means of the computation of the topological degree of Tid on a well chosen set Ω ⊂ R N,+ . Now we take two positive constants c 2 and C 2 so that c 2 < c1 and C 2 > C1 where c1 and C1 are the constants obtained in Lemma 6.1. Let Ω be the following open set

Ω := v ∈ R N,+ | c 2 ≤ N i=1 v i ≤ C 2
and let us compute deg(T -Id, Ω, 0). By Lemma 6.1 for all s ∈ [0, 1] in H(s, v)v = 0 on ∂Ω. Therefore using that H(., .) is an homotopy, we conclude that deg(T -Id, Ω, 0) = deg(H(1, .) -Id, Ω, 0) = deg(H(0, .) -Id, Ω, 0). By construction, from Section 4.1, one can check that deg(H(0, .) -Id, Ω, 0) = 0 since the map T 0 has an unique stable and positive fixed point. Thus deg(T -Id, Ω, 0) = 0 which shows that T has a fixed point in Ω.

THE GENERAL CASE: ASYMPTOTIC BEHAVIOUR

In this section we prove Theorem 1.4. That is to say, under the extra assumption (1.6) we analyse the asymptotic behaviour of the solution v(t) when for all i the interaction Ψ i can be expressed like: Ψ i (v) = α(v) + ǫψ i (v) with ψ i uniformly bounded. To obtain the asymptotic behaviour in this case, we follow the strategy developed in Subsection 4.2. Namely, we start by showing some a priori estimates on the solution, then we analyse the convergence by of a Lyapunov functional. For convenience we dedicate a subsection to each essential part of the proof.

A priori estimate.

We start by establishing some useful differential inequalities. Namely we show that Lemma 7.1. Assume that r i , (µ ij ), Ψ i satisfies (1.2), (1.3) and (1.6). Assume further that Ψ i (v) = α(v) + ǫψ i (v) with ψ i uniformly bounded. Then there exists ǫ 0 > 0 so that for all 0 ≤ ǫ ≤ ǫ 0 there exists ω + ∈ R N , ω + positive and a positive real γ so that

(i) d N i=1 ω+ i v i dt ≤ 1 K (α(ω + ) -α(v)) N j=1 ω+ i v i d N i=1 ω+ i v i dt ≥ 1 K (α(γ ω+ ) -α(v)) N j=1 ω+ i v i (ii) dE(v) dt ≤ - N i,j=1 µ ij ω+ i ω+ j v i ω+ i - v j ω+ j 2 + 2 K (α(ω + ) -α(v))E(v) dE(v) dt ≥ - N i,j=1 µ ij ω+ i ω+ j v i ω+ i - v j ω+ j 2 + 2 K (α(γ ω+ ) -α(v))E(v) (iii)
d dt log E(v) ( N i=1 ω+ i v i ) 2 ≤ - 1 E(v) N i,j=1 µ ij ω+ i ω+ j v i ω+ i - v j ω+ j 2 + 2 K (α(ω + ) -α(γ ω+ )) d dt log E(v) ( N i=1 ω+ i v i ) 2 ≥ - 1 E(v) N i,j=1 µ ij ω+ i ω+ j v i ω+ i - v j ω+ j 2 + 2 K (α(γ ω+ ) -α(ω + ))
Proof: First, we observe that (iii) can be straightforwardly obtained by combining (i) and (ii). So we deal only with (i) and (ii).

Let us denote σ := ǫ ψ ∞ . Then since v is positive, from (1.1) it follows that

dv i dt ≤ (r i + σ -α(v))v i + N j=1 µ ij (v j -v i ), dv i dt ≥ (r i -σ -α(v))v i + N j=1 µ ij (v j -v i ).
Let ω+ and ω-be the stationary solutions of the corresponding equations

dω + i dt = (r i + σ -α(ω + ))ω + i + N j=1 µ ij (ω + j -ω + i ), dω - i dt = (r i -σ -α(ω -))ω - i + N j=1 µ ij (ω - j -ω - i ).
Now, let us fix ǫ small enough so that ω± exists. Then by arguing as in the proof of Lemma 3.2, we obtain

d dt N i=1 ω+ i v i ≤ 1 K (α(ω + ) -α(v)) N j=1 ω+ i v i , d dt N i=1 ω- i v i ≥ 1 K (α(ω -) -α(v)) N j=1 ω- i v i , dE(v) dt ≤ - N i,j=1 µ ij ω+ i ω+ j v i ω+ i - v j ω+ j 2 + 2 K (α(ω + ) -α(v))E(v), dE(v) dt ≥ - N i,j=1 µ ij ω- i ω- j v i ω- i - v j ω- j 2 + 2 K (α(ω -) -α(v))E(v).
Note that by Lemma 4.3, ω± are positive eigenvectors of the matrices R + M ± σId. Thus ω are eigenvectors of the matrix R + M associated to the principal eigenvalue of R + M . Since R + M is irreducible, the eigenspace associated to the principal eigenvalue is unidimensional. So, we have ω + = γω -for some positive γ. Hence (i) and (ii) hold true since

N i,j=1 µ ij ω- i ω- j v i ω- i - v j ω- j 2 = N i,j=1 µ ij ω+ i ω+ j v i ω+ i - v j ω+ j 2 .
Next, we derive some a priori estimates for the solution v for an interaction Ψ as in Theorem 1.4. We first prove some sharp a priori estimates for stationary solution vǫ of (1.1). Lemma 7.2. Assume that r i , (µ ij ), Ψ i satisfies (1.2), (1.3) and (1.6). Assume further that Ψ i (v) = α(v) + ǫψ i (v) with ψ i uniformly bounded. Then there exists c1 < C1 and ǫ 1 so that for all 0 ≤ ǫ ≤ ǫ 1 and for any positive stationary solution vǫ of (1.1), we have

c1 ≤ N i=1 vǫ,i < C1 .
Moreover, there exists r 0 so that for all ǫ ≤ ǫ 1 , vǫ ∈ Q √ ǫr0 (ω + ). Furthermore, for any nonnegative initial datum v i (0) not identically zero there exists two constants c2 (v(0)), C2 (v(0)) so that for all 0 ≤ ǫ ≤ ǫ 1 the solution v ǫ satisfies for all t,

c2 ≤ β(v ǫ (t)) := N i=1 vǫ,i v ǫ,i (t) ≤ C2 .

Proof:

Let us first observe that for ǫ ≤ ǫ 0 by replacing v ǫ by vǫ in (i) of Lemma 7.1, we get

0 ≤ 1 K (α(ω + ) -α(v ǫ )) N j=1 ω+ i vǫ,i , (7.1) 0 ≥ 1 K (α(γ ω+ ) -α(v ǫ )) N j=1 ω+ i vǫ,i . (7.2)
From the proof of Lemma 4.3, we also see that

(7.3) α(ω + ) = K(λ p + σ), α(ω -) = K(λ p -σ),
where λ p is the positive principal eigenvalue of the matrix R+M . By using the monotonicity of α, we deduce that the maps σ → ω ± are monotone. Moreover, we have 0 ≤ α(ω + )α(ω -) ≤ 2σK.

To obtain c1 and C1 we argue as follows. Let us fix ǫ 1 := min{ǫ 0 , λp 4 ψ ∞ }. Then by (7.3), for all ǫ ∈ [0, ǫ 1 ] we have (7.4) α(ω

+ ) ≤ α(ω + ǫ1 ) = 5Kλ p 4 , α(ω -) ≥ α(ω - ǫ1 ) = 3Kλ p 4 .
Now thanks to the assumptions (1.2)-(1.3) satisfied by α, there exists R α , c α , k α and κ α so that :

For all x ∈ R N,+ \ Q Rα (0), c α N i=1 x i kα ≤ α(x), (7.5)
For all i, and for all

x ∈ Q Rα (0), α(x) ≤ κ α N j=1 |x j |. (7.6)
So for ǫ ∈ [0, ǫ 1 ], by combining (7.1), (7.2),(7.3),(7.4), (7.5) and (7.6) we achieve

N j=1 vǫ,j ≤ sup 5Kλ p 4c α 1 kα , R α =: C1 , (7.7) N j=1 vǫ,j ≥ min R α , 3Kλ p 4κ α =: c1 . (7.8)
To obtain a more precise estimate on vǫ , we argue as follows. Let us decompose vǫ := λ ǫ ω+ +h ǫ where h ǫ is orthogonal to ω+ . Then by replacing v ǫ by vǫ in (iii) of Lemma 7.1, and using that

|α(ω + ) -α(ω -)| ≤ 2σK we get 0 ≤ - 1 E(v ǫ ) N i,j=1 µ ij ω+ i ω+ j h ǫ,i ω+ i - h ǫ,j ω+ j 2 + 4σ, (7.9) 0 ≥ - 1 E(v ǫ ) N i,j=1 µ ij ω+ i ω+ j h ǫ,i ω+ i - h ǫ,j ω+ j 2
-4σ. (7.10) From (7.7) and using the functional inequality, Lemma 5.3, we deduce that

(7.11) E(h ǫ ) ≤ 4σ C2 1 C(ω + ) .
Combining the latter estimate with (7.7) and the positivity of ω+ and vǫ , we have the estimate

λ ǫ ≤ C1 N i=1 ωi 1 + 2 N σ C(ω) ,
where ω is the stationary solution with ǫ = 0. Let R 0 := C1 1 + 2 N σ C(ω) , and choose ǫ 1 smaller if necessary to have R 0 ≤ 2 C1 . Next consider the set QR0 (0) ⊂ Q 2 C1 (0). From the above estimates on λ ǫ and h ǫ , since α is Lipschitz continuous in Q 2 C1 (0) there exists κ 0 so that for all ǫ,

|α(λ ǫ ω+ ) -α(λ ǫ ω+ + h ǫ )| ≤ κ 0 N E(h ǫ ),
which combine with (7.11) enforces (7.12)

|α(λ ǫ ω+ ) -α(λ ǫ ω+ + h ǫ )| ≤ 2 C1 κ 0 N σ C(ω) .
Thus from (7.1), (7.2) and (7.12) we deduce that

(7.13) |α(ω + ) -α(λ ǫ ω+ )| ≤ 2Kσ + 2 C1 κ 0 N σ C(ω) ≤ C √ σ,
with C independent of ǫ. Observe that by construction there exists two positives constants 0 < ι 0 < 1 < ι 1 so that for all ǫ ≤ ǫ 1 , there exists ι ǫ ∈ (ι 0 , ι 1 ) so that ω+ = ι ǫ ω. Recall now that by assumption ∇α > 0, then the real map s → α(sω) is smooth (C 1 (R)) and increasing. It is therefore an homeomorphism in R + and a local diffeomorphism in R + . So by the Inverse Function Theorem, we deduce that for all s, t ∈ (0, C),

|s -t| ≤ k ι 0 |α(sω + ) -α(tω + )|,
where

C := sup ι 1 ι 0 , C1 N i=1 ωi 1 + 2 N σ C(ω) k := 1 min s∈(0,C) (< ∇(α(sω)), ω >) .
In particular, we have

|s -1| ≤ k ι 0 |α(sω + ) -α(ω + )|,
which combined with (7.13) enforces

|λ ǫ -1| ≤ k ι 0 C √ σ.
Hence from the latter estimate and (7.11) we have for all ǫ ≤ ǫ 1 , vǫ ∈ Q C √ ǫ (ω + ). Next, we derive an uniform upper bound for β(v ǫ ) when ǫ ∈ [0, ǫ 1 ]. In the sequel of this proof, for convenience we drop the subscript ǫ on v.

First, we observe that by Lemma 3.2 we have

d dt N i=1 v i vi = 1 K N i=1 (Ψ i (v) -Ψ i (v))v i vi , = 1 K (α(v) -α(t)) N i=1 v i vi + ǫ K N i=1 (ψ i (v) -ψ i (v))v i vi .
Since the functions ψ i are uniformly bounded, we achieve

d dt N i=1 v i vi ≤ 1 K (α(v) + 2ǫ ψ ∞ -α(t)) N i=1 v i vi .
By using (7.1) and (7.4), it follows

d dt N i=1 v i vi ≤ 1 K 7Kλ p 4 -α(t) N i=1 v i vi .
Again using that α satisfies the assumptions (1.2)-(1.3), there exists R α , c α , k α so that for all x ∈ R N,+ \ Q Rα (0),

(7.14) c α N i=1 x i kα ≤ α(x).
Now, let us assume that for some t > 0, v(t) ∈ R N,+ \ QRα (0) otherwise the proof is done since we have

β(v) ≤ max i∈{1,...,N } vi N j=1 |v j | ≤ C1 R α ,
where C1 is the bound obtained above. Let Σ be the following set

Σ := {t ∈ R + | N (t) > R α }.
From (7.14), (7.7) and using that the v i are non negative, we see that for all t ∈ Σ

α(t) ≥ c α N (t) kα ≥ c α 1 C1 kα β(t) kα .
Therefore, with c 0 := c α 1 C1

kα we have for t ∈ Σ

dβ(t) dt ≤ 1 K 7Kλ p 4 -c 0 β kα (t) β(t).
Using the logistic character of the above equation, we can check that

β(v(t)) ≤ sup β(v(0)), sup x∈QR α (0) β(x), 7Kλ p 4 c 0 1 kα .
Thus, by using (7.7), we achieve for all ǫ ∈ [0, ǫ 1 ]

β(v ǫ (t)) ≤ C2 := sup β(v(0)), C1 R α , C1 7Kλ p 4c α 1 kα
.

To obtain the lower bound for β(v ǫ ) we argue as follows. First let us observe that by taking ǫ 1 smaller if necessary, since ω+ → ω as ǫ → 0 and vǫ ∈ Q C √ ǫ (ω + ), there exists a positive constant c 0 independent of ǫ ∈ [0, ǫ 1 ] so that for all stationary solution vǫ we have min i∈{1,...,N } vǫ,i ≥ c 0 . Now by (7.8), for all ǫ ∈ [0, ǫ 1 ] we have (7.15)

β(v ǫ ) ≥ min i∈{1,...,N } vǫ,i N i=1 v ǫ,i ≥ c 0 N i=1 v ǫ,i .
Therefore to obtain an uniform lower bound for β(v ǫ ) it is enough to obtain an uniform lower bound for N ǫ := N i=1 v ǫ,i . From (1.1) by summing over all i and by using the definition of Ψ and the boundedness of the ψ i we deduce that N ǫ satisfies the following inequality

dN ǫ dt ≥ 1 K (α(v ǫ ) -2ǫ ψ ∞ -α(t)) N ǫ .
Thanks to (7.1) and (7.4), we have

dN ǫ dt ≥ 1 K Kλ p 4 -α(t) N ǫ .
By reproducing the argumentation of the proof of Lemma 2.1 and by using Remark 2.2 we can check that (7.16)

N ǫ (t) ≥ min 1, Kλ p 4κ 1 , N (0) 2 ,
where κ 1 denotes the Lipschitz constant of the function α in the unit cube. Hence, by collecting (7.15) -(7.16) we achieve for all ǫ ≤ ǫ 1 and all t > 0,

β(v ǫ (t)) ≥ c1 N min 1, Kλ p 4κ 1 , N j=1 v i (0) 2 =: c2 .
Remark 7.3. Note that from the above argumentation, using the Logistic character of the equations, we can get that for all ǫ ≤ ǫ 1 and all initial data v(0) ≥ 0, there exists t 0 so that for all t ≥ t 0 we have

1 2 c1 N min 1, Kλ p 4κ 1 ≤ β(v) ≤ 2 sup C1 R α , C1 7Kλ p 4c α 1 kα
.

Lastly, we obtain some uniform control on a continuous set of homeomorphisms

Ψ v (s) := N i=1 Ψ i (sv)v 2 i where v ∈ U ⊂ R N,+ .
Namely, we show that Lemma 7.4. Assume that r i , (µ ij ), Ψ i satisfies (1.2), (1.3) and (1.6). Assume further that

Ψ i (v) = α(v) + ǫψ i (v) with α ∈ C 1 loc satisfying (1.
2), (1.3)and ψ i ∈ C 1 loc uniformly bounded. Then there exists ǫ 2 and τ 0 > 0 so that for for all ǫ ≤ ǫ 2 and for all vǫ stationary solution of (1.1) we have

c3 ≤ Ψ vǫ (1) ≤ C3 ≤ 2 C3 ≤ Ψ vǫ (1 + τ 0 ).
Moreover there exists ǫ 3 and k > 0 so that for all ǫ ≤ ǫ 3 we have for all vǫ stationary solution of (1.1) and t, s ∈ (0, 1 + τ 0 )

|t -s| ≤ k| Ψ vǫ (t) -Ψ vǫ (s)|.

Proof:

Recall that from the proof of Lemma 7.2 for all ǫ ≤ ǫ 1 , for any stationary solution of (1.1) vǫ , we have α(ω -) ≤ α(v ǫ ) ≤ α(ω + ). So, for all ǫ ≤ ǫ 1 and for all stationary solution of (1.1) we have (7.4) and Lemma 7.2 we get

< vǫ , vǫ > (α(ω -) -σ) ≤ Ψ vǫ (1) ≤ (α(ω + ) + σ) < vǫ , vǫ >, where σ = ǫ ψ ∞ . Let us fix ǫ 2 := min{ǫ 1 , λp 4K ψ ∞ }. From
c3 := c2 1 Kλ p 2N ≤ Ψ vǫ (1) ≤ 3 C2 1 Kλ p 2 =: C3 ,
for all ǫ ≤ ǫ 2 and for any stationary solution of (1.1). By using to the monotonicity of the map σ → ω± and Lemma 7.2, we can choose ǫ 2 smaller if necessary to achieve for any ǫ ≤ ǫ 2 and any stationary solution vǫ , s 0 ω-ǫ1 ≤ s 0 vǫ ≤ s 0 ω+ ǫ1 for all s 0 > 0.

The latter inequalities imply that we have for all ǫ ≤ ǫ 2 and for any stationary solution vǫ ,

Ψ vǫ (s 0 vǫ ) ≥ α(s 0 ω- ǫ1 ) - λ p 4K .
Let us fix s 0 such that α(s 0 ω-ǫ1 ) -

λ p 4K = 2 C3 .
This is always possible since α is monotone increasing and lim µ→∞ α(µω - ǫ1 ) = +∞. By construction, s 0 > 1, since Ψ vǫ is monotone increasing and we have

Ψ vǫ (s 0 ) ≥ 2 Ψ vǫ (1).
Let us denote τ 0 := s 0 -1.

Now since for each i the function Ψ i satisfies (1.2), (1.3) and that α and the

ψ i are C 1 loc (R N ), the function Ψ vǫ (s) := N i=1 Ψ i (sv ǫ )v 2 ǫ,i is C 1 loc (R)
and monotone increasing. Therefore, for any stationary solution of (1.1) vǫ , the function Ψ vǫ is a R + homeomorphism.

Next, we check that for a fixed ǫ and a fixed stationary solution vǫ , the homeomorphism Ψ vǫ is a C 1 diffeomorphism on (0, 1 + τ 0 ) → Ψ vǫ ((0, 1 + τ 0 )). Thanks to the Inverse Function Theorem, to show that Ψ vǫ is a local C 1 diffeomorphism it is sufficient to prove that for all s ∈ (0, 1 + τ 0 ), Ψ ′ vǫ (s) = 0. By a straightforward computation we have:

(Ψ vǫ ) ′ (s) =< ∇α(sv ǫ ), vǫ > E(v) + ǫ N i=1 < ∇ψ i (sv ǫ ), v > v2 ǫ,i .
For all s ∈ (0, 1 + τ 0 ), and for any stationary solution vǫ ∈ Q (1+τ0) C1 (0) we have:

|∇ψ i (sv ǫ )| ≤ sup v∈Q (1+τ 0 ) C1 (0) |∇ψ i (v)| := ζ 1 , ∇α(sv ǫ ) ≤ sup v∈Q (1+τ 0 ) C1 (0) ∇α(v) := ζ 2 > 0, ∇α(sv ǫ ) ≥ inf v∈Q (1+τ 0 ) C1 (0) ∇α(v) := ζ 3 > 0.
Therefore, by Lemma 7.2 we deduce that

(Ψ vǫ ) ′ (s) ≥ c3 1 N ζ 3 -ǫζ 1 C3 1 .
By choosing ǫ ≤ ǫ 3 := min ǫ 2 ,

c3 1 ζ3 2N ζ1 C3 1
, we get for all ǫ, s ∈ (0, 1 + τ 0 ) and for any stationary solution of (1.1) vǫ ,

0 < c3 1 ζ 3 2N ≤ (Ψ vǫ ) ′ (s) ≤ C3 1 ζ 2 + ζ 3 2N .
From the the latter a priori bounds, we see that for all ǫ ≤ ǫ 3 , s ∈ (0, 1 + τ 0 ) and for any stationary solution of (1.1) vǫ , we get the following a priori estimate

1 C3 1 ζ 2 + ζ3 2N ≤ Ψ -1 vǫ ′ [ Ψ vǫ (s)] = 1 (Ψ vǫ ) ′ (s) ≤ 2N c2 1 ζ 3 .
Hence, we deduce that for all ǫ ≤ ǫ 3 , s, t ∈ (0, 1 + τ 0 ) and for any stationary solution of (1.1) vǫ , we have

|s -t| ≤ k| Ψ vǫ (s) -Ψ vǫ (t)|, with k := 2N c2 1 ζ 3 .

Asymptotic Behaviour.

We are now in position to obtain the asymptotic behaviour of the solution v ǫ (t) as t goes to +∞ for ǫ ∈ [0, ǫ * ], where ǫ * is to be determined later on.

Let us fix ǫ ∈ [0, ǫ 1 ] where ǫ 1 is obtained in Lemma 7.2 and let vǫ be a stationary solution of (1.1). For simplicity we denote <, > the standard scalar product in R N .

As in the proof of Lemma 4.7, we start by observing that since vǫ = 0 we can write v ǫ (t) := λ(t)v ǫ + h(t) with for all t, < h, vǫ >= 0. In the sequel of this subsection, for more clarity in the presentation we drop the subscript ǫ on v and v.

First, we note that from this decomposition we can derive the following equalities:

λ < v, v >= N i=1 v i vi , (7.17) dE(h) dt = dE(v) dt -2λλ ′ < v, v > . (7.18)
By Lemma 7.2 and (7.17), we can check that for all ǫ ≤ ǫ 1 and for all t > 0, we have

(7.19) c2 C2 1 ≤ λ(t) ≤ N C2 c2 1 .
Similarly, by using (7.19), Lemma 7.2 and

N i=1 |h i | ≤ λ(t) N i=1 vi + N i=1 v i , we deduce from E(v) = E(h) + λ 2 E(v) that E(h) ≤ N C2 2 c2 1 (N + 1), (7.20) N i=1 |h i | ≤ N C2 c1 1 + C1 c1 . (7.21)
By (7.19),(7.20),(7.21), Lemma 7.2,and by using the Lipschitz regularity of Ψ i and the Cauchy-Schwartz inequality we can check that for some constant C > 0 independent of ǫ and v

N i=1 |Ψ i (λ(t)v) -Ψ i (v(t))|v 2 i ≤ C E(h), (7.22) N i=1 |(ψ i (v) -ψ i (v))v i h i | ≤ C E(h). ( 7
|ψ i (v) -ψ i (v)||h i |v i ≤ N i=1 κ i   N j=1 |v j -v j |   |h i |v i , where κ i are the Lipschitz constant of ψ i on the set Q R0 (0) with R 0 := N C2 C1 c2 1 1 + 2 C1
c1 . From the decomposition of v and by using the Cauchy-Schwartz inequality, we get

N i=1 |(ψ i (v) -ψ i (λ(t)v + h))v i h i | ≤ κ   |1 -λ(t)| N j=1 vj + N j=1 |h j |   N i=1 |h i |v i (7.24) ≤ N κ E(h)E(v) |1 -λ(t)| E(v) + E(h) , (7.25)
where κ := sup i∈{1,...,N } κ i . Therefore, by (7.20) and Lemma 7.2, the inequality (7.23) holds true for some positive constant C independent of ǫ ≤ ǫ 1 . A similar argumentation holds to get (7.22).

Next, we show that Lemma 7.5. There exists ǫ * ≤ min{ǫ 0 , ǫ 1 , ǫ 3 }, so that for all ǫ ≤ ǫ * , E(h ǫ (t)) → 0 as t → +∞.

Assume the lemma holds true, then we can conclude the proof of Theorem 1.4 by arguing as follows.

By combining (7.17) and Lemma 3.2, we achieve

(7.26) λ ′ (t) < v, v >= d dt N i=1 v i vi = 1 K N i=1 Γ i v i vi .
Now by using E(h) → 0, we deduce that h → 0 as t → ∞ and from the latter equality we are reduced to analyse the ODE

λ ′ (t)E(v) = λ(t) K N i=1 (Ψ i (v) -Ψ i (λ(t)v))v 2 i + o(1), where o(1) := N i=1 (Ψ i (λ(t)v) -Ψ i (v))v i v i ,
which by (7.22)

′ (t) = λ(t) KE(v) (θ -Ψ(λ(t))) + λ(t) KE(v) o(1),
which can be rewritten

(7.28) λ ′ (t) = λ(t) KE(v) (θ + o(1) -Ψ(λ(t))),
where θ :=

N i=1 Ψ i (v)v 2 i and Ψ(s) := N i=1 Ψ i (sv)v 2 i . By construction, Ψ ∈ C 1 loc (R)
and is monotone increasing.

The above ODE is of logistic type with a perturbation o(1) → 0 with a non negative initial datum. Therefore, by a standard argumentation, we see that λ(t) converges to λ > 0 where λ is the unique solution of Ψ( λ) = θ. By construction, we have Ψ(1) = θ, so we deduce that λ = 1. Hence, λ(t) → 1 and we can conclude that v converges to v.

Let us now turn our attention to the proof of the Lemma 7.5.

Proof of Lemma 7.5:

First, let us denote

Γ i := Ψ i (v) -Ψ i (v).
Then, by combining (7.17), (7.18)and Lemma 3.2, we achieve

(7.29) λ ′ (t) < v, v >= d dt N i=1 v i vi = 1 K N i=1 Γ i v i vi and dE(h) dt = - N i,j=1 µ ij vi vj h j vj - h j vj 2 + 2 K N i=1 Γ i h 2 i + 2λ K N i=1 Γ i h i vi , = - N i,j=1 µ ij vi vj h j vj - h j vj 2 + 2 K N i=1 Γ i v i h i .
Therefore using the definition of Ψ i and with the notation

γ i := ψ i (v) -ψ i (v), we have dE(h) dt = - N i,j=1 µ ij vi vj h j vj - h j vj 2 + 2 K (α(v) -α(v))E(h) + 2ǫ K N i=1 γ i h 2 i + 2ǫλ(t) K N i=1 γ i vi h i , which implies that (7.30) dE(h) dt ≤ - N i,j=1 µ ij vi vj h j vj - h j vj 2 + 2 K (α(v) + 2ǫ ψ ∞ -α(v))E(h) + 2ǫλ(t) K N i=1 γ i vi h i .
By (7.29), using the definition of Ψ i we also have

d dt N i=1 vi v i = 1 K (α(v) -α(v)) N i=1 vi v i + ǫ K N i=1 γ i vi v i , ≥ 1 K (α(v) -2ǫ ψ ∞ -α(v)) N i=1 vi v i . By Lemma 7.2, for all ǫ ≤ ǫ 1 , c2 ≤ β(v) = N i=1 vi v i ≤ C2 with c2 and C2 independent of ǫ. So we have d dt log(β(v(t))) ≥ 1 K (α(v) -2ǫ ψ ∞ -α(v)),
which combined with (7.30) The proof now will follow several steps:

Step One: Since by (7. Step Two: First, we define some constant quantities: By the previous step, we have that for all ǫ ≤ ǫ * , all t ≥ t 0 E(h) ≤ ǫδ 0 .

δ 0 := 2 C2 c2 4C 4 1 + N C2
We claim that Claim 7.7. For all ǫ ≤ ǫ * there exists t ǫδ0 such that for all t ≥ t ǫδ0 E(h(t)) ≤ ǫδ 0 2 .

Proof:

First, we can check that for ǫ ≤ ǫ * there exists t * so that for all t ≥ t * |1λ(t)| ≤ 2kCǫδ 0 .

Indeed, by (7.28)

4 2 .

 42 |1λ(t)| E(h),whereC 5 := 2 K 4 ψ ∞ + N 2 κ C2 c1 and C 4 := 2 C2N κ K . Recall that C 1 (v) is an eigenvalue, therefore for ǫ small enough, say ǫ ≤ ǫ 4 , since v ∈ Q C √ ǫ (ω + ) and ω+ → ω as ǫ → 0 one has C 1 (v) ≥ C(ω + ) Withthe latter estimate and by choosing ǫ ≤ min(ǫ 1 , ǫ 4 ) smaller if necessary, we get (7.33) dE(h) dt -E(h) d dt log(β 2 (v(t))) ≤ -C 1 (ω + ) 4 E(h) + ǫC 4 |1λ(t)| E(h).

c2 1 C 1

 11 (ω +) .

c2 1 C 1 (

 11 ǫ * := min ǫ 1 , ǫ 2 , ǫ 3 , ǫ 4 C, k are respectively defined in (7.22),(7.23) and in the Lemma 7.4.

  .23) Indeed by (7.19)-(7.21) and Lemma 7.2, λv and h are uniformly bounded and we have

	N
	i=1

  By Lemma 7.2, for all ǫ ≤ ǫ * ≤ ǫ 1 we have c2 ≤ N i=1 vi v i ≤ C2 with c2 and C2 independent of ǫ. Therefore by (7.17) c1 E(v) ≤ λ(t) ≤ C2 E(v)for all t > 0. Thus λ satisfies an ODE of the form λ

		and (7.23) satisfies
	(7.27)	|o(1)| ≤	C(1 + ǫ) K	E(h) → 0	as t → +∞.

  where C 1 (ǫ) is the second largest eigenvalue of some associated linear problem. Now, we estimate the last term of the above inequality. Recall that by (7.25) we have

									leads to
	dE(h) dt	≤ -	N i,j=1	µ ij vi vj	h j vj	-	h j vj	2	+	d dt	log(β 2 (v(t)))E(h) +	8ǫ ψ ∞ K	E(h) +	2ǫλ(t) K	N i=1	γ i vi h i .
	By using the functional inequality, Lemma 5.3, and rearranging the terms in the above inequality
	we get
	(7.31)	dE(h) dt	-E(h)	d dt	log(β 2 (v(t))) ≤ -C 1 (v ǫ ) +	8ǫ ψ ∞ K	E(h) +	2ǫλ(t) K	N i=1	γ i vi h i ,
	By combining the above estimate and (7.31), we achieve
	(7.32)		dE(h) dt	-E(h)	d dt	log(β

N i=1 |γ i vi h i | ≤ N κ E(h)E(v) |1λ(t)| E(v) + E(h) . 2 (v(t))) ≤ (-C 1 (v ǫ ) + ǫC 5 ) E(h) + ǫC

  For all ǫ ≤ min(ǫ 1 , ǫ 4 ), there exists t 0 so that for all t ≥ t 0 we have and we can check that there exists t 0 > 0 so that E(h(t 0 )) ≤ 2 (v(t)) is a decreasing function which by Lemma 7.2, is bounded from below. Moreover, we have for all t > 0 E(h(t)) > 2. Therefore F converges as t tends to +∞ and dF dt → 0. Thus for t large enough, we get the contradiction Let us denote t * := inf Σ. So at t * we haveE(h(t * )) = 2 (v(t)) is a decreasing function of t for all t ∈ Σ. By arguing on each connected component of Σ, we can check that for all t ∈ Σ

	Thus F (t) := log	E(h) β 2 ǫ4C4 1+	N C2 c2 1
													C1(ω + )
	-Let Σ be the set Σ := 1 2 ǫC 4 1 +	N C2 c2 1    t > t 0 | E(h(t)) > 2 ≤ E(h) d dt log ǫ4C4 1+ E(h) β 2 (v(t)) N C2 c2 1 C1(ω + )	≤ ǫC 4 1 +    . Assume that Σ is non empty N C2 . c2 1
	otherwise the claim is proved since C2 N C2 c2 > 1. ǫ4C4 1+ c2 1 C1(ω + ) . Again, by dividing(7.34) by E(h) and rearranging the terms, on the set Σ we have
	(7.36)			E(h)	d dt	log		E(h) β 2 (v(t))	≤ -	C 1 (ω + ) 4	E(h) + ǫC 4 1 +	N C2 c2 1	≤ 0.
	Thus log	E(h) β 2 E(h(t)) ≤	c2 β(v(t))	ǫ4C 4 1 + N C2 c2
												E(h) ≤ 2	C2 c2	ǫ4C 4 1 + N C2 c2 1 C 1 (ω + )	.
	Hence, since C2 c2 > 1 we get for all t ≥ t 0 ,
												E(h) ≤ 2	c2 C2	ǫ4C 4 1 + N C2
												E(h) ≤ 2ǫ	C2 c2	4C 4 1 + N C2 c2 1 C 1 (ω + )
	Indeed, by (7.17), and Lemma 7.2 we have
	(7.34)		dE(h) dt	-E(h)	d dt	log(β 2 (v(t))) ≤ -	C 1 (ω + ) 4	E(h) + ǫC 4 1 +	N C2 c2 1	E(h)
													N C2 c2 1 C1(ω + ) ǫ4C4 1+	. If not, then for all
	t > 0 E(h(t)) > 2 get that	N C2 c2 1 C1(ω + ) ǫ4C4 1+	and by dividing (7.34) by E(h) and rearranging the terms, we
	(7.35)	E(h)	d dt	log	E(h) β 2 (v(t))	≤ -	C 1 (ω + ) 4	E(h) + ǫC 4 1 +	N C2 c2 1	< -ǫC 4 1 +	N C2 1 c2	.

[START_REF] O; Diekmann | The dynamics of adaptation: An illuminating example and a hamilton-jacobi approach[END_REF]

we have |1λ(t)| ≤ 1 + N C2 c2 1

for all t, we claim that Claim 7.6.

1 C 1 (ω + )

.

Therefore by using Lemma 7.2 for t ≥ t * we have

  Cǫδ 0 for t ≥ t 0 . Let λ ±Cǫδ0 be the solution of the ODE Cǫδ 0 -Ψ vǫ (λ ±Cǫδ0 (t))).Therefore λ ±Cǫδ0 → λ±Cǫδ0 where λ±Cǫδ0 are the unique positive solutions ofΨ vǫ ( λ±Cǫδ0 ) = Ψ vǫ (1) ± Cǫδ 0 .Thanks to the strict monotonicity of Ψ vǫ , we also haveλ-2Cǫδ0 < λ-Cǫδ0 < λCǫδ0 < λ2Cǫδ0 ,where λ±2Cǫδ0 are the unique positive solutions ofΨ vǫ ( λ±2Cǫδ0 ) = Ψ vǫ (1) ± 2Cǫδ 0 . Since ǫ ≤ ǫ * ≤ c3 4Cδ0 , it follows that c3 2 ≤ Ψ vǫ (1) ± 2Cǫδ 0 ≤ 3 C32 and therefore, by Lemma (7.4), we have(7.42) 0 < λ±2Cǫδ0 < 1 + τ 0 .By (7.42) and Lemma 7.4, since ǫ ≤ ǫ * ≤ ǫ 3 , we deduce that for t ≥ t * ,|1λ(t)| ≤ k sup{| Ψ vǫ (1) -Ψ vǫ ( λ-2Cǫδ0 )|, | Ψ vǫ (1) -Ψ vǫ (λ +2Cǫδ0 )|} ≤ 2kCǫδ 0 .From the latter estimate, by using (7.33), we see that for t ≥ t * + 2ǫ 2 C 4 kCδ 0 E(h).By following the argumentation ofStep one, we can show that there exists t ǫδ0 such that for t ≥ t ǫδ0 we have Since for all t ≥ t ǫδ0 , by arguing as in the proof of Claim 7.7, we show that for all ǫ ≤ ǫ * there exists t ǫ δ 0By reproducing inductively the above argumentation, for all ǫ ≤ ǫ * we can construct a sequence (t n ) n∈N so that for all t ≥ t n we have E(h(t)) ≤ ǫδ 0 2 n . Hence, for all ǫ ≤ ǫ * we deduce that

	since ǫδ 0 < d 4kC .
	and (7.27), we have λ ′ (t) = KE(v) λ(t) λ ′ ±Cǫδ0 (t) = λ ±Cǫδ0 (t) KE(v) 4Cδ0 , for t ≥ t 0 we have with o(1) ≤ (7.41) (7.40) 3c 3 4 ≤ Ψ vǫ (1) ± Cǫδ 0 ≤ (θ + o(1) -Ψ(λ(t))), 4 . 5 C3 ( Ψ vǫ (1) ± Since ǫ ≤ c3 Step Three: E(h(t)) ≤ ǫδ 0 , 2 2 t ≥ t ǫ δ 0 E(h) ≤ ǫδ 0 4 .	2	so that for all
	lim t→∞	E(h(t)) → 0.
	Now, recall that for t ≥ t 0 , λ(t) satisfies λ ′ (t) ≤ λ(t) KE(v) ( Ψ vǫ (1) + Cǫδ 0 -Ψ vǫ (λ(t))),
	λ ′ (t) ≥ ( Ψ (7.43) λ(t) KE(v) dE(h) dt -E(h) d dt log(β 2 (v(t))) ≤ -E(h) E(h) ≤ C 1 (ω + ) 4 2kCǫ 2 δ 2 0 . d Hence, we have for all t ≥ t ǫδ0 E(h) ≤ ǫδ 0 2 ,

vǫ (1) -Cǫδ 0 -Ψ vǫ (λ(t))).

Thus, by a standard argumentation, we can show that for t ≥ t * we have

λ-2Cǫδ0 ≤ λ(t) ≤ λ+2Cǫδ0 .

Therefore, for t ≥ t * we have

|1λ(t)| ≤ sup{|1 -λ-2Cǫδ0 |, |1λ +2Cǫδ0 |}.

Acknowledgements. J. Coville thanks the member of the project ERBACE of INRIA for early discussion on this subject. F. Fabre is supported by the project VirAphid of the Agence Nationale de la Recherche (project ANR-10-STRA-0001).

A CASE OF INTEREST

In this section we analyse more precisely the dynamics of the solution v of (1.1) when the interactions Ψ i take the form Ψ i (v) := N j=1 r j v j . We prove the Theorem 1.2 that we recall below.

Theorem 5.1. Assume that the interactions Ψ i take the form Ψ i (v) = N j=1 r j v j then for any nonnegative initial datum v(0) not identically zero, the solution v(t) of (1.5) converges exponentially fast to its unique equilibrium. That is to say there exists two positive constants C 1 and C 2 so that

Before proving this Theorem we establish two auxiliary results that for convenience we present in two separate subsections. We prove Theorem 1.2 at the end of this section.

Study of the evolution of the total population.

Let us denote N (t) = N i=1 v i the total population. A straightforward computation shows that for the interactions Ψ i of the form Ψ i (v) := N j=1 r j v j we see that N (t) satisfies the equation :

which written with the new variable P (t) = K -N (t) takes the form

The dynamic of the above equation is strongly related to the behaviour of α(t) and we have

Lemma 5.2. For any nonnegative initial datum v(0) not identically zero, N (t) converges exponentially fast toward its unique equilibrium K. Moreover N satisfies identically (5.4) min{N min (t), N max (t)} ≤ N (t) ≤ max{N min (t), N max (t)},

where N min and N max are the solutions of the logistic equations:

with respectively ξ -= min{r 1 , . . . , r N } and ξ + = max{r 1 , . . . , r N }

Proof:

Assume for the moment that (5.4) holds then the convergence exponentially fast to K is a straightforward consequence of (5.3). Indeed by (5.4) we deduce that R min min(N min , N max ) ≤ α(s) ≤ R max min(N min , N max ) where R max := max{r 1 , . . . , r N } and R min := min{r 1 , . . . , r N }. Therefore α(s) > C 0 , since N min and N max converge to K.

To obtain (5.4) we investigate the following three cases, N (0) = K, N (0) < K and N (0) > K.

In the first case N (0) = K, we see that (5.4) holds true trivially, since N max = N min = N (t) ≡ K for all t. Let us now investigate the two other situations. The argumentation in both situation being similar we expose only the case N (0) < K.

In this situation, N min and N max being the solutions of logistic equations they are increasing functions. Moreover wa have N min (t) ≤ N max (t) < K for all t. On another hand, by continuity of N (t), there exists also t 1 > 0 so that N (t) < K on [0, t 1 ). Furthermore on (0, t 1 ) we can see that N (t) satisfies the following differential inequalities