Bilel Elbetch

How does

Keywords: Population Dynamics, Nonlinear diffusion, Migration rate, Source-sink, Logistic growth, Global stability, Slow-fast systems, Tikhonov's theorem Mathematics Subject Classification: 37N25 92D25 34D23 34D15 dispersion . . . . . . . . . . . .

How does nonlinear asymmetric dispersal affect the dynamics of a population and the coexistence of species in Two-patch source-sink heterogeneous environment?

Bilel Elbetch

To cite this version:

Bilel Elbetch. How does nonlinear asymmetric dispersal affect the dynamics of a population and the coexistence of species in Two-patch source-sink heterogeneous environment?. 2023.

Email : elbetchbilel@gmail.com

Introduction

Population dynamics is a wide field of mathematics, which contains many problems, for example fragmentation of population and the effect of migration in the general dynamics of population. Bibliographies can be found in the work of Levin [START_REF] Levin | Spatial patterning and the structure of ecological communities[END_REF][START_REF]Patch Dynamics[END_REF] and Holt [START_REF] Holt | Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution[END_REF]. There are ecological situations that motivate the representation of space as a finite set of patches connected by migrations, for instance an archipelago with bird population and predators. It is an example of insular bio-geography. A reference work on mathematical models is the book of Levin et al. [START_REF]Patch Dynamics[END_REF], whereas Hanski and Gaggiotti [START_REF]Ecology, Genetics and Evolution of Metapopulations[END_REF] give a more ecological account of the subject. The standard question in this type of biomathematical problems, is to study the effect of migration on the general population dynamics, and the consequences of fragmentation on the persistence or extinction of the population.

An ecological model is used to describe population changes in two habitats, both occupied by the same species. One habitat is of high quality and allows a population to increase, leading to a surplus. This is the source. The other habitat is of low quality, leading to a deficit that ordinarily would lead to the habitat being abandoned. This is the sink. The source-sink model was first proposed by biologist Crick [START_REF] Crick | Diffusion in Embryogenesis[END_REF]. In population ecology, the source-sink model is used to describe how variation in habitat quality may affect the population growth or decline of organisms.

In 2019, Wu et al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF] studied the following two-patch source-sink model:

       dx 1 dt = r 1 x 1 1 - x 1 K 1 + (x 2 -sx 1 ), dx 2 dt = r 2 x 2 -1 - x 2 K 2 + (sx 1 -x 2 ), (1)
where x 1 and x 2 represent population densities of the species in patch 1 and 2, respectively. The parameters r i > 0 and K i > 0 are respectively the intrinsic growth rate and the carrying capacity of patch i. Parameter represents the dispersal intensity while the parameter s reflects the dispersal asymmetry. The authors show that the dispersal asymmetry can lead to either an increased total size of the population in two patches, a decreased total size with persistence in the patches, or even extinction in both patches. They show also that for a large growth rate of the species in the source and a fixed dispersal intensity: (i) If the asymmetry is small, the population would persist in both patches and reach a density higher than that without dispersal and the population approaches its maximal density at an appropriate asymmetry, (ii) If the asymmetry is intermediate, the population persists in both patches but reaches a density less than that without dispersal, (iii) If the asymmetry is large, the population goes to extinction in both patches.

Arino et al. [START_REF] Arino | Number of Source Patches Required for Population Persistence in a Source-Sink Metapopulation with Explicit Movement[END_REF] also studied a source-sink model of n patches, where the source patch follows a logistic growth rate, and the sink patch follows exponential decay and they proved the existence of a threshold number of source patches such that the population potentially becomes extinct below the threshold and persistent above the threshold.

In [START_REF] Allen | Persistence and extinction in single-species reaction-diffusion models[END_REF], Allen considered the n-patch general model given by the following equations:

dx i dt = r i x i 1 - x i K i + Γ i (x), i = 1, . . . , n, (2)
where r i and K i are positive constants; x = (x 1 , . . . , x n ) T where x i represents the population density in the i-th patch. The function Γ i represent one of the three types of different mechanisms. The mechanism for linear diffusion is given by:

Γ i (x) = n j=1,j =i ij (x j -θ ij x i ) , , i = 1, . . . , n, (3)
where ij and θ ij are positive constants. Dispersal by linear diffusion implies that the species is able to move to all locations within its environment with equal probability.

The mechanism for biased diffusion is given by:

Γ i (x) = n j=1,j =i ij x i (x j -θ ij x i ) , , i = 1, . . . , n, (4)
where ij and θ ij are positive constants. Note that, the term 'biased' means that the diffusion rate is a function of population density. The diffusion rate is regulated by population density, increasing for large populations and decreasing for small populations. The third type of mechanism, is the directed diffusion which is formulated by Gurney and Nisbet [START_REF] Gurney | The regulation of inhomogeneous populations[END_REF], given by:

Γ i (x) = n j=1,j =i ij x 2 j -θ ij x 2 i , , i = 1, . . . , n, (5)
where ij and θ ij are positive constants. Dispersal by directed diffusion implies that the individuals move from high population concentration to low ones, i.e., the movement is a function of species density. For more information on the biological interpretation and also the continuous version of those types of diffusion, we refer the readers to [START_REF] Allen | Persistence and extinction in Lotka-Volterra reaction-diffusion equations[END_REF] and [START_REF] Gurney | The regulation of inhomogeneous populations[END_REF]. The objective of the work of Allen [START_REF] Allen | Persistence and extinction in single-species reaction-diffusion models[END_REF] is to study the effect of different types of the dispersion on the persistence and extinction of the species. The persistence and extinction behavior is completely determined in a two-patch model [START_REF] Allen | Persistence and extinction in single-species reaction-diffusion models[END_REF](3) for n = 2 ( see Theorem 1 in [START_REF] Allen | Persistence and extinction in single-species reaction-diffusion models[END_REF]). For model (2)(4), Allen [2, Theorem 2] showed that a population modelled with biased diffusion is always persistent and in fact represents a strongly persistent population. For more details on the results of persistence and extinction, see Theorem 3 of [START_REF] Allen | Persistence and extinction in single-species reaction-diffusion models[END_REF] for n-patch model (2)(4) and Proposition 1 of [START_REF] Allen | Persistence and extinction in single-species reaction-diffusion models[END_REF] for 2-patch case; Theorem 4 of [START_REF] Allen | Persistence and extinction in single-species reaction-diffusion models[END_REF] for the n-patch model (2)(5).

In [START_REF] Elbetch | Nonlinear diffusion in the multi-patch logistic model[END_REF], Elbetch et al. considered the following two-patch model coupled by nonlinear terms of migration:

       dx 1 dt = r 1 x 1 1 - x 1 K 1 + θ 2 x 2 2 -θ 2 x 1 2 , dx 2 dt = r 2 x 2 1 - x 2 K 2 + θ 1 x 1 2 -θ 2 x 2 2 . (6)
For system [START_REF] Arino | Number of Source Patches Required for Population Persistence in a Source-Sink Metapopulation with Explicit Movement[END_REF], the authors given a complete classification of the model parameter space as to whether nonlinear dispersal is beneficial or detrimental to the sum of two carrying capacities. They studied also the model [START_REF] Arino | Number of Source Patches Required for Population Persistence in a Source-Sink Metapopulation with Explicit Movement[END_REF] in the case where one growth rate is much larger than the second one. For general information on the effect of asymmetric and nonlinear diffusion on the total population abundance for the n-patch model, the reader is referred to [START_REF] Elbetch | Nonlinear diffusion in the multi-patch logistic model[END_REF].

Our aim of the present paper, is to study the effect of the nonlinear asymmetric migration i.e. when the individuals move from high population concentration between patches, on the total population and on the persistence/ extinction for two-patch source-sink model ( Model ( 26)), and we compare our results with those obtained by Wu et al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF] for two-patch source-sink model with linear asymmetric dispersal (Model ( 7)), and also the results of Arditi et al. [START_REF] Arditi | Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation[END_REF][START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF] for two-patch logistic model.

This paper is organized as follows: In Section 2, some proprieties of Two-patch source-sink model with linear asymmetric dispersal (1) have been recalled as a function of the two parameters θ 1 and θ 2 , Theorem 2.1 for the global dynamic, Theorem 2.2

for the effect of linear asymmetric dispersal on the dynamic of the total equilibrium population in two patches, Proposition 2.1 give the perfect mixing formula and our first result for this model is given in Theorem 2.3, which I studied the model in two time scales different used Tykhonov's theorem [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF][START_REF] Lobry | On Tykhonov's theorem for convergence of solutions of slow and fast systems[END_REF][START_REF] Tikhonov | Systems of differential equations containing small parameters in the derivatives[END_REF][START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF]. In Section 3, we introduce the two-patch source-sink model coupled by nonlinear asymmetric terms of migration. In Subsection 3.1, we prove their global stability in the interior of the positive cone. Next, in Subsection 3.2, we study the behavior of the model in the case when the migration rate goes to infinity using singular perturbation arguments and Tykhonov's theorem [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF][START_REF] Lobry | On Tykhonov's theorem for convergence of solutions of slow and fast systems[END_REF][START_REF] Tikhonov | Systems of differential equations containing small parameters in the derivatives[END_REF][START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF] ( see Theorems 3.1 and 3.2 ). In Subsection 3.3, we compare the total equilibrium population with the carrying capacity of source patch for all parameter space by using the same method as in the works of Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF] and Wu et al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF] ( see Theorem 3.3 and Corollary 3.2 ). In subsections 3.4 and 3. give also the detailed proof of the existence and stability of the equilibrium point of [START_REF] Allen | Persistence and extinction in Lotka-Volterra reaction-diffusion equations[END_REF]. In Appendices B and C, we have calculated the derivatives of the total equilibrium population of (7), T * 1 , and also of (26), T * 2 respectively, with respect of .

Some preliminary results for Two-patch source-sink model with linear dispersion

In this section, we consider the 2-patch system with source-sink dynamics given by:

       dx 1 dt = r 1 x 1 1 - x 1 K 1 + (θ 2 x 2 -θ 1 x 1 ) , dx 2 dt = r 2 x 2 -1 - x 2 K 2 + (θ 1 x 1 -θ 2 x 2 ) , (7)
where x 1 and x 2 represent population densities of the species in patch 1 and 2, respectively. Patch 1 is assumed to be the source but patch 2 is the sink, i.e. r 1 , r 2 > 0.

The parameters α i := r i /K i are the intraspecific competition degree. Parameter represents the dispersal intensity. We denote θ 2 the migration rate from source patch to the sink patch and θ 1 from sink patch to source patch, the dispersal is symmetric if θ 1 = θ 2 . This system is studied in [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF]. We recall some essential results of [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF] as function of the parameters θ 1 and θ 2 . Note that, Wu et al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF] studied the model ( 1)

as function of s = θ 1 /θ 2 .
First of all, let's start by recalling the global dynamics of System (7).

Global dynamics

We consider the following regions in the set of parameters θ 1 and θ 2 , denoted D 0 , D 1 and D 2 depicted in Figure 1 and defined by:

               D 0 = (θ 1 , θ 2 ) : θ 2 ≥ r 2 r 1 θ 1 , D 1 = (θ 1 , θ 2 ) : r 2 r 1 θ 1 < θ 2 < r 2 r 1 r 2 + r 1 θ 1 , D 2 = (θ 1 , θ 2 ) : θ 2 ≥ a 2 r 1 r 2 + r 1 θ 1 . (8)
0

D 1 D 0 D 2 θ 2 = r2 r1 θ 1 θ 1 θ 2 θ 2 = r2 r1r2+ r1 Figure 1.
Global stability of Model [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF]. In D 0 and D 1 the system has unique equilibrium E * 1 ( ) which is GAS. In the region D 2 , the system has the origin as unique equilibrium which is GAS.

The global dynamic of System ( 7) is described as follows.

Theorem 2.1 (Prop. 5.5 in [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF]). Consider Model [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF]. Then, if (θ 1 , θ 2 ) ∈ D 0 ∪ D 1 , System (7) admits unique equilibrium in R 2 \{0} denoted E * 1 ( ), which is is globally asymptotically stable (GAS), and if (θ 1 , θ 2 ) ∈ D 2 , then the origin is GAS.

Total population abundance

In this section, we recall the comparison given in [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF]Proposition 5.11], between the total equilibrium population

T * 1 ( ) = x * 1 ( ) + x * 2 ( ), E * 1 ( ) = (x * 1 ( ), x * 2 ( )),
of [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF] and carrying capacity K 1 , by analyzing the stable positive equilibrium E * 1 ( ).

Note that, when there is no dispersal (i.e., = 0), the total equilibrium population is

T * 1 (0) = K 1 .
We consider the regions in the set of the parameters θ 1 and θ 2 , denoted L 0 , L 1 , L 2 , L 3 and L 4 , depicted in Figure 2 and defined by:

                             If r 2 ≥ r 1 then      L 0 = (θ 1 , θ 2 ) : θ2 θ1 ≤ r2 r1 , L 1 = (θ 1 , θ 2 ) : θ2 θ1 > r2 r1 . If r 2 < r 1 then            L 2 = (θ 1 , θ 2 ) : θ2 θ1 ≤ r2 r1 , L 3 = (θ 1 , θ 2 ) : r2 r1 < θ2 θ1 < α2(K1+K2) r1-r2 , L 4 = (θ 1 , θ 2 ) : θ2 θ1 ≥ α2(K1+K2) r1-r2 . (9)
Case r 2 ≥ r 1 .

0

L 1 L 0 θ 1 θ 2 θ2 θ1 = r2 r1 Case r 2 < r 1 . 0 L 4 L 3 L 2 θ 1 θ 2 θ2 θ1 = α2(K1+K2) r1-r2 θ2 θ1 = r2 r1 Figure 2.
Qualitative properties of source-sink model [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF]. In L 0 and L 1 the effect is detrimental with extinction in two patches for L 0 and persistence for L 1 . In L 2 and L 3 , the effect is beneficial for < 0 and detrimental for > 0 with persistence of the population in the region L 2 and extinction in the region L 3 . In L 4 , patchiness has a beneficial effect on the total equilibrium population.

Theorem 2.2. The total equilibrium population described by [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF] satisfies the following properties (1) If r 2 ≥ r 1 , let L 0 and L 1 be defined by [START_REF] Elbetch | The multi-patch logistic equation[END_REF]. Then we have: θ2r1r2 θ1r2-θ2r1 , such that:

0 < T * 1 ( ) ≤ K 1 If < * , T * 1 ( ) = 0 If ≥ * . ( 10
) • if (θ 1 , θ 2 ) ∈ L 1 then 0 < T * 1 ( ) ≤ K 1 for all ≥ 0.
(2) If r 2 < r 1 , let L 2 , L 3 and L 4 be defined by [START_REF] Elbetch | The multi-patch logistic equation[END_REF]. Then we have:

(a) if (θ 1 , θ 2 ) ∈ L 2 then T * 1 ( ) > K 1 for < 0 and T * 1 ( ) < K 1 for all > 0 ,
where

0 = (r 1 -r 2 ) (K 1 + K 2 ) (θ 2 (r 1 -r 2 ) + θ 1 α 1 (K 1 + K 2 )) (α 1 -1 + α 2 -1 )
,

with α i = r i /K i . (11) Moreover, there is * ≥ 0 such that T * 1 ( ) = 0 for all ≥ * . (b) if (θ 1 , θ 2 ) ∈ L 3 then we have T * 1 ( ) ≥ K 1 If ≤ * , 0 < T * 1 ( ) < K 1 If > * . ( 12
) (c) if (θ 1 , θ 2 ) ∈ L 4 , then T * 1 ( ) ≥ K 1 for all ≥ 0.
Proof. All the results were established by Wu at al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF]Proposition 5.11]. Note that, the explicit expression (11) of 0 was not given in [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF].

In biological terms, the results of the previous theorem for source-sink patch-model shows that, the nonlinear asymmetry dispersal can lead to an increased total size of the species in two patches, a decreased total size with persistence in the patches, and even extinction in both patches. Comparing these results with that of [START_REF] Arditi | Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation[END_REF][START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF] for 1-source 1-source patch model, we deduce that the existence of a sink patch among the two patches, can cause an extinction of the total population in the two patches.

Let us explain the result of Theorem 2.2 in the particular case where the dispersion is symmetric i.e. θ1 θ2 = 1. Note that, this case is not initialized by Wu et al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF]. We consider the regions in the set of the parameters r 1 and r 2 , denoted Λ 0 , Λ 1 and Λ 2 , depicted in Figure 3 and defined by:

         Λ 0 = {(r 1 , r 2 ) : r 2 ≥ r 1 } , Λ 1 = (r 1 , r 2 ) : r 1 < r 2 < K2 K1+2K2 r 1 , Λ 2 = (r 1 , r 2 ) : r 2 ≥ K2 K1+2K2 r 1 . (13)
Corollary 2.1. Consider the model [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF] with θ 1 = θ 2 = 1. Let Λ 0 , Λ 1 and Λ 2 be defined by [START_REF] Elbetch | Effect of dispersal in single-species discrete diffusion systems with source-sink patches[END_REF]. The total equilibrium population T * 1 of (7) satisfies the following properties

(1) if (r 1 , r 2 ) ∈ Λ 0 then, T * 1 ( ) ≤ K 1 for all . More over, there is * = r1r2 r2-r1 , such that:

0 < T * 1 ( ) ≤ K 1 If < * , T * 1 ( ) = 0 If ≥ * . ( 14
) 0 Λ 0 Λ 1 Λ 2 θ 1 θ 2 r 2 = r 1 r 2 = K2 K1+2K2 r 1 Figure 3.
Qualitative properties of source-sink model [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF] with θ 1 = θ 2 = 1. In Λ 0 , the effect is detrimental with extinction in two patches. In Λ 1 , the effect is beneficial for < * * and detrimental for > * * with persistence of the population. In Λ 2 , patchiness has a beneficial effect on the total equilibrium population.

(2) if (r 1 , r 2 ) ∈ Λ 1 then, there is * * > 0, such that

T * 1 ( ) ≥ K 1 If ≤ * * , 0 < T * 1 ( ) < K 1 If > * * . ( 15
)
Moreover, T * 1 ( ) → T * 1 (∞) > 0 when → ∞. (3) if (r 1 , r 2 ) ∈ Λ 2 , then T * 1 ( ) ≥ K 1 for all ≥ 0. Moreover, if r1 r2 = K2 K1+2K2 , then T * 1 (∞) = K 1 for all .
Proof. Consequence direct of the theorem 2.2.

Perfect mixing

In the case of perfect mixing (i.e when → ∞), we have the following result [34, Proposition 5.10]:

Proposition 2.1. We have:

T * 1 (∞) := lim →∞ T * 1 ( ) =    (θ 1 + θ 2 ) θ 2 r 1 -θ 1 r 2 θ 2 2 r 1 /K 1 + θ 2 1 r 2 /K 2 if θ 1 /θ 2 < r 1 /r 2 , 0 otherwise. (16)
Or, as function of s = θ1 θ2 , (16) becomes

T * 1 (∞) =    (1 + s) r 1 -sr 2 r 1 /K 1 + s 2 r 2 /K 2 if s < r 1 /r 2 , 0 otherwise. ( 17
)
We can use the theory of singular perturbations and Tikhonov's theorem [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF][START_REF] Lobry | On Tykhonov's theorem for convergence of solutions of slow and fast systems[END_REF][START_REF] Tikhonov | Systems of differential equations containing small parameters in the derivatives[END_REF][START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF] to obtain a better understanding of the behaviour of the system (7) in the case of perfect mixing and we find again the formula [START_REF] Freedman | Global stability and predator dynamics in a model of prey dispersal in a patchy environment[END_REF]. In the following theorem, we present our first result in this work.

Theorem 2.3. Consider the system [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF]. Let (x 1 (t, ), x 2 (t, )) be the solution of the system (7) with initial condition (x 0 1 , x 0 2 ) satisfying x 0 i ≥ 0 for i = 1, 2. Let Z(t) be the solution of the logistic equation

   dX dt = rX 1 - X (θ 1 + θ 2 )K , X(0) = x 0 1 + x 0 2 , (18)
where r = θ2r1-θ1r2

2 and K = θ2r1-θ1r2 θ 2 2 α1+θ 2 1 α2
, with α i = ri Ki . Then, when → ∞, we have

x 1 (t, ) + x 2 (t, ) = Z(t) + o (1), uniformly for t ∈ [0, +∞) (19)
and, for any t 0 > 0, we have

     x 1 (t, ) = θ 2 θ 1 + θ 2 Z(t) + o (1), x 2 (t, ) = θ 1 θ 1 + θ 2 Z(t) + o (1) uniformly for t ∈ [t 0 , +∞). ( 20
)
Proof. Let X(t, ) = x 1 (t, ) + x 2 (t, ). We rewrite the system (7) using the variables (X, x 1 ). One obtains:

       dX dt = r 1 x 1 1 - x 1 K 1 -r 2 (X -x 1 ) 1 + X -x 1 K 2 , dx 1 dt = r 1 x 1 1 - x 1 K 1 + (-(θ 1 + θ 2 )x 1 + θ 2 X) . (21)
When → ∞, ( 21) is a slow-fast system, with one slow variable, X, and one fast variable x 1 . According to Tikhonov's theorem [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF][START_REF] Lobry | On Tykhonov's theorem for convergence of solutions of slow and fast systems[END_REF][START_REF] Tikhonov | Systems of differential equations containing small parameters in the derivatives[END_REF][START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF] we consider the dynamics of the fast variable in the time scale τ = t. One obtains

dx 1 dτ = 1 r 1 x 1 1 - x 1 K 1 + -(θ 1 + θ 2 )x 1 + θ 2 X.
In the limit → ∞, we find the fast dynamics

dx 1 dτ = -(θ 1 + θ 2 )x 1 + θ 2 X. (22)
The slow manifold is formed by the equilibrium points of the fast equation [START_REF] Gurney | The regulation of inhomogeneous populations[END_REF], which

given by:

x * 1 = θ 2 θ 1 + θ 2 X. (23)
Since x * 1 is locally asymptotically stable (LAS) for the system [START_REF] Gurney | The regulation of inhomogeneous populations[END_REF], the theorem of Tikhonov ensures that after a fast transition toward the slow manifold, the solutions of ( 21) are approximated by the solutions of the reduced model which is obtained by

dX dt = r 1 θ 2 θ 1 + θ 2 X     1 - θ 2 θ 1 + θ 2 X K 1     -r 2 θ 1 θ 1 + θ 2 X     1 + θ 1 θ 1 + θ 2 X K 2     , (24)
which gives the equation [START_REF] Gao | Fast diffusion inhibits disease outbreaks[END_REF]. If θ 2 r 1 > θ 1 r 2 , (18) admits X * = (θ 1 + θ 2 )K as a positive equilibrium point, which is LAS in the positive axis, and if

θ 2 r 1 ≤ θ 1 r 2 , ( 18
)
admits the origin as unique equilibrium point, which is LAS. The approximation given by Tikhonov's theorem holds for all t ≥ 0 for the slow variable and for all t ≥ t 0 > 0 for the fast variable, where t 0 is small as we want. Therefore, let Z(t) be the solution of the reduced model ( 24) of initial condition Z(0) = X(0, ) = x 0 1 + x 0 2 , then, when → ∞, we have the approximations ( 19) and ( 20).

Wu et al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF] proved that large dispersal intensity (i.e., → ∞), the intermediate asymmetry θ 1 /θ 2 can lead to population density higher than that without dispersal, and extremely small asymmetry is still favorable, while extremely large asymmetry is unfavorable: (i) When the dispersal asymmetry is small, the species can approach a density larger than that without dispersal, while it reaches its maximum value at [START_REF] Freedman | Global stability and predator dynamics in a model of prey dispersal in a patchy environment[END_REF]. We have:

an intermediate asymmetry θ1 θ2 = r1-r2 2α2(K1+K2) . (ii) When θ 1 /θ 2 is
T * 1 (+∞)        > K 1 if θ1 θ2 < r1-r2 α2(K1+K2) , = K 1 if θ1 θ2 = r1-r2 α2(K1+K2) , < K 1 if θ1 θ2 > r1-r2 α2(K1+K2) . (25)
Moreover, T * 1 (+∞) approaches its maximum value

θ 2 2 r 2 K 1 K 1 + K 2 4(θ 2 2 r 1 K 2 + θ 2 1 r 2 K 1 ) r 1 -r 2 α 2 (K 1 + K 2 ) 2 at θ1 θ2 = r1-r2 2α2(K1+K2) .

Two-patch source-sink model with nonlinear dispersion

In this section, we consider the two-patch system with source-sink dynamics coupled by nonlinear terms of migration ( see Fig. 4), which can be written in the following form:

       dx 1 dt = r 1 x 1 1 - x 1 K 1 + θ 2 x 2 2 -θ 1 x 1 2 , dx 2 dt = r 2 x 2 -1 - x 2 K 2 + θ 1 x 1 2 -θ 2 x 2 2 , (26)
where x 1 and x 2 represent population densities of the species in patch 1 and 2, respectively. Patch 1 is assumed to be the source but patch 2 is the sink, i.e. r 1 , r 2 > 0.

The parameters α i := r i /K i > 0 are the intraspecific competition degree. Parameter represents the dispersal intensity. We denote θ 1 the migration rate from source patch 1 to the sink patch 2 and θ 2 from sink patch 2 to source patch 1, the dispersal is

symmetric if θ 1 = θ 2 . θ 1 x 2 1 θ 2 x 2 2 dx 1 dt = r 1 x 1 1 - x 1 K 1 dx 2 dt = r 2 x 2 -1 - x 2 K 2 Figure 4.
A Tow-patch source-sink coupled system

Global stability

According to Appendix A, model ( 26) has always a unique positive equilibrium, again

denoted by E * 2 ( ) := (x * * 1 ( ), x * * 2 (
)) which given by the positive intersection between the ellipse E 1 and the hyperbola H ( see Fig. 5), i.e satisfies to

       r 1 x * * 1 ( ) 1 - x * * 1 ( ) K 1 -r 2 x * * 2 ( ) 1 + x * * 2 ( ) K 2 = 0, x * * 2 ( ) = 1 θ 2 θ 1 (x * * 1 ) 2 ( ) - r 1 x * * 1 ( ) 1 -x * * 1 ( ) K1 (27)
Note that, the origin is always unstable. The equilibrium

E * 2 is GAS in R 2 \ {0} (see Appendix A).
Comparing these stability results with those of Wu et al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF] for the system (1), we conclude that the origin cannot be globally asymptotically stable as in the case of linear diffusion. Biologically speaking, dispersal with high concentration between patches ensures persistence in the two patches.

We thus define the total equilibrium population at the positive equilibrium under dispersal rate , i.e.

T * 2 ( ) = x * * 1 ( ) + x * * 2 ( ), (28)
as the total realized asymptotic population abundance.

Our aim in this work is to study Two-patch source-sink model coupled by nonlinear asymmetric migration terms ( same type of dispersion ( 5)) given by [START_REF] Levin | Dispersion and Population Interactions[END_REF]. In particular, we are interested in studying the effect of nonlinear dispersion on the dynamics of population and coexistence of species, more precisely, is it possible, depending on the migration rate, that the total equilibrium population be larger than the carrying capacity K 1 ? Note that, this last question has been studied by many researches, see [START_REF] Arditi | Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation[END_REF][START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF][START_REF] Elbetch | The multi-patch logistic equation[END_REF][START_REF] Elbetch | The multi-patch logistic equation with asymmetric migration[END_REF][START_REF] Elbetch | Nonlinear diffusion in the multi-patch logistic model[END_REF][START_REF] Freedman | Mathematical Models of Population Interactions with Dispersal II: Differential Survival in a Change of Habitat[END_REF] for effect of linear diffusion on the total biomass with logistic growth, [START_REF] Elbetch | Nonlinear diffusion in the multi-patch logistic model[END_REF] for nonlinear diffusion, [START_REF] Arino | Number of Source Patches Required for Population Persistence in a Source-Sink Metapopulation with Explicit Movement[END_REF][START_REF] Elbetch | Effect of dispersal in single-species discrete diffusion systems with source-sink patches[END_REF][START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF] for Source-Sink patch model and [START_REF] Guo | Global stability of the endemic equilibrium of multigroup SIR epidemic models[END_REF][START_REF] Gao | A Multipatch Malaria Model with Logistic Growth Populations[END_REF] for susceptible-infected-susceptible (SIS) patch-model).

In the next section, we looked the system [START_REF] Levin | Dispersion and Population Interactions[END_REF] when the migration rate goes to infinity, in other words, when there is no restriction whatsoever on travel.

• • E * 2 ( ) O x 1 x 2 K 1 E 1 H (a) O x 1 E * 2 ( ) • x 2 (b) Figure 5. (a)
The ellipse E 1 and H for some parameters of the model. The equilibrium points are the intersection in the positive cone between E 1 and H , this intersections contains the origin and a second positive point E * 2 ( ). (b) Possible configurations for the isoclines of the system (26) (in red for x 1 and in blue for x 2 ) for certain parameters. The equilibrium points are the intersection between these two isoclines: the origin and the positive equilibrium E * 2 ( ).

Fast dispersal

Let we denote by s := θ1 θ2 , the dispersal asymmetry. In this case, we can rewritten the model [START_REF] Levin | Dispersion and Population Interactions[END_REF] as follow:

       dx 1 dt = r 1 x 1 1 - x 1 K 1 + x 2 2 -sx 1 2 , dx 2 dt = r 2 x 2 -1 - x 2 K 2 + sx 1 2 -x 2 2 . ( 29
)
Let X(t, ) = x 1 (t, ) + x 2 (t, ). We rewrite the system (29) using the variables (X, x 1 ).

One obtains:

       dX dt = r 1 x 1 1 - x 1 K 1 -r 2 (X -x 1 ) 1 + X -x 1 K 2 , dx 1 dt = r 1 x 1 1 - x 1 K 1 + (1 -s)x 2 1 -2Xx 1 + X 2 . ( 30
)
When → ∞, (30) is a slow-fast system, with one slow variable, X, and one fast variable x 1 . According to Tikhonov's theorem [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF][START_REF] Lobry | On Tykhonov's theorem for convergence of solutions of slow and fast systems[END_REF][START_REF] Tikhonov | Systems of differential equations containing small parameters in the derivatives[END_REF][START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF] we consider the dynamics of the fast variable in the time scale τ = t. One obtains

dx 1 dτ = 1 r 1 x 1 1 - x 1 K 1 + (1 -s)x 2 1 -2Xx 1 + X 2 . dx 1 dτ = (1 -s)x 2 1 -2Xx 1 + X 2 . ( 31
)
We propose to study the behavior of the model ( 29) distinguishing three following cases: 0 < s < 1, s = 1, and s > 1.

First, let we start by the symmetric case (s = 1).

Theorem 3.1. Consider the system (29) with s = 1. Let (x 1 (t, ), x 2 (t, )) be the solution of the system (29) with initial condition

(x 0 1 , x 0 2 ) satisfying x 0 i ≥ 0 for i = 1, 2.
Let Z(t) be the solution of the logistic equation

   dX dt = rX 1 - X 2K , X(0) = x 0 1 + x 0 2 , (32)
where r = r1-r2 2 and K = r1-r2 α1+α2 , with α i = ri Ki . Then, when → ∞, we have

x 1 (t, ) + x 2 (t, ) = Z(t) + o (1), uniformly for t ∈ [0, +∞) (33)
and, for any t 0 > 0, we have

       x 1 (t, ) = 1 2 Z(t) + o (1), x 2 (t, ) = 1 2 Z(t) + o (1) uniformly for t ∈ [t 0 , +∞). ( 34
)
Proof. The slow manifold is formed by the equilibrium points of the fast equation ( 31) with s = 1, which given by:

x * * 1,1 = 0 and x * * 1,2 = 1 2 X. (35)
Since x * * 1,2 = 1 2 X is LAS for the system (31), the theorem of Tikhonov ensures that after a fast transition toward the slow manifold, the solutions of (30) are approximated by the solutions of the reduced model which is obtained by replacing x * * 1,2 = 1 2 X into the dynamics of the slow variable, that is:

dX dt = r 1 1 2 X 1 - X 2K 1 -r 2 1 2 X 1 + X 2K 2 , ( 36
)
which gives the equation [START_REF] Tikhonov | Systems of differential equations containing small parameters in the derivatives[END_REF]. If r 1 > r 2 , (32) admits X * = 2K as a positive equilibrium point, which is LAS in the positive axis, and if r 1 ≤ r 2 , (32) admits the origin as unique equilibrium point, which is LAS. The approximation given by Tikhonov's theorem holds for all t ≥ 0 for the slow variable and for all t ≥ t 0 > 0 for the fast variable, where t 0 is small as we want. Therefore, let Z(t) be the solution of the reduced model (36) of initial condition Z(0) = X(0, ) = x 0 1 + x 0 2 , then, when → ∞, we have the approximations [START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF] and [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF].

Note that, in the case of perfect mixing, Approximation [START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF] shows that: if r 1 > r 2

then the total population behaves like the unique logistic equation ( 32) and then, when t and tend to ∞, the total population x 1 (t, ) + x 2 (t, ) tends toward 2 r1-r2 α1+α2 where 32) has the origin as unique equilibrium, and then, when t and tend to ∞, the total population x 1 (t, ) + x 2 (t, ) tends toward 0.

α i = r i /K i . If r 1 ≤ r 2 , then Equation (
Next, we consider the asymmetric case s = 1:

Theorem 3.2. Consider the system (29). Assume that s = 1. Let (x 1 (t, ), x 2 (t, ))

be the solution of the system (29) with initial condition (x 0 1 , x 0 2 ) satisfying x 0 i ≥ 0 for i = 1, 2. Let U (t) be the solution of the following equation

dX dt =        RX 1 - X (1 - √ s)K if 0 < s < 1, RX 1 - X (1 + √ s)K if s > 1, ( 37
)
where

X(0) = x 0 1 + x 0 2 , R = r1- √ sr2 1+ √ s and K = r1- √ sr2
α1+sα2 with α i = ri Ki . Then, when → ∞, we have

x 1 (t, ) + x 2 (t, ) = U (t) + o (1), uniformly for t ∈ [0, +∞) (38)
and, for any t 0 > 0, we have

(x 1 (t, ), x 2 (t, )) =            1 1 + √ s U (t) + o (1), √ s 1 + √ s U (t) + o (1) if 0 < s < 1 1 1 - √ s U (t) + o (1), √ s 1 - √ s U (t) + o (1) if s > 1,
(39) uniformly for t ∈ [t 0 , +∞).

Proof. The slow manifold is formed by the equilibrium points of the fast equation [START_REF] Smith | The Theory of the Chemostat : Dynamics of Microbial Competition[END_REF], which given by: [START_REF] Smith | The Theory of the Chemostat : Dynamics of Microbial Competition[END_REF], the theorem of Tikhonov ensures that after a fast transition toward the slow manifold, the solutions of (30) are approximated by the solutions of the reduced model which is obtained by replacing x * * 1,1 into the dynamics of the slow variable, that is:

x * * 1,1 = √ s -1 s -1 X and x * * 1,2 = - 1 + √ s s -1 X. ( 40
) If s > 1, then x * * 1,1 > 0 and x * * 1,2 < 0. As dx1 dτ > 0 for all x * * 1,2 < x 1 < x * * 1,1 and dx1 dτ < 0 for all x 1 < x * * 1,2 and x 1 > x * * 1,1 then x * * 1,1 is LAS and x * * 1,2 is unstable . Since x * * 1,1 is LAS for the system
dX dt = r 1 √ s -1 s -1 X     1 - √ s -1 s -1 X K 1     -r 2 s - √ s s -1 X     1 + s - √ s s -1 X K 2     = RX 1 - X (1 + √ s)K ,
which gives the equation (37).

If s < 1, then 0 < x * * 1,1 < x * * 1,2 . As dx1 dτ > 0 for all x * * 1,1 < x 1 < x * * 1,2 and dx1 dτ < 0 for all x 1 < x * * 1,1 and x 1 > x * * 1,2 then x * * 1,2 is LAS and x * * 1,1 is unstable. Since x * * 1,2
is LAS for the system [START_REF] Smith | The Theory of the Chemostat : Dynamics of Microbial Competition[END_REF], the theorem of Tikhonov ensures that after a fast transition toward the slow manifold, the solutions of ( 30) are approximated by the solutions of the reduced model which is obtained by replacing x * * 1,2 into the dynamics of the slow variable, that is:

dX dt = r 1 √ s + 1 1 -s X     1 - √ s + 1 1 -s X K 1     -r 2 s + √ s s -1 X     1 + s + √ s s -1 X K 2     = RX 1 - X (1 - √ s)K ,
which gives the equation (37

). If r 1 > √ sr 2 (37) admits X * = (1 + √ s)K if s > 1 and X * = (1 - √ s)K if s < 1
as a positive equilibrium point, which is LAS in the positive axis, and if r 1 ≤ √ sr 2 , (37) admits the origin as unique equilibrium point, which is LAS. The approximation given by Tikhonov's theorem holds for all t ≥ 0 for the slow variable and for all t ≥ t 0 > 0 for the fast variable, where t 0 is small as we want. Therefore, let U (t) be the solution of the reduced model (36) of initial condition U (0) = X(0, ) = x 0 1 + x 0 2 , then, when → ∞, we have the approximations (38) and (39).

Note that, in the case of perfect mixing, Approximation (38) shows that: if r 1 > √ sr 2

then the total population behaves like the unique equation (37) and then, when t and tend to ∞, the total population T * 2 (t, ) tends toward (1

+ √ s) r1- √ sr2 α1+sα2 if s > 1 and tends toward (1 - √ s) r1- √ sr2 α1+sα2 if s < 1 where α i = r i /K i . If r 1 ≤ √ sr 2 ,

then Equation

(37) has the origin as unique equilibrium, and then, when t and tend to ∞, the total population T * 2 (t, ) = x 1 (t, ) + x 2 (t, ) tends toward 0.

As a corollary of the previous theorems we obtain the following result which describes the total equilibrium population when → ∞ for all possible value of s:

Corollary 3.1. We have:

                                 If r 2 ≥ r 1 then T * 2 (+∞) =        (1 - √ s) r 1 - √ sr 2 r1 K1 + s r2 K2 if 0 < s < r 2 1 r 2 2 , 0 if s ≥ r 2 1 r 2 2 . If r 2 < r 1 then T * 2 (+∞) =                  (1 - √ s) r 1 - √ sr 2 r1 K1 + s r2 K2 if 0 < s < 1, (1 + √ s) r 1 - √ sr 2 r1 K1 + s r2 K2 if 1 ≤ s < r 2 1 r 2 2 , 0 if s ≥ r 2 1 r 2 2 .
(41)

Effect of nonlinear dispersal on total population

In this section, Our aim is to compare the total equilibrium population

T * 2 ( ) = x * * 1 ( ) + x * * 2 ( ), E * 2 ( ) = (x * * 1 ( ), x * * 2 ( )),
of ( 26) and carrying capacity K 1 , by analyzing the stable positive equilibrium E * 2 ( ).

Note that, when there is no dispersal (i.e., = 0), the total equilibrium population is

T * 2 (0) = K 1 .
We study the effect of nonlinear asymmetric dispersal on the total equilibrium population for the two-patch source-sink system [START_REF] Levin | Dispersion and Population Interactions[END_REF]. Using the method graphic of Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF] and of Wu et al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF], we describe the position affects the equilibrium E * 2 ( ) when the migration rate varies from zero to infinity, we will give the condition whether T * 2 is greater or smaller than the carrying capacity T * 2 (0) = K 1 .

We prove there are only five cases as in the two-patch source-sink model with linear diffusion can occur. We consider the regions in the set of the parameters θ 1 and θ 2 , denoted ∆ 0 , ∆ 1 , ∆ 2 , ∆ 3 and ∆ 4 , depicted in Figure 6 and defined by:

                               If r 2 ≥ r 1 then      ∆ 0 = θ 1 , θ 2 ) : θ2 θ1 ≤ r 2 2 r 2 1 , ∆ 1 = (θ 1 , θ 2 ) : θ2 θ1 > r 2 2 r 2 1 . If r 2 < r 1 then              ∆ 2 = (θ 1 , θ 2 ) : θ2 θ1 ≤ r 2 2 r 2 1 , ∆ 3 = (θ 1 , θ 2 ) : r 2 2 r 2 1 < θ2 θ1 < α2(K1+K2) r1-r2 2 , ∆ 4 = (θ 1 , θ 2 ) : θ2 θ1 ≥ α2(K1+K2) r1-r2 2 . (42)
We have the following results which gives the conditions under which fragmentation and nonlinear asymmetrical migration can lead to a total equilibrium population greater or smaller than the sum of the carrying capacities:

Theorem 3.3. The total equilibrium population T * 2 of (26) satisfies the following properties (1) If r 2 ≥ r 1 , let ∆ 0 and ∆ 1 be defined by (42). Then we have:

Case r 2 ≥ r 1 . 0 ∆ 1 ∆ 0 θ 1 θ 2 θ2 θ1 = r 2 2 r 2 1 Case r 2 < r 1 . 0 ∆ 4 ∆ 3 ∆ 2 θ 1 θ 2 θ2 θ1 = α2(K1+K2) r1-r2 2 θ2 θ1 = r 2 2 r 2 1 Figure 6.
Qualitative properties of source-sink model [START_REF] Levin | Dispersion and Population Interactions[END_REF]. In ∆ 0 and ∆ 1 , the effect is detrimental, with extinction in two patches for ∆ 0 , and persistence for ∆ 1 . In ∆ 2 and ∆ 3 , the effect is beneficial for < 0 and detrimental for > 0 with persistence of the population in the region ∆ 3 , and extinction in the region ∆ 2 . In ∆ 4 , patchiness has a beneficial effect on the total equilibrium population.

(a) if (θ 1 , θ 2 ) ∈ ∆ 0 then T * 2 ( ) ≤ K 1 for all ≥ 0. Moreover, T * 2 ( ) -→ 0 when -→ +∞. (b) if (θ 1 , θ 2 ) ∈ ∆ 1 then 0 < T * 2 ( ) ≤ K 1 for all ≥ 0. Moreover, T * 2 ( ) -→ T * 2 (∞) > 0 when -→ +∞.
(2) If r 2 < r 1 , let ∆ 2 , ∆ 3 and ∆ 4 be defined by (42). Then we have:

(a) if (θ 1 , θ 2 ) ∈ ∆ 2 then, there is 0 > 0, such that: T * 2 ( ) > K 1 for < 0 and T * 2 ( ) < K 1 for all > 0 . Moreover, T * 2 ( ) -→ 0 when -→ +∞. (b) if (θ 1 , θ 2 ) ∈ ∆ 3 then, there is > 0, such that: T * 2 ( ) > K 1 for < and T * 2 ( ) < K 1 for all > . Moreover, T * 2 ( ) -→ T * 2 (∞) > 0 when -→ +∞. (c) if (θ 1 , θ 2 ) ∈ ∆ 4 , then T * 2 ( ) ≥ K 1 for all ≥ 0. Moreover, if θ2 θ1 = α2(K1+K2) r1-r2 2 , then T * 2 → T * 2 (∞) = K 1 .
Proof. Recall that, the equilibrium points of ( 26) are the non-negative intersection between the ellipse E 1 and hyperbola H . There are two equilibrium points.

The first is the trivial point (0, 0) and the second is a non trivial point noted

E * ( ) := (x * * 1 ( ), x * * 2 (
)) whose position depend on migration rate ( see Fig. 5-(a)).

As our study is limited in the positive cone, then we are interested only in the positive branch of H . The hyperbola H ( shown in blue in Fig. 5-(a)) depend on the migration rate . It always passes through the origin and the point

B := K 1 , θ1 θ2 K 1 .
Notice that, the hyperbola H intersect the axis (Ox 1 ) at 0 and a second positive point

x 1 = r1/ θ1+α1/ which always smaller than K 1 .

When → 0, the left branch of hyperbola H tend to the vertical line x 1 = 0 and the right branch into the vertical line H 0 :

x 1 = K 1 . Moreover, E 1 ∩ H 0 = {(K 1 , 0)}.
In the case when → ∞, the hyperbola H tend to the oblique line

H ∞ : x 2 = θ1 θ2 x 1 . Moreover, E 1 ∩ H ∞ = (0, 0), θ 2 √ θ 2 r 1 - √ θ 1 r 2 θ 2 r 1 /K 1 + θ 1 r 2 /K 2 , θ 1 √ θ 2 r 1 - √ θ 1 r 2 θ 2 r 1 /K 1 + θ 1 r 2 /K 2 .
The tangent space of the ellipse E 1 at O and A = (K 1 , 0) is T O E 1 : x 2 = r1 r2 x 1 , and

T A E 1 : x 2 = -r1 r2 x 1 respectively. Case (1) -(a). 0 ∆ E 1 H θ 1 θ 2 H ∞ Case (1) -(b). 0 ∆ E 1 B • θ 1 θ 2 H H ∞ Figure 7
. This illustrates the cases (1)-a and (1)-b in Theorem 3.3. Case (a) illustrate an example with (θ 1 , θ 2 ) ∈ ∆ 0 and the case (b) illustrate an example with (θ 1 , θ 2 ) ∈ ∆ 1 . The ellipse E 1 and the hyperbola H are shown in red and blue respectively for some parameters of the model ( 26), the straight lines ∆ and H∞ are shown in green and orange respectively. As the migration intensity increases from 0 to ∞, the equilibrium point E * 2 ( ) moves counterclockwise along the ellipse from A = (K 1 , 0) to O in the case (1)-a, from A to B in the case (1)-b. The total equilibrium population is always smaller than K 1 for all ≥ 0.

To facilitate comparison between the total equilibrium population T * 2 ( ) and

T * 2 (0) = K 1 ,
we define a straight line ∆ :

x 1 + x 2 = K 1 .
If the intersection of the ellipse E 1 and the hyperbola H , i.e., the equilibrium (x * * 1 ( ), x * * 2 ( )), is on or below the line ∆, then T * 2 ( ) ≤ T * 2 (0), whereas if the intersection is above the line ∆, then In the case when r 1 > r 2 , direct calculation finds that the ellipse E 1 and the line ∆ have two intersections:

T * 2 ( ) ≥ T * 2 (0). We see very simply that dispersal is favorable to T * 2 ( ) if E * 2 ( ) is above ∆, unfavorable if below ∆. The equilibrium point E * 2
A = (K 1 , 0), and C = α 2 K 1 + K 2 α 1 + α 2 , r 1 -r 2 α 1 + α 2 .
We denote by Σ the straight line joint the origin and C. The slope of Σ is equal to r1-r2 α2(K1+K2) . We distinguish three cases relative position of the points O, B = E * 2 (∞), and C, or equivalently, the three lines T O E 1 , H ∞ and Σ whose slopes respectively are

r 1 r 2 , θ 1 θ 2 and r 1 -r 2 α 2 (K 1 + K 2 )
.

By the method graphic of Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF] ( see also [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF]), we conclude the complete proof ( see Figures 8 and9). Case ( 2)

Case (2) -(a). 0 ∆ C • E 1 H θ 1 θ 2 H ∞ Case (2) -(b). 0 ∆ E 1 C B • • θ 1 θ 2 H H ∞ Figure 8.
-(c). 0 ∆ E 1 H θ 1 θ 2 H ∞ Case (d). 0 ∆ E 1 C = B • θ 1 θ 2 H H ∞ Figure 9
. This illustrates the case (2)-c in Theorem 3.3. As the migration intensity increases from 0 to ∞, the equilibrium point E * 2 ( ) moves counterclockwise along the ellipse from A = (K 1 , 0) to B. The total equilibrium point is always greater than K 1 for all , because it belongs to the ellipse and the limit point B is above ∆. The figure (d) is a particular case of ( 2)-(c), where

T * 2 → K 1 = T * 2 (∞) for θ 2 θ 1 = α 2 (K 1 +K 2 ) r 1 -r 2 2 .
Speaking biologically, the result of the previous theorem show that the nonlinear asymmetric dispersal can lead to either an increased total size of the population in two patches, a decreased total size with persistence in the patches, or even extinction in both patches when the growth of migration goes to infinity. Let us explain the result of Theorem 3.3 in the particular case where the dispersion is symmetric i.e. θ1 θ2 = 1. In this case, the system (26) becomes

       dx 1 dt = r 1 x 1 1 - x 1 K 1 + x 2 2 -x 1 2 , dx 2 dt = r 2 x 2 -1 - x 2 K 2 + x 1 2 -x 2 2 , ( 43
)
Corollary 3.2. Consider the model (43). Let Λ 0 , Λ 1 and Λ 2 be defined by [START_REF] Elbetch | Effect of dispersal in single-species discrete diffusion systems with source-sink patches[END_REF]. The total equilibrium population T * 2 of (43) satisfies the following properties

(1) if (r 1 , r 2 ) ∈ Λ 0 then, T * 2 ( ) ≤ K 1 for all . Moreover, T * 2 ( ) → 0 when → ∞.
(2) if (r 1 , r 2 ) ∈ Λ 1 then, there is * * > 0, such that

T * 2 ( ) ≥ K 1 If ≤ * * , 0 < T * 2 ( ) < K 1 If > * * . ( 44
)
Moreover, T * 2 ( ) → T * 2 (∞) > 0 when → ∞. (3) if (r 1 , r 2 ) ∈ Λ 2 , then T * 2 ( ) ≥ K 1 for all ≥ 0. Moreover, if r1 r2 = K2 K1+2K2 , then T * 2 (∞) = K 1 .
Proof. Consequence direct of the theorem 3.3.

Comparing the effects of linear and nonlinear asymmetric dispersion on the persistence and extinction of population, we remark that, the nonlinear symmetry between patches assure that: if the derivative of the total equilibrium population at = 0 is negative, then the population goes to extinction in two patches. Comparing this situation with the nonlinear asymmetric case, we obtain that we can have the derivative of the total equilibrium population at = 0 is negative without the population goes to extinction in two patches. The second conclusion, in the nonlinear symmetric case, when there is a * such that T * 2 ( * ) = K 1 , then the population persistent in two patches unlike in the asymmetric case, i.e. we can have T * 2 ( ) = K 1 for certain value of migration rate , but, the population goes to extinction in two patches. As conclusion, the nonlinear dispersal without the condition of symmetry, increases the probability of extinction of population in the patches.

In the remainder of this section, we study the effect of the rapid growth of the population in the source patch and rapid death of the population in the sink patch on the dynamics of the total equilibrium population and on the coexistence of the population in both patches. Note that, these situations were not examined in [START_REF] Arditi | Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation[END_REF][START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF][START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF].

Here, we study the total equilibrium population as a function of the migration rate in the case where the growth (resp. death) rate is much larger than the death (resp. growth) rate. In particular, we explicitly calculate the total equilibrium in the both situations, its derivative in the absence of the migration, its limit for large migration rate and we compare the total equilibrium population with the carrying capacity of the source patch. First, we start by the following situation:

The death rate is much larger than the growth rate

In this part, we consider the two-patch model [START_REF] Levin | Dispersion and Population Interactions[END_REF] and we assume that the death rate r 2 is much larger than the growth rate r 1 . One can write the model in the following way:

       dx 1 dt = r 1 x 1 1 - x 1 K 1 + θ 2 x 2 2 -θ 1 x 2 1 , dx 2 dt = r 2 η x 2 -1 - x 2 K 2 + θ 1 x 2 1 -θ 2 x 2 2 , (45)
where η is assumed to be a small positive number. First, we have the following result:

Theorem 3.4. Let (x 1 (t, η), x 2 (t, η)) be the solution of System (45) with initial condition (x 0 1 , x 0 2 ) satisfying x 0 i ≥ 0 for i = 1, 2. Let z(t) be the solution of the differential equation

dx 1 dt = r 1 x 1 1 - x 1 K 1 -θ 1 x 2 1 =: ϕ(x 1 ), (46)
with initial condition z(0) = x 0 1 . Then, when η → 0, we have

x 1 (t, η) = z(t) + o η (1), uniformly for t ∈ [0, +∞) (47)
and, for any t 0 > 0, we have

x 2 (t, η) = o η (1), uniformly for t ∈ [t 0 , +∞). (48)
Proof. When η → 0, System (45) is a slow-fast system, with one slow variable, x 1 , and one fast variable, x 2 . Tikhonov's theorem [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF][START_REF] Lobry | On Tykhonov's theorem for convergence of solutions of slow and fast systems[END_REF][START_REF] Tikhonov | Systems of differential equations containing small parameters in the derivatives[END_REF][START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF] prompts us to consider the dynamics of the fast variables in the time scale τ = 1 η t. One obtains

dx 2 dτ = r 2 x 2 -1 - x 2 K 2 + η(θ 1 x 2 1 -θ 2 x 2 2 ). (49)
In the limit η → 0, we find the fast dynamics

dx 2 dτ = r 2 x 2 -1 - x 2 K 2 . (50)
The slow manifold is given by the equilibrium of System (50), i.e. x 2 = 0, which is LAS in the positive axis. When η goes to zero, Tikhonov's theorem ensures that after a fast transition toward the slow manifold, the solutions of (45) converge to the solutions of the reduced model (46), obtained by replacing x 2 = 0 into the dynamics of the slow variable.

The differential equation ( 46) admits as a positive equilibrium

x * * 1 ( , 0 + ) := r 1 K 1 r 1 + θ 1 K 1 . (51)
As ϕ(x 1 ) > 0 for all 0 ≤ x 1 < x * * 1 ( , 0 + ) and ϕ(x 1 ) < 0 for all x 1 > x * * 1 ( , 0 + ) then, the equilibrium x * * 1 ( , 0 + ) is LAS in the positive axis, so, the approximation given by Tikhonov's theorem holds for all t ≥ 0 for the slow variable and for all t ≥ t 0 > 0 for the fast variable, where t 0 is as small as we want. Therefore, let z(t) be the solution of the reduced model (46) of initial condition z(0) = x 0 1 , then, when η → 0, we have Approximations (47) and (48).

We have the following result which gives the conditions for which patchiness is beneficial or detrimental in System (45) when goes to zero.

Theorem 3.5. The total equilibrium population x * * 1 ( , 0 + ) of System (45) when η → 0, given by (51) is decreasing. Moreover, 0 < x * * 1 ( , 0 + ) < K 1 for and x * * 1 ( , 0 + ) → 0 as → ∞.

Proof. The derivative of (51) is given by:

dx * * 1 d ( , 0 + ) = - r 1 K 1 2 θ 1 (r 1 + θ 1 K 1 ) 2 , (52)
which is negative for all .

The growth rate is much larger than the death rate

In this part, we consider the two-patch model ( 26) and we assume that the growth rate r 1 is much larger than the death rate r 2 . On can write the model in the following way:

       dx 1 dt = r 1 η x 1 1 - x 1 K 1 + θ 2 x 2 2 -θ 1 x 2 1 , dx 2 dt = r 2 x 2 -1 - x 2 K 2 + θ 1 x 2 1 -θ 2 x 2 2 , (53)
where η is assumed to be a small positive number. We prove the following result:

Theorem 3.6. Let (x 1 (t, η), x 2 (t, η)) be the solution of System (53) with initial condition (x 0 1 , x 0 2 ) satisfying x 0 i ≥ 0 for i = 1, 2. Let z(t) be the solution of the differential equation

dx 2 dt = r 2 x 2 -1 - x 2 K 2 + (θ 1 K 2 1 -θ 2 x 2 2 ) =: ψ(x 2 ), (54)
with initial condition z(0) = x 0 2 . Then, when η → 0, we have

x 2 (t, η) = z(t) + o η (1), uniformly for t ∈ [0, +∞) (55)
and, for any t 0 > 0, we have

x 1 (t, η) = K 1 + o η (1), uniformly for t ∈ [t 0 , +∞). ( 56
)
Proof. When η → 0, System (53) is a slow-fast system, with one slow variable, x 2 , and one fast variable, x 1 . Tikhonov's theorem [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF][START_REF] Lobry | On Tykhonov's theorem for convergence of solutions of slow and fast systems[END_REF][START_REF] Tikhonov | Systems of differential equations containing small parameters in the derivatives[END_REF][START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF] prompts us to consider the dynamics of the fast variables in the time scale τ = 1 η t. One obtains

dx 1 dτ = r 1 x 1 1 - x 1 K 1 + η(-θ 1 x 2 1 + θ 2 x 2 2 ). ( 57
)
dx 1 dτ = r 1 x 1 1 - x 1 K 1 . (58)
The slow manifold is given by the equilibrium of System (58), i.e. x 1 = K 1 , which is LAS in the positive axis. When η goes to zero, Tikhonov's theorem ensures that after a fast transition toward the slow manifold, the solutions of (53) converge to the solutions of the reduced model (54), obtained by replacing x 1 = K 1 into the dynamics of the slow variable.

The differential equation (54) admits as a positive equilibrium

x * * 2 ( , 0 + ) := 1/2 -r 2 K 2 + r 2 2 K 2 2 + 4 K 2 θ 1 K 1 2 r 2 + 4 2 K 2 2 θ 1 K 1 2 θ 2 r 2 + K 2 θ 2 . ( 59
)
As ψ(x 2 ) > 0 for all 0 ≤ x 2 < x * * 2 ( , 0 + ) and ψ(x 2 ) < 0 for all x 2 > x * * 2 ( , 0 + ) then, the equilibrium x * * 2 ( , 0 + ) is LAS in the positive axis, so, the approximation given by Tikhonov's theorem holds for all t ≥ 0 for the slow variable and for all t ≥ t 0 > 0 for the fast variable, where t 0 is as small as we want. Therefore, let z(t) be the solution of the reduced model (54) with initial condition z(0) = x 0 2 , then, when η → 0, we have Approximations ( 55) and (56).

As a corollary of the previous theorem, we have the following result which give the limit of the total equilibrium population T * 2 ( , 0 + ) of System (53) when η goes to zero:

Corollary 3.3. We have:

T * 2 ( , 0 + ) := K 1 + 1/2 -r 2 K 2 + r 2 2 K 2 2 + 4 K 2 θ 1 K 1 2 r 2 + 4 2 K 2 2 θ 1 K 1 2 θ 2 r 2 + K 2 θ 2 . ( 60
)
In the following proposition, we calculate the derivative of T * 2 ( , 0 + ) at = 0 and the formula of perfect mixing (i.e. when → ∞) of the total equilibrium population defined by (60).

Proposition 3.1. Consider the total equilibrium population (60). Then,

dT * 2 d (0, 0 + ) = θ 1 K 2 1 r 2 , (61)
and

T * 2 (+∞, 0 + ) = (1 + √ s)K 1 = √ θ 1 + √ θ 2 √ θ 2 K 1 . (62)
Proof. The derivative of the total equilibrium population T * 2 ( , 0 + ) defined by (60) with respect to is:

dT * 2 d ( , 0 + ) = -1/2 -r 2 K 2 + r 2 2 K 2 2 + 4 K 2 θ 1 K 1 2 r 2 + 4 2 K 2 2 θ 1 K 1 2 θ 2 K 2 θ 2 (r 2 + K 2 θ 2 ) 2 (63) + 1/4 4 K 2 θ 1 K 1 2 r 2 + 8 K 2 2 θ 1 K 1 2 θ 2 (r 2 + K 2 θ 2 ) r 2 2 K 2 2 + 4 K 2 θ 1 K 1 2 r 2 + 4 2 K 2 2 θ 1 K 1 2 θ 2
In particular, the derivative of the total equilibrium population at = 0 is given by the formula (61).

By taking the limit of (60) when → ∞, we get that the total equilibrium population T * 2 ( , 0 + ) tends to (62).

We have the following result which gives the conditions for which patchiness is beneficial or detrimental in model ( 53) when η goes to zero.

Theorem 3.7. Consider the total equilibrium population T * 2 ( , 0 + ) given by (60).

Then, T * 2 ( , 0 + ) ≥ K 1 , for all ≥ 0.

Proof. First, we try to solve the equation T * 2 ( , 0 + ) = K 1 with respect to , the solutions of this last equation give the points of intersection between the curve of the total equilibrium population → T * 2 ( , 0 + ) and the straight line → K 1 . For any ≥ 0, we have

T * 2 ( , 0 + ) = K 1 ⇐⇒1/2 -r 2 K 2 + r 2 2 K 2 2 + 4 K 2 θ 1 K 1 2 r 2 + 4 2 K 2 2 θ 1 K 1 2 θ 2 r 2 + K 2 θ 2 = 0 ⇐⇒ r 2 2 K 2 2 + 4 K 2 θ 1 K 1 2 r 2 + 4 2 K 2 2 θ 1 K 1 2 θ 2 = r 2 K 2 ⇐⇒4 K 2 θ 1 K 1 2 (r 2 + K 2 θ 2 ) = 0 ⇐⇒ = 0.
Therefore, since dT * 2 d (0, 0 + ) > 0, the curve of the total equilibrium population intersects the straight line → K 1 in a unique point which is (0, K 1 ). Therefore,

T * 2 ( , 0 + ) ≥ K 1 , for all ≥ 0.
Biologically speaking, from Sections 3.4 and 3.5, we conclude that, the rapid increase in the population in the source patch results in persistence in the both patches with increased total size population, and the rapid decrease in the sink population results in extinction in both patches.

Numerical simulations

In this section, we give some numerical examples which illustrates the results of 17) and (41) respectively. In the figure 10, we plot the graphs of the functions ψ 1 ( shown in red) and ψ 2 ( shown in green), as a function of the dispersal asymmetry parameter s for some parameters of the model. Its clear that ψ 1 , ψ 2 → K 1 when s → 0 and ψ 1 , ψ 2 → 0 when s → ∞. The derivative of ψ 1 with respect to s is given by:

dψ 1 ds (s) =    - (-r 2 α 2 + α 2 r 1 ) s 2 + (2 α 2 r 1 + 2 r 2 α 1 ) s -r 1 α 1 + r 2 α 1 (α 1 + s 2 α 2 ) 2 if s < r 1 /r 2 , 0 otherwise. ( 64
)
We have dψ 1 ds (s) = 0 if and only if,

s 1 = -α2r1+r2α1+ √ α2 2 r1 2 +r2 2 α1 2 +r2 2 α2α1+α2r1 2 α1 α2(-r2+r1)
or

s 2 = -α2r1-r2α1+ √ α2 2 r1 2 +r2 2 α1 2 +r2 2 α2α1+α2r1 2 α1 α2(-r2+r1)
. If r 1 ≤ r 2 then, s 2 < 0 < s 1 , so, the derivative of ψ 1 is negative for all s < r 1 /r 2 , since s 1 > r 1 /r 2 ( see figure 10-(a)). If The derivative of ψ 2 with respect to s is given by:

r 1 > r 2 , then s 1 < 0 < s 2 ,
                         If r 2 ≥ r 1 then dψ 2 ds (s) =        -r1-r2 √ s 2 √ s(α1+sα2) - (1- √ s)r2 2 √ s(α1+sα2) - (1- √ s)(r1-r2 √ s)α2 (α1+sα2) 2 if 0 < s < r 2 1 r 2 2 , 0 if s ≥ r 2 1 r 2 2 . If r 2 < r 1 then dψ 2 ds (s) =              -r1-r2 √ s 2 √ s(α1+sα2) - (1- √ s)r2 2 √ s(α1+sα2) - (1- √ s)(r1-r2 √ s)α2 (α1+sα2) 2 if 0 < s < 1, r1-r2 √ s 2 √ s(α1+sα2) - (1+ √ s)r2 2 √ s(α1+sα2) - (1+ √ s)(r1-r2 √ s)α2 (α1+sα2) 2 if 1 ≤ s < r 2 1 r 2 2 , 0 if s ≥ r 2 1 r 2 2 . (65)
If r 2 ≥ r 1 , then the derivative of ψ 2 is negative for all 0 < s < r 2 In Table 1, we give the space of parameters (r 1 , r 2 , K 1 , K 2 ) ∈ R 4 + .

Table 1. Parameters values of the four cases used in Fig. 10. . . The graphs of ψ 1 (red) and ψ 2 (green) as function of s for the parametrs given in Table 1. The function ψ 1 is continous for all s. If r 2 < r 1 , at symmetric case (s = 1), ψ 2 is discontinous as showen in case (b), (c) and (d).

and the line K 1 , for the sets of parameters (θ 1 , θ 2 ) ∈ ∆ i ∩ L i for i = 0, .., 4, where L i and ∆ i are given by ( 9) and (42) respectively, choosing in Table 2. 16) and (41) respectivly . . 7) and ( 26) respectively as a function of the migration rate for the sets of the parameter values given in Table 1

Figure

r 1 r 2 K 1 K 2 θ 1 θ 2 dT * 1 d (0) T * 1 (∞) dT * 2 d (0) T * 2 (∞)
with (θ 1 , θ 2 ) ∈ ∆ 0 ∩ L 0 . The horizontal line shown in blue is K 1 .
Depending on the values of the migration rate , as shown in the figures [START_REF] Elbetch | Effect of dispersal in Two-patch environment with Richards growth on population dynamics[END_REF], ( 12), ( 13), ( 14), ( 15) and ( 16), in which we have the following examples of inequalities between the total equilibrium populations T * 1 , T * 2 and also the value K 1 which is again represented by the horizontal blue line. 7) and ( 26) respectively as a function of the migration rate for the sets of the parameter values given in Table 1 with (θ 1 , θ 2 ) ∈ ∆ 1 ∩ L 1 . The horizontal line shown in blue is K 1 . . 7) and ( 26) respectively as a function of the migration rate for the sets of the parameter values given in Table 1 with (θ 1 , θ 2 ) ∈ ∆ 2 ∩ L 2 . The horizontal line shown in blue is K 1 . . 7) and ( 26) respectively as a function of the migration rate for the sets of the parameter values given in Table 1 with (θ 1 , θ 2 ) ∈ ∆ 3 ∩ L 3 . The horizontal line shown in blue is K 1 . . 7) and ( 26) respectively as a function of the migration rate for the sets of the parameter values given in Table 1 with (θ 1 , θ 2 ) ∈ ∆ 4 ∩ L 4 . The horizontal line shown in blue is K 1 . . 7) and ( 26) respectively as a function of the migration rate for the sets of the parameter values given in Table 1 with

the condition θ 2 θ 1 = α 2 (K 1 +K 2 ) r 1 -r 2 = α 2 (K 1 +K 2 ) r 1 -r 2 2
. The horizontal line shown in blue is K 1 . . 16) and (41) respectivly. 7) and ( 26) respectively as a function of the migration rate for the sets of the parameter values given in Table 3. The horizontal line shown in blue is K 1 .

Figure Case

r 1 r 2 K 1 K 2 dT * 1 d (0) T * 1 (∞) dT * 2 d (0) T * 2 (∞) Case A 0.25 1
A: K 1 = 0.5 < 1 Case B: K 1 = 1 Case C: K 1 = 2 > 1

Case

A: K 1 = 0.5 < 1 Case B: K 1 = 1 Case C: K 1 = 2 > 1 Figure 18.
Total equilibrium populations T * 1 (shown in green) and T * 2 (shown in red) of the system ( 7) and ( 26) respectively as a function of the migration rate for the sets of the parameter values given in Table 3. The horizontal line shown in blue is K 1 . (c) if K 1 > 1, then, there is a * > 0 such that T * 1 ( ) ≤ T * 2 ( ) for all 0 ≤ ≤ * , and T * 1 ( ) > T * 2 ( ) for all > * .

Case

A: K 1 = 0.5 < 1 Case B: K 1 = 1 Case C: K 1 = 2 > 1

Conclusion

This work was aimed to find out whether the total equilibrium population of the two patches source-sink connected by nonlinear asymmetric migration can be greater than the carrying capacity of the source isolated patch, i.e. at equilibrium. The system has a unique positive equilibrium, which furthermore is globally asymptotically stable in the interior of the positive cone, and the origin as trivial equilibrium which is unstable, unlike in the symmetric case, which can be globally asymptotically stable ( see Prop.

5.5 in [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF] and Theorem 2.1).

In Section 3.2, we considered the particular case of perfect mixing, i.e. when the diffusion rate goes to infinity, that is, individuals may travel freely between two patches.

We compute the total equilibrium population in that case, and by perturbation arguments, we proved that the dynamics in this ideal case provides a good approximation for the case when the diffusion rate is large. Rigorous analysis on the model demonstrate a mechanism by which the nonlinear dispersal asymmetry can lead to either an increased total size of the species in two patches, a decreased total size with persistence in the patches, or even extinction in both patches for large migration rates. In Subsections 3.3 and 3.4, we study the total equilibrium population of the source-sink patch model ( 26) as a function of the diffusion rate in the case where the growth (resp. death) rate is much larger than the death (resp. growth) rate. Some numerical simulations are given which we confirm and extend our results. We also compare our present results with the result of previous work [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF] for linear asymmetry dispersal.

Finally, comparison of the present results with those of Wu et al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF] suggests that this nonlinear asymmetric dispersal mechanism has the same impact as in the case of linear asymmetric dispersal. However, comparing the conditions ( 9) and (42) shows that this nonlinear asymmetry dispersal can have a strong quantitative influence on the effect of dispersal. In the case symmetric θ 1 = θ 2 , the conditions ( 9) and (42) are the same, and becomes [START_REF] Elbetch | Effect of dispersal in single-species discrete diffusion systems with source-sink patches[END_REF].

Some questions remain open, for examples, how our results generalize to situations with more than two patches i. e. the generalization of the present results to the three-patch source-sink model with nonlinear asymmetric migration. In particular, the global stability, the behavior of the model for large migration rate and the influence of nonlinear asymmetric dispersal on total population abundance. The mathematical extension to n patches (n > 2) is probably very intricate and is a challenge for further work. I think this question is difficult and requires a lot of work and mathematical tools.

Proposition A.1. The positive cone R 2 + is positively invariant for the system [START_REF] Levin | Dispersion and Population Interactions[END_REF].

Proof. Suppose that, at a given time t, one of the state variables of the system [START_REF] Levin | Dispersion and Population Interactions[END_REF] is at a boundary of R 2 + , meaning that at least one population is at 0. We suppose first that x 1 = 0, and x 2 ≥ 0, then the dynamics of x 1 is given by dx1 dt = θ 2 x 2 2 ≥ 0, and, if

x 2 = 0, and x 1 ≥ 0, then we have dx2 dt = θ 1 x 2 1 ≥ 0. So each trajectory initiated at a boundary of R 2 + either remains at the boundary or goes to the interior of R 2 + . According to [START_REF] Smith | The Theory of the Chemostat : Dynamics of Microbial Competition[END_REF]Proposition B.7,page 267], no trajectory comes out of R 2 + . Therefore, R 2 + is positively invariant for [START_REF] Levin | Dispersion and Population Interactions[END_REF].

The equilibrium of the system [START_REF] Levin | Dispersion and Population Interactions[END_REF] is the solutions of the following algebraic system:

       0 = r 1 x 1 1 - x 1 K 1 + (θ 2 x 2 2 -θ 1 x 2 1 ), 0 = -r 2 x 2 1 + x 2 K 2 + (θ 1 x 2 1 -θ 2 x 2 2
).

(A1)

The sum of the two equations of (A1) shows that the equilibrium points are in a ellipse noted E 1 , which its equation is given by:

E 1 : ϕ(x 1 , x 2 ) := r 1 x 1 1 - x 1 K 1 -r 2 x 2 1 + x 2 K 2 = 0. (A2)
The ellipse E 1 passes through the points (0, 0), (K 1 , 0), (0, -K 2 ) and A := (K 1 , -K 2 ).

Note that, it is independent of migration rate and θ i (shown in red in Fig. 5-(a)).

Solving the first equation of system (A1) for x 2 yields a hyperbola noted H defined by

H : h (x 1 ) := 1 θ 2 θ 1 x 2 1 - r 1 x 1 1 - x 1 K 1 .
As our study is limited in the positive cone, then we are interested only in the positive branch of H . The hyperbola H ( shown in blue in Fig. 5-(a)) depend on the migration rate . It always passes through the origin and the point B := K 1 , θ1 θ2 K 1 .

So, the equilibrium points are the non-negative intersection between the curves E 1 Remark A.1. The isoclines of the system (26) are given by the two equations:

           H ,1 (x 1 ) = - r 1 θ 2 x 1 1 - x 1 K 1 + θ 1 θ 2 x 2 1 , H ,2 (x 2 ) = r 2 θ 1 x 2 1 + x 2 K 2 + θ 2 θ 1 x 2 2 .
The equilibrium points are the intersection between these two isoclines: the origin and the positive equilibrium E * 2 ( ) (see Fig. 5-(b)).

1 Introduction 2 2 5 2. 1 5 2. 2 6 2. 3

 12515263 Some preliminary results for Two-patch source-sink model with linear dispersion Global dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total population abundance . . . . . . . . . . . . . . . . . . . . . . . . Perfect mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

  ( ) is always in ellipse, then, for = 0, the equilibrium point states at A, and when increases, E * 2 ( ) describes an arc of the ellipse and ends at point E * (∞). Suppose r 1 ≤ r 2 . Then the slope of T O E 1 , -r1 r2 ≥ -1, which means that the ellipse E 1 is below the line ∆ in the first quadrant and T * 2 ( ) < T * 2 (∞), as shown in Fig. 7. If the slope of H ∞ , θ1 θ2 , is bigger than the slope of T O E 1 , i.e. θ1 θ2 ≥ r1 r2 , the ellipse is below the line ∆ in the first quadrant. Thus, there is * , such that, when 0 < ≤ * , the equilibrium E * 2 ( ) changes from point A to O. When ≥ * , we have E * 2 ( ) -→ O and T * 2 ( ) -→ 0 when → ∞ ( see Fig. 7-(a)). If θ1 θ2 < r1 r2 , the ellipse and the line H ∞ has an intersection B. Then we have T * 2 ( ) → T * 2 (∞) > 0, as shown in Fig. 7-(b).

  Figure 8. This illustrates the cases (2)-a and (2)-b in Theorem 3.3. As the migration intensity increases from 0 to ∞, the equilibrium point E * 2 ( ) moves counterclockwise along the ellipse from A = (K 1 , 0) to O in the case (a) from A to B in the case (b), passing through the point C which is the other point of intersection between the ellipse and the line ∆. The dispersal is favorable when E * 2 ( ) is between A and C, and unfavorable when E * 2 ( ) is between C and B .

Proposition 2 . 1 ,

 21 Corollary 3.1, and also the results of Theorem 3.3. First, we start by given some examples of the graphs of the functions s → ψ 1 (s) := T * 1 (+∞) and s → ψ 2 (s) := T * 2 (+∞) defined by (

  therefore, the derivative of ψ 1 is positive for all 0 < s < s 2 and negative for s > s 2 since s 2 < r 1 /r 2 ( see Fig.10-(b), (c), (d)).

2 ( 2 , 1 r 2 2(

 2212 see Fig.(10)-(a)). If r 2 < r 1 , then the derivative of ψ 2 is negative for all 0 < s < 1 and zero for all s ≥ the sign of dψ2 ds is difficult to study.We give two examples where the derivative of ψ 2 has a negative sign ( see Fig. (10)-(b) and Fig. (10)-(c) ) and a non-constant sign for 1 < s < r 2 see Fig. (10)-(d)).

Figure r 1 r 2 K 1 Figure 10

 2110 Figure r 1 r 2 K 1 K 2 10-a 1 2 2 3 10-b 2 1 2 3 10-c 4 1 1 2 10-d 7 1 1 2

FigFigure 11 .

 11 Fig. 11 1 3 0.5 1.5 5 11 -1.76 0 -0.83 0 Fig. 12 1 3 0.5 1.5 0.5 11 -1.67 0.45 -0.83 0.22 Fig. 13 1.5 1 0.5 5 12 4 2.00 0 0.99 0 Fig. 14 1.5 1 0.5 0.083 1 4 0.16 0.42 0.08 0.29 Fig. 15 7 1 0.5 6 1 14 0.42 0.58 0.21 0.6 Fig. 16 12 2 0.5 0.125 2 2 0.41 0.5 0.20 0.5

Figure 12 .

 12 Figure 12. Total equilibrium population T * 1 (shown in green) and T * 2 (shown in red) of the system (7) and (26) respectively as a function of the migration rate for the sets of the parameter values given in Table1with (θ 1 , θ 2 ) ∈ ∆ 1 ∩ L 1 . The horizontal line shown in blue is K 1 . .

Figure 13 .

 13 Figure 13. Total equilibrium population T * 1 (shown in green) and T * 2 (shown in red) of the system (7) and (26) respectively as a function of the migration rate for the sets of the parameter values given in Table1with (θ 1 , θ 2 ) ∈ ∆ 2 ∩ L 2 . The horizontal line shown in blue is K 1 . .

Figure 14 .

 14 Figure 14. Total equilibrium population T * 1 (shown in green) and T * 2 (shown in red) of the system (7) and (26) respectively as a function of the migration rate for the sets of the parameter values given in Table1with (θ 1 , θ 2 ) ∈ ∆ 3 ∩ L 3 . The horizontal line shown in blue is K 1 . .

Figure 15 .

 15 Figure 15. Total equilibrium population T * 1 (shown in green) and T * 2 (shown in red) of the system (7) and (26) respectively as a function of the migration rate for the sets of the parameter values given in Table1with (θ 1 , θ 2 ) ∈ ∆ 4 ∩ L 4 . The horizontal line shown in blue is K 1 . .

Figure 16 .

 16 Figure 16. Total equilibrium population T * 1 (shown in green) and T * 2 (shown in red) of the system (7) and (26) respectively as a function of the migration rate for the sets of the parameter values given in Table1 with

Figure 17 .

 17 Figure 17. Total equilibrium population T * 1 (shown in green) and T * 2 (shown in red) of the system (7) and (26) respectively as a function of the migration rate for the sets of the parameter values given in Table3. The horizontal line shown in blue is K 1 .

Figure 19 .

 19 Figure 19. Total equilibrium populations T * 1 (shown in green) and T * 2 (shown in red) of the system (7) and (26) respectively as a function of the migration rate for the sets of the parameter values given in Table3. The horizontal line shown in blue is K 1 .

2 andH

 2 . There are two equilibrium points. The first is the trivial point (0, 0) and the second is a non trivial point noted E * ( ) := (x * * 1 ( ), x * * 2 ( )) whose position depend on migration rate ( see Fig.5-(a)).

Table 2 .

 2 Parameters values of the six cases used in Fig. 11, 12, 13, 14, 15 and 16. The derivatives

	dT * 1 d (0),

Table 3 .

 3 Parameters values of the three case used in Fig.17, 18 and 19. The derivatives

	dT * 1 d (0) and	dT * 2 d (0)

Table 2, the total equilibrium population T * 1 is smaller than T * 2 , and both smaller than K 1 . In addition, there is a migration rate 0 0.84 such that T * 1 ( ) = 0 for all ≥ 0.84. The total equilibrium T * 2 goes to zero when migration rate goes to infinity.

This example prove that, linear asymmetric dispersal can give extinction of population in both patches for small values of migration rate and this extinction is faster compared to the extinction for the nonlinear asymmetric dispersal, which product for the large migration rate.

In the figure 12, that is plotted for the parameter values given in the second line of Table 2. We have, the total equilibrium population T * 1 is smaller than T * 2 for all 0 ≤ ≤ 0.82, and T * 1 is greater than T * 2 for all ≥ 0.82, and both smaller than K 1 . For 0.82, T * 1 (0.82) = T * 2 (0.82) 0.47. The difference between the case of the figure 11 and 12, is that we have the persistence of the population in two patches for nonlinear symmetric and asymmetric dispersal.

In the figure 13, that is plotted for the parameter values given in the third line of Table 2. We have, the total equilibrium population T * 1 is greater than T * 2 for all 0 ≤ ≤ 0.03, and T * 1 is smaller than T * 2 for all ≥ 0.03, for 0.03, T * 1 (0.03) = T * 2 (0.03) 0.52. There are two migration rates 1 0.05 and 2 0.12 such that T * 1 (0.05) = T * 2 (0.12) = 0.5. The total equilibrium T * 1 ( ) = 0 for all ≥ 0.26.

In the figure 14, that is plotted for the parameter values given in the forth line of In the figure 15, that is plotted for the parameter values given in the five line of Table 2, the total equilibrium population T * 1 is greater than T * 2 for all 0 ≤ ≤ 0.08, T * 1 is smaller than T * 2 for all ≥ 0.08, and both greater than K 1 . The figure 16 is an example which that T * 1 , T * 2 → 0.5 = K 1 .

Symmetric dispersion

In this part, we consider the case of symmetric migration i.e. when θ 1 = θ 2 = 1.

We plot the total equilibrium populations → T * 1 ( ) and → T * 2 ( ) as a function of migration rate . We give some examples of the three cases given in Corollaries 2.1 and 3.2. We plot in the figures 17, 18 and 19, the curves of the total equilibrium population T * 1 , T * 2 and the line K 1 , for the sets of parameters choosing in Table 3.

Depending on the values of the migration rate , as shown in the figures [START_REF] Gao | How does dispersal affect the infection size?[END_REF], [START_REF] Gao | Fast diffusion inhibits disease outbreaks[END_REF]and [START_REF] Guo | Global stability of the endemic equilibrium of multigroup SIR epidemic models[END_REF], in which we have the following three examples of inequalities and equality between the total equilibrium populations T * 1 , T * 2 and also the value K 1 which is again represented by the horizontal blue line. We propose here to plot three figures, and each figure contain three cases corresponding to the three values of capacity:

As the two derivatives equals for K 1 = 1, we choose to plot the figures for values smaller than, equal to and greater than K 1 = 1.

In the figure 17, that is plotted for the parameter values given in the three first lines of Table 3. In case A, for K 1 < 1, the total equilibrium population T * 1 is smaller than T * 2 , and both smaller than K 1 . In case B, for K 1 = 1, the total equilibrium population In the figure 18, that is plotted for the parameter values given in the three second lines of Table 3. In case A, for K 1 < 1, there is a 0 0.2, such that, the total equilibrium population T * 1 is greater than T * 2 for all 0 ≤ ≤ 0.2, T * 1 is smaller than T * 2 for all ≥ 0.2 and both greater than K 1 . For 0.2, we have T * 1 (0.2) T * 2 (0.2) 0.59.

In case B, for

for all positive. In case C, for K 1 > 1, there is a 0 0.04, such that, the total equilibrium population T * 1 is smaller than T * 2 for 0 ≤ ≤ 0.2, and T * 1 is greater than

The same situations are product in the figure 19. Note that, according to these examples of figures, the passage for values of K 1 lower than 1 to values higher than 1, the positions of the curves of T * 1 and T * 2 change, and are sames for the particular value of capacity K 1 = 1. In the case when θ 1 = θ 2 = 1, the total equilibrium population T * 1 satisfies:

and T * 2 :

It is very difficult to prove analytically these numerical results, and I think, according to numerical simulations, for all symmetric migration, we only have these configurations, so we can give the following conjecture:

Conjecture 4.1. Consider the models [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF] and [START_REF] Levin | Dispersion and Population Interactions[END_REF] with

T * 2 be the total equilibrium populations of ( 29) and (43) respectively. Let Λ 0 , Λ 1 and Λ 2 be defined by [START_REF] Elbetch | Effect of dispersal in single-species discrete diffusion systems with source-sink patches[END_REF]. The total equilibrium populations T * 1 and T * 2 satisfied to the following properties

and

and

(

and (

and

Appendix A. Equilibria and stability of [START_REF] Levin | Dispersion and Population Interactions[END_REF] In this section, our goal is to prove the global stability of the system (26). In the absence of migration, i.e. the case where = 0, the system (26) admits (K 1 , 0) as a non trivial equilibrium point, which furthermore is globally asymptotically stable (GAS) in the positive cone R 2 + . The problem is whether the equilibrium continues to be positive and globally stable for any > 0 or not. We first prove the non negativity of the solutions of System [START_REF] Levin | Dispersion and Population Interactions[END_REF]. We have the following proposition:

In the following, our aim is to show the global stability of the equilibrium E * ( ).

For this, we need some results. First, for the non-negativity and boundedness of the solutions of the system (26), we have the following result:

Lemma A.1. For any non-negative initial condition, the solutions of the system (26) remain bounded, for all t ≥ 0. Moreover, the set

where ξ * 1 = min {r 1 , r 2 } and ξ * 2 = r 1 K 1 + r 2 K 2 , is positively invariant and is a global attractor for the system [START_REF] Levin | Dispersion and Population Interactions[END_REF].

Proof. To show that all solutions are bounded, we consider the quantity defined by

. So, we have

For all r i and K i , we have the inequality:

Substituting Equation (A6) into (A3), we get

for all t ≥ 0, which gives

Hence,

Therefore, the solutions of System ( 26) are positively bounded and defined for all t ≥ 0. From (A5), it can be deduced that the set Σ is positively invariant and it is a global attractor for the system [START_REF] Levin | Dispersion and Population Interactions[END_REF].

Recall that, a matrix A = (a ij ) is called cooperative if a ij ≥ 0 for all i = j. Let we consider the autonomous system:

where ẋ denote the derivative of x, Φ = (Φ 1 , . . . ,

Recall that, for general cooperative systems, we have the following result, which guarantees that all solutions converge to equilibria/infinity without periodic oscillation (see page 28 Hofbauer and Sigmund [24]). The solutions of a two-dimensional cooperative system converge either to an equilibrium or to infinity. We have the following result:

Proof. The two-dimensional system (26) is cooperative. Indeed, its Jacobian matrix evaluate at E * 2 ( ) is given by:

where

Therefore, κ 1 < 0 and κ 2 < 0. This implies that tr(J(E * 2 ( ))) = κ 1 + κ 2 < 0, where tr means the trace.

It's clear that, in the figures 5-(b), at the equilibrium E * 2 ( ), we have:

, which gives -

Hence by the Routh-Hurwitz criteria for stability, the real parts of the eigenvalues value of the Jacobian matrix J(E * 2 ( )) are negative, proving that E * 2 ( ) is asymptotically stable. Lemma A.1 imply that there cannot be any non-trivial closed paths lying in the interior of the positive quadrant and hence the asymptotic stability must be global.

Appendix B. Derivative of the total equilibrium population T * 1 at = 0 Proposition B.1. The derivative of the total equilibrium population T * 1 at = 0 is given by:

In particular, dT * 1 d (0) = 0 if and only if, r 1 = r 2 .

Proof. The equilibrium point E * 1 ( ) of (7) satisfies the system

Dividing the first and the second equation by r1 K1 x * 1 ( ) and r 2 + r2 K2 x * 2 ( ) respectively, one obtains

Hence, the total equilibrium population T * 1 is given by

By differentiating the equation (B4) at = 0, we get:

which gives (B1), since x * 1 (0) = K 1 and x * 2 (0) = 0.

Note that, the derivative (B1) is dependent on the parameters r 1 , r Appendix C. Derivative of the total equilibrium population T * 2 at = 0 Proposition C.1. The derivative of the total equilibrium population T * 2 at = 0 is given by:

In particular, dT * 2 d (0) = 0 if and only if, r 1 = r 2 .

Proof. The equilibrium point E * 2 ( ) of (26) satisfies the system Note that, the derivative (C1) is dependent on the parameters r 1 , r


Éric Le Ferrand

Fabiola Henri

Benjamin Lecouteux

Emmanuel Schang

Application of Speech Processes for the Documentation of Kréyòl Gwadloupéyen

In recent times, there has been a growing number of research studies focused on addressing the challenges posed by low-resource languages and the transcription bottleneck phenomenon. This phenomenon has driven the development of speech recognition methods to transcribe regional and Indigenous languages automatically. Although there is much talk about bridging the gap between speech technologies and field linguistics, there is a lack of documented efficient communication between NLP experts and documentary linguists. The models created for low-resource languages often remain within the confines of computer science departments, while documentary linguistics remain attached to traditional transcription workflows. This paper presents the early stage of a collaboration between NLP experts and field linguists, resulting in the successful transcription of Kréyòl Gwadloupéyen using speech recognition technology.

Introduction

The fields of descriptive and documentary linguistics concentrate on gathering information and describing language phenomena. This work is typically performed on small, Indigenous, and regional languages that have a limited number of speakers. The linguist's process typically involves recording raw speech, either spontaneous or elicited, transcribing the recordings, translating them, and conducting an analysis. In this pipeline, the transcription becomes the data, but transcribing raw speech is a time-consuming task and is often seen as a bottleneck when a large amount of speech is collected but only a small portion is used.

Speech technologies have been viewed as a solution to this bottleneck issue by automatically annotating raw speech collections. Regular automatic speech recognition (ASR) has proven to be challenging due to the lack of data available in most languages to train robust models. However, alternative methods, such as spoken term detection, phone recognition, and the use of universal models, offer new possibilities for collaboration between field linguists and NLP experts.

We present here an application of speech processing on raw field linguistics recordings in Kréyòl Gwadloupéyen. Our objective has two parts: firstly, to exhibit the capability of a wav2vec and CTCbased system for our target language, and secondly, to illustrate how the transcription output can be valuable and utilised by field linguists.

Background

Fieldwork technologies

In the past decade, there have been ongoing discussions about developing technology for the purpose of linguistic fieldwork [START_REF] Gessler | Closing the NLP gap documentary linguistics and NLP need a shared software infrastructure[END_REF][START_REF] Gauthier | Collecter Transcrire Analyser: quand la machine assiste le linguiste dans son travail de terrain[END_REF][START_REF] Moeller | SayMore, a tool for language documentation productivity[END_REF]. The main argument has been to adapt emerging technologies such as smartphones for fieldwork. The recent improvement of speech recognition for low-resource languages has also been seen as a way to mitigate the transcription bottleneck [START_REF] Nikolaus | Documentary and descriptive linguistics[END_REF] automatically transcribing large amount of untranscribed speech data (e.g. [START_REF] Foley | Building speech recognition systems for language documentation: The CoEDL Endangered Language Pipeline and Inference System (ELPIS)[END_REF][START_REF] Shi | Leveraging end-to-end asr for endangered language documentation: An empirical study on Yolóxochitl Mixtec[END_REF][START_REF] Adams | Userfriendly automatic transcription of low-resource languages: Plugging ESPnet into ELPIS[END_REF].

Looking at the role of technologies in the current linguistics fieldwork workflow, only a few tools are still widely used (e.g. [START_REF] Boersma | Praat, a system for doing phonetics by computer, version 3.4[END_REF][START_REF] Wittenburg | ELAN: a professional framework for multimodality research[END_REF]. The other projects involving tools design often end up discontinued [START_REF] Bird | Aikuma: A mobile app for collaborative language documentation[END_REF][START_REF] Gauthier | LIG-Aikuma: A mobile app to collect parallel speech for under-resourced language studies[END_REF] or stayed at the prototype stage [START_REF] Lane | A computational model for interactive transcription[END_REF][START_REF] Éric | Fashioning local designs from generic speech technologies in an Australian Aboriginal community[END_REF][START_REF] Bettinson | Developing a suite of mobile applications for collaborative language documentation[END_REF]. Leveraging speech technologies for scaling up language documentation has had limited impact as well, probably because of lack of data available for low-resource languages to build robust models (Gupta and Boulianne, 2020a,b).

The recent expansion of speech recognition models based on wav2vec2.0 [START_REF] Conneau | Unsupervised cross-lingual representation learning for speech recognition[END_REF] combined with CTC algorithms (e.g. [START_REF] Macaire | Automatic speech recognition and query by example for creole languages documentation[END_REF] open new opportunities for low-resource languages. Such an architecture is not restricted by a language model and can produce tokens out of vocabulary.

Kréyòl gwadloupéyen

Kréyòl gwadloupéyen is spoken on Guadeloupe Island and in mainland France by approximately 800 000 speakers. Kréyòl gwadloupéyen was born in the colonial context from the contact between French settlers and African slaves in the French West Indies (see [START_REF] Prudent | Des baragouins à la langue antillaise[END_REF], [START_REF] Chaudenson | La créolisation: théorie, applications, implications[END_REF] among others). It has historically been stigmatised and viewed as a "lesser" form of language compared to French, the language of the colonisers. In terms of language use, Kréyòl gwadloupéyen is the primary language of daily communication for a large part of the population of Guadeloupe, particularly in informal settings. French, on the other hand, is used in formal and official contexts, such as in schools, government institutions, and the media. In this context of diglossia [START_REF] Jeannot | L'enseignement du français en contexte diglossique Guadeloupéen: état des lieux et propositions. Former les enseignants du XXIème siècle dans toute la francophonie[END_REF], code-mixing is frequent, which is an obvious challenge for ASR systems. In short, creole languages share most of their lexicon with the dominant language (the lexifier language), while their grammar is significantly different from the grammar of the lexifier. The origins of the grammatical differences might be a matter of debate (see [START_REF] Salikoko S Mufwene | Jargons, pidgins, creoles, and koines: What are they?[END_REF][START_REF] Velupillai | Pidgins, creoles and mixed languages[END_REF] among others). To give only one example of the distance and similarities of French and Kréyòl gwadloupéyen, see ( 1 The NSF-IRES 1952568: Experimental linguistics in the Caribbean seeks to provide students with an international experience conducting linguistic research on low-resource and under-described creole languages like Kréyòl gwadloupéyen. During this 5-7 weeks program, fellows investigate a linguistic phenomenon in Gwadloupéyen on the ba-sis of raw data (spontaneous speech or directed interviews) they collect to contribute to the description and documentation of the language. As previously noted, one of biggest challenges for field linguists and even more so, for the NSF-IRES fellows, remains time invested with transcriptions. Often, these recordings are unexploited for lack of time, adding to the issue of under-description. Only 60min of the approximately 10 hours of recordings collected in 2022 was transcribed, and this only after the program had ended. Notwithstanding code switching/mixing, the fellows' unfamiliarity with the language's phonology made the transcription exercise arduous and lengthier.

Automations

Data

The ASR experiments are based on the work of [START_REF] Macaire | Automatic speech recognition and query by example for creole languages documentation[END_REF], who used a 60-minute-long speech corpus of spontaneous speech in Kréyòl gwadloupéyen for training.

The testing data consist of several hours of raw, unsegmented, and untranscribed speech recorded during a 2022 fieldwork. The speech is spontaneous and sparse across the recording, with overlapping speech, laughters, silences, and random noises spread across the collection. The speech segments are also not necessarily in Kréyòl, and even if the limit between French and Kréyòl gwadloupéyen is not clear, some segments are clearly in French and even English. One 1-hour-long recording was selected, which, after some verification, contains a majority of segments in Kréyòl.

Preprocessing

Speech processing systems generally expect short utterances of clear speech, so the type of data described previously is not usable as is and needs to be preprocessed. Following the ideas of the sparse transcription model [START_REF] Bird | Sparse transcription[END_REF], we used auditok1 , a Voice Activity Detection tool, to filter out non-speech segments. This tool works in an unsupervised fashion, with detection based on the energy of the audio signal. Although more accurate VAD tools are available, auditok provides a good baseline for this preliminary study.

ASR and Self-supervised Learning

Self-supervised learning (SSL) is the task of learning powerful representations from huge unlabeled data to recognise and understand patterns from a less common problem. These models allows to improve performance on downstream tasks for ASR in low-resource contexts [START_REF] Baevski | Effectiveness of self-supervised pretraining for speech recognition[END_REF][START_REF] Kawakami | Learning robust and multilingual speech representations[END_REF]. These work are based on the Wav2Vec2.0 [START_REF] Baevski | wav2vec 2.0: A framework for self-supervised learning of speech representations[END_REF] model. It builds context representations from continuous speech representations and dependencies are obtained by the self-attention mechanism across the entire sequence of latent representations end-toend. In [START_REF] Conneau | Unsupervised cross-lingual representation learning for speech recognition[END_REF], multilingual pretraining of Wav2Vec2.0 model on 53 languages with more than 56k hours of unlabeled speech data (XLSR-53) has shown to construct better speech representations for cross-lingual transfer. It is in this context that we consider fine-tuning this model on creole languages. In [START_REF] Evain | LeBenchmark: A Reproducible Framework for Assessing Self-Supervised Representation Learning from Speech[END_REF], several Wav2Vec2.0 models (LeBenchmark) specific to French language were pretrained. We propose to fine-tune these models on creole languages. Results are generated with a Connectionist Temporal Classification (CTC) beam search decoder [START_REF] Graves | Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks[END_REF]. CTC is an algorithm that assign a probability for any Y given an X. In our case X represent the acoustic features generated by LeBenchmark and Y the items in the orthographic transcription. The combination of LeBenchmark and CTC allowed us to produce an orthographic transcription of every speech segment provided by the VAD algorithm.

Evaluation

A gold standard has been created by the second author using the transcription automatically generated. We computed a Character Error Rate (CER) and a Word Error Rate (WER) on a set of 549 utterances. WER and CER calculates the percentage of items (words or character) that are incorrectly recognised in relation to the total number of items in a reference transcript. We obtained a CER of 0.45 and a WER of 0.728. We present in figure 1 the distribution of the WER and CER per utterances. To improve the visibility of the figure, We removed 5 examples that were too high. Although the overall results may be deemed suboptimal, the boxplot analysis reveals that a considerable proportion of utterances exhibit a WER of less than 50%. This suggests that a significant number of the generated utterances remain usable for downstream applications.

While evaluating a speech recognition system, its usability is often only based on the WER and CER . The results obtained are not groundbreaking but our collaboration between NLP scientists and linguists could help us understand how the system created is useful, how it can be exploited and how it can be improved.

Code-mixing: An under-resourced language is generally in contact with a widely spoken language. In our case, because French is the official language of Guadeloupe island and because some of the linguists involved in the data collection were English speakers, Gwadloupéyen, French and English were intertwined in the recordings. Non-Gwadloupéyen segments were then transcribed with the Gwadloupéyen norms. It seems unlikely to automatically differentiate French and Gwadloupéyen segments due to their lexical similarity. However, recent language diarisation tool could help us to filter out English segments (e.g. [START_REF] Liu | End-to-End Language Diarization for Bilingual Code-Switching Speech[END_REF].

Voice Activity Detection: VAD was highly accurate and saved time by filtering out non-speech segments. A few inaccuracies have however been mentioned specifically for segments starting with non-voiced consonants. the algorithm also tended to over-segment some segments that belonged together.

Automatic transcription: The quality of the transcriptions generated was not uniform across the recording (cf. Figure 1). While some transcriptions were not exploitable at all, others happen to be helpful support for transcription. On one hand, some of the utterance had a WER closed to 0 which allowed us to just copy paste the generated transcription to the gold standard with minor corrections. On the other, for utterances with more errors, the transcription could help to more clearly identify what is said.

Transcription errors: Besides the errors due to code mixing, most of the errors of the systems were due to oversegmentation of tokens. However, this type of errors could be mitigated by plugging a language model at the end of the CTC system. Another error noticed was the difficulty of the system to correctly identify the nasals which are usually recognised as orals (cf. Table 1).

Conclusion

We have detailed the first stage of a joint effort between field linguists and NLP experts to aid in transcribing Kréyòl Gwadloupéyen field linguistic data. Our approach involved using a voice ac-comments gold standard automatic generation the final nasal is recognised as two orals zot matinike gwadloupeyen zoln patinike gwadloup ee the sentence was French deux saison deu sezon segmentation error zo kay an grante jo kay angrandte segmentation errors and nasal confusion matinik e gwadloupeyen martini ke gwadelou pe ent segmentation error se limajiner a sa se limaj jener a sa segmentation and transcription errors byen pale de bonda nou kay soukre bonda mye fame de gonda nou ka ai soucebo Our initial work has prompted us to consider possibilities beyond conventional metrics such as WER and CER and to explore how even a transcription with a high error rate can still be useful. These early results have led us to question the relevance of standard metrics for evaluating a transcription system that can output words out of vocabulary. While a naive approach would be to assume that an automatically generated transcription is simply a starting point for post-editing and corrections (Bird, 2020, p.2), we have found that it can offer support for creating a gold standard and help transcribers better identify the content of a recording, especially when they are not confident in the target language. Moreover, the errors made by the system have increased our understanding of the requirements for a speech recognition system, potentially leading to improved recording quality in the future.

Moving forward, we will look to improve the output of the system. This will involve utilising an overlapping speech detector to eliminate noisy utterances, employing a language model to prevent token hyper-segmentation, and gradually improving the quality of the training data to enhance the transcription.

Figure 1 :

 1 Figure 1: WER and CER distributions

Table 1 :

 1 Examples of transcriptions

https://auditok.readthedocs.io/en/latest/