Bilel Elbetch
email: elbetchbilal@gmail.

Ali Moussaoui
email: moussaoui@

NONLINEAR DIFFUSION IN THE MULTI-PATCH LOGISTIC MODEL *

Keywords: Population Dynamics, Logistic equation, Nonlinear diffusion, Slow-fast systems; Tikhonov's theorem, Perfect mixing AMS subject classifications. 37N25, 92D25, 34D23, 34D15

In this work, we study a multi-patch model, where the patches are coupled by nonlinear asymmetrical migration terms, and each patch follows a logistic law. First, used the theory of a cooperative differential system, we prove the global stability of the model. Next, in the case of perfect mixing, i.e when the migration rate tends to infinity, the total population follows a logistic law with a carrying capacity which in general is different from the sum of the n carrying capacities, and depends on the migration terms. Second, we determine, in some particular cases, 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. Finally, for the two-patch model, we give a complete classification of the model parameter space as to whether nonlinear dispersal is beneficial or detrimental to the sum of two carrying capacities.

1. Introduction. There are many factors affecting the growth and the general dynamics of population. One such important factor is the dispersal amounts and in random ways. These dispersal can cause disturbances to the various ecosystems as well as to the persistence or extinction of organisms. Bibliographies can be found in the work of Levin [START_REF] Levin | Dispersion and population interactions[END_REF][START_REF] Levin | Spatial patterning and the structure of ecological communities[END_REF], Holt [START_REF] Holt | Population dynamics in two patch environments: some anomalous consequences of an optimal habitat distribution[END_REF], Allen [START_REF] Allen | Persistence and extinction in Lotka-Volterra reaction-diffusion equations[END_REF][START_REF] Allen | Persistence and extinction in single-species reaction-diffusion models[END_REF][START_REF] Allen | Persistence, extinction, and critical patch number for island populations[END_REF], and Gurney and Nisbet [START_REF] Gurney | The regulation of inhomogeneous populations[END_REF].

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

(1.1)

dx i dt = r i x i 1 - x i K i + Υ i (x), i = 1, . . . , n,
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 Υ i represent one of the three types of different mechanisms. The mechanism for linear diffusion is given by:

(1.2) Υ i (x) = n j=1,j̸ =i β ij (x j -γ ij x i ) , , i = 1, . . . , n,
where β ij and γ 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:

(1.3) Υ i (x) = n j=1,j̸ =i β ij x i (x j -γ ij x i ) , , i = 1, . . . , n,
where β ij and γ 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:

(1.4) Υ i (x) = n j=1,j̸ =i β ij x 2 j -γ ij x 2 i , , i = 1, . . . , n,
where β ij and γ 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 Lotka-Volterra reaction-diffusion equations[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 (1.1)(1.2) for n = 2 ( see Theorem 1 in [START_REF] Allen | Persistence and extinction in Lotka-Volterra reaction-diffusion equations[END_REF]). For model (1.1)(1.3), Allen [1, 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 Lotka-Volterra reaction-diffusion equations[END_REF] for n-patch model (1.1)(1.3) and Proposition 1 of [START_REF] Allen | Persistence and extinction in Lotka-Volterra reaction-diffusion equations[END_REF] for 2-patch case; Theorem 4 of [START_REF] Allen | Persistence and extinction in Lotka-Volterra reaction-diffusion equations[END_REF] for the n-patch model (1.1)(1.4).

Lu and Takeuchi [START_REF] Lu | Global asymptotic behavior in single-species discrete diffusion systems[END_REF] have also considered the multi-patch logistic model with nonlinear diffusion terms, i.e the system (1.1) (1.4). They obtained the sufficient and necessary conditions for the system to be globally stable ( see Theorem 3 in [START_REF] Lu | Global asymptotic behavior in single-species discrete diffusion systems[END_REF]).

DeAngelis and Zhang [START_REF] Deangelis | Effects of dispersal in a non-uniform environment on population dynamics and competition: a patch model approach[END_REF], DeAngelis et al. [START_REF] Deangelis | Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems[END_REF] and Zhang et al. [START_REF] Zhang | Effects of dispersal on total biomass in a patchy heterogeneous system: analysis and experiment[END_REF] have considered the model (1.1) with (1.5) Υ i (x) = n j=1,j̸ =i

(x i-1 -2x i + x i+1 ), i = 1, • • • , n,
where we denote x 0 = x n and x n+1 = x 1 , allowing the patches to join in a circle so that the same relationships hold between x i , x i-1 and x i+1 for all values of i. An interesting result of (1.1)(1.5) is that in the case r i = K i , for i = 1, • • • , n, the total population at steady state satisfies

(1.6) n i=1 x * i > n i=1 K i .
Our aim in this work is to study the model of n patches coupled by nonlinear migration terms ( same type of dispersion (1.4)). In particular, we are interested in studying the effect of nonlinear dispersion on the dynamics of population, and to compare some results for the nonlinear dispersion with those obtained for the linear dispersion in [START_REF] Elbetch | The multi-patch logistic equation[END_REF][START_REF] Elbetch | The multi-patch logistic equation with asymmetric migration[END_REF] for the model (1.7). In 2021, Elbetch et al. [START_REF] Elbetch | The multi-patch logistic equation with asymmetric migration[END_REF] have considered the model of multi-patch logistic growth, coupled by asymmetric linear migration terms (1.7)

dx i dt = r i x i 1 - x i K i + β n j=1,j̸ =i (θ ij x j -θ ji x i ) , i = 1, • • • , n,
where n is the number of patches in the system. The parameters r i and K i are respectively the intrinsic growth rate and the carrying capacity of patch i. The term

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on the right hand side of the system (1.7) describes the effect of the linear migration between the patches, where β is the migration rate and Θ := (θ ij ) is the matrix representing the migrations between the patches. Note that, the system (1.7) is studied also by Elbetch et al. [START_REF] Elbetch | The multi-patch logistic equation[END_REF] and Takeuchi [START_REF] Takeuchi | Cooperative systems theory and global stability of diffusion models[END_REF] in the case when the matrix Θ is symmetric. We recall that, when the matrix of migration Θ is irreducible, System (1.7) admits a unique positive equilibrium which is globally asymptotically stable (GAS), see [7, Theorem 2.2], [6, Theorem 1] or [14, Theorem 6.1], when β → ∞, this equilibrium tend to

i δ i r i i δ 2 i α i (δ 1 , . . . , δ n ),
where α i = ri Ki and (δ 1 , . . . , δ n ) T the vector which generate the vector space ker Θ ( see Section 3).

In [START_REF] Elbetch | The multi-patch logistic equation[END_REF][START_REF] Elbetch | The multi-patch logistic equation with asymmetric migration[END_REF], Elbetch et al. have answered in some particular cases of the model (1.7) to the following important question: Is it possible, depending on the migration rate, that the total equilibrium population be larger than the sum of the capacities i K i ? This question is of ecological importance since the answer gives the conditions under which the linear dispersal is either beneficial or detrimental to total equilibrium population. Note that, this question has been studied by many researches ( see [START_REF] Arditi | In 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] Auger | Increase of maximum sustainable yield for fishery in two patches with fast migration[END_REF][START_REF] Deangelis | Persistence and stability of seed-dispersel species in a patchy environment[END_REF][START_REF] Deangelis | Dispersal and heterogeneity: single species[END_REF][START_REF] Deangelis | Effects of dispersal in a non-uniform environment on population dynamics and competition: a patch model approach[END_REF][START_REF] Deangelis | Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems[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] Freedman | Mathematical Models of Population Interactions with Dispersal II: Differential Survival in a Change of Habitat[END_REF][START_REF] Freedman | Global stability and predator dynamics in a model of prey dispersal in a patchy environment. Non linear Anal[END_REF][START_REF] Freedman | Mathematical Models of Population Interactions with Dispersal I: Stability of two habitats with and without a predator[END_REF][START_REF] Gao | Fast diffusion inhibits disease outbreak[END_REF][START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF][START_REF] Zhang | Effects of dispersal on total biomass in a patchy heterogeneous system: analysis and experiment[END_REF] and [START_REF] Gao | How does dispersal affect the infection size?[END_REF][START_REF] Gao | A multipatch malaria model with logistic growth[END_REF] and on susceptibleinfected-susceptible (SIS) patch-model). They proved that, if all the patches do not differ with respect to the intrinsic growth rate (i.e., r 1 = . . . = r n ), then the effect of linear migration is always detrimental. In the case when (K 1 , . . . , K n ) T ∈ ker Θ ( if the matrix Θ is symmetric, the condition (K 1 , . . . , K n ) T ∈ ker Γ means that the patches do not differ with respect to the carrying capacity ), linear migration has no effect on the total equilibrium population. An example when the effect of linear migration is always beneficial, is in the case when Θ is symmetric and all the patches do not differ with respect to the the parameter α = r/K quantifying intraspecific competition (i.e., α 1 = . . . , α n ) ( see also [START_REF] Elbetch | The multi-patch logistic equation with asymmetric migration[END_REF]Prop. 4.2]).

This paper is organized as follows: in Section 2, we introduce the mathematical model and we give some definitions and notations. Next, in Section 3, we study the asymptotic behavior of the stability modulus of the matrix A given by (2.7), and we prove that the stability modulus of A is strictly decreasing as function of the migration rate and it is always negative. This result is used in Section 4 to prove the global stability of the model (2.1). In Section 5, we study the behavior of the system (2.1) in the case when the migration rate goes to infinity by direct method and also by using perturbation arguments. In Section 6, we compare the total equilibrium population with the sum of the n carrying capacities for some parameter space, and by using the same method as Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF], we give a complete analysis of two-patch case in Section 7. In Section 8, two-patch model where one growth rate is much larger than the second one is considered, we compare the total equilibrium population with the sum of two capacities in this case. We perform some numerical simulations to illustrate our results in this paper and give a brief concluding remark.

2. The mathematical model. We consider the model of multi-patch logistic growth, coupled by nonlinear migration terms given by:

(2.1)

dx i dt = r i x i 1 - x i K i + β n j=1 (θ ij x 2 j -θ ji x 2 i ), i = 1, • • • , n,
where x i is the population in patch i, the parameters r i and K i are respectively the intrinsic growth rate and the carrying capacity patch i, n is the number of the patches

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in the system. The parameter β represents the dispersal rate of the population; θ ij ≥ 0 denotes the flux between patches j and i, for j ̸ = i. Note that if θ ij > 0 there is a flux of migration between patches j and i and if θ ij = 0 there is no migration. The system (2.1) of differential equations can be written:

(2.2)

dx i dt = r i x i 1 - x i K i + β n j=1 θ ij x 2 j , i = 1, • • • , n, where (2.3) θ ii = - n j=1,j̸ =i θ ji , i = 1, • • • , n
denotes the outgoing flux of patch i. We denote by Θ the following matrix

(2.4) Θ =      -k̸ =1 θ k1 θ 12 . . . θ 1n θ 21 -k̸ =2 θ k2 . . . θ 2n • . . . . . . . . . θ n1 . . . θ n,n-1 -k̸ =n θ kn      .
We call Θ the movement matrix of the system (2.1). Its columns sum to 0 since the diagonal elements θ ii are defined by (2.3). The matrix

(2.5) Θ 0 := Θ -diag(θ 11 , . . . , θ nn )
which is the same as the matrix Θ, except that the diagonal elements are 0, is called the connectivity matrix. It is the adjacency matrix of the weighted directed graph G, which has exactly n vertices (the patches), and there is an arrow from patch j to patch i precisely when θ ij > 0, with weight θ ij assigned to the arrow.

The system (2.1) can be also rewritten in matrix form as follow:

(2.6)

dx dt = diag(r 1 , . . . , r n )x + Ax 2 ,
where x = (x 1 , . . . , x n ) T , x 2 := (x 2 1 , . . . , x 2 n ) T and A the matrix defined by (2.7)

A := βΘ -diag (α 1 , . . . , α n ) .
where α i = ri Ki . First, we start by giving some definitions.

Definition 2.1. A matrix M = (m ij ) is called cooperative if m ij ≥ 0 for all i ̸ = j.
Recall that the differential system

dx dt = F (x),
is said to be cooperative, if its jacobian matrix is cooperative, i.e., for all i ̸ = j;

(∂F i /∂x j ) ≥ 0, for all x positive.

Definition 2.2. The stability modulus of a matrix M is given by

(2.8) s(M ) = max {Re(λ) : λ is an eigenvalue of M } .
Note that, the matrix A is a non linear part of the system (2.1) and it is a cooperative matrix.

3. Asymptotic behavior of the stability modulus of the matrix A . Using some results proven by Gao and Dong [START_REF] Gao | Fast diffusion inhibits disease outbreak[END_REF] on SIS patch-model. we give in this section some proprieties of the stability modulus of the matrix A defined by (2.7), which is a non linear part of the differential system (2.1). First for all, we recall that, if Θ is irreducible, then 0 is a simple eigenvalue of Θ and all non-zero eigenvalues of Θ have negative real part. Moreover, the kernel of the matrix Θ is generated by a positive vector ( see Lemma 2 in [START_REF] Arino | Number of Source Patches Required for Population Persistence in a Source-Sink Metapopulation with Explicit Movement[END_REF]). In all of this paper, we denote by δ := (δ 1 , . . . , δ n ) T this positive vector. For the existence , uniqueness, and positivity of δ see Lemma In Lemma 2.1 of Guo et al. [START_REF] Guo | Global stability of the endemic equilibrium of multigroup SIR epidemic models[END_REF] gives explicit formulas of the components of the vector δ, with respect of the coefficients of Θ as follow:

(3.2) δ k = T ∈T k (i,j)∈E(T ) θ ij , k = 1, . . . , n,
where T k is the set of all directed trees of n vertices rooted at the k-th vertex, and E(T ) denotes the set of arcs in a directed tree T .

Proposition 3.1. The stability modulus λ * (β) := s(A), with

A = βΘ -diag {α 1 , . . . , α n } satisfies 1. λ * is strictly decreasing in β ∈ [0, +∞[. 2. λ * → -max {α 1 , . . . , α n } as β → 0. 3. λ * → λ * ∞ := - n i=1 α i δ i n i=1 δ i as β → ∞. In addition, if the matrix Θ is symmetric, then λ * ∞ = -1 n n i=1 α i .
Proof. The proof is derived from Corollary 3.5 in Gao and Dong [START_REF] Gao | Fast diffusion inhibits disease outbreak[END_REF]. Indeed, it was shown by Gao and Dong [START_REF] Gao | Fast diffusion inhibits disease outbreak[END_REF] for a SIS patch model that, the function β → s(F -D + βΘ) is strictly decreasing in β ∈ [0, ∞[ where F and D be a two positive diagonal matrix. Therefore, for F = I and D = diag(α 1 , . . . , α n ), we get

0 > d dβ s(I-diag(α 1 , . . . , α n )+βΘ) = d dβ (1 + s(-diag(α 1 , . . . , α n ) + βΘ)) = dλ * dβ (β).
It is clear that limit λ * tend to -max {α 1 , . . . , α n } as β → 0.

When β → ∞, by Corollary 3.5 in Gao and Dong [START_REF] Gao | Fast diffusion inhibits disease outbreak[END_REF] for the choice F -D = -diag(α 1 , . . . , α n ) we obtain the limit λ * ∞ of λ * . In particular, if Θ is symmetric, one has δ i = 1 for all i.

As corollary of the previous proposition, we have the result:

Corollary 3.2. Consider the matrix A defined by (2.7), then s(A) < 0.

4. Global dynamics. In this section, our goal is to study the dynamics of the system (2.1). Note that, in the absence of migration, i.e the case where β = 0, the system (2.1) admits (K 1 , . . . , K n ) as a non trivial equilibrium point, which furthermore is GAS, and the origin as trivial equilibrium which is unstable. The problem is whether or not, the equilibrium continues to be exist and GAS for any β > 0. First, we prove the following result:

Proposition 4.1. The positive cone R n + is positively invariant for the system (2.1).

Proof. Suppose that, at a given time t, one of the state variables of the system (2.1) is at a boundary of R n + , meaning that at least one population is at 0. We suppose that x i = 0, and x j ≥ 0 for all j ̸ = i, then the dynamics of x i is given by

dx i dt = β j̸ =i θ ij x 2 j ≥ 0.
So each trajectory initiated at a boundary of R n + either remains at the boundary or goes to the interior of R n + . According to [30, Proposition B.7, page 267], no trajectory comes out of R n + . Therefore, R n + is positively invariant for (2.1).

To establish the boundedness of solutions of (2.1), we have the following result:

Proposition 4.2. For any non negative initial condition, the solutions of the system (2.1) remain non negative and positively bounded. Moreover, the set

(4.1) Λ = (x 1 , . . . , x n ) ∈ R n : 0 ≤ n i=1 x i ≤ ξ * 2 ξ * 1
is positively invariant and is a global attractor for (2.1), where ξ * 1 = min 1≤i≤n r i and

ξ * 2 = n i=1 r i K i .
Proof. To show that all solutions are bounded, we consider the quantity defined by X T (t) = n i=1 x i (t). So, we have

(4.2) ẊT (t) = n i=1 r i x i (t) 1 - x i (t) K i .
For all r i , K i ∈ R * + , we have the following inequality:

(4.3) r i x i 1 -xi Ki ≤ r i (K i -x i ).
Substituting Equation (4.3) into (4.2), we get

(4.4) ẊT (t) ≤ -ξ * 1 X T (t) + ξ * 2 for all t ≥ 0, which gives (4.5) X T (t) ≤ X T (0) - ξ * 2 ξ * 1 e -ξ * 1 t + ξ * 2 ξ * 1 , for all t ≥ 0. Hence, (4.6) X T (t) ≤ max X T (0), ξ *
Therefore, the solutions of system (2.1) are positively bounded and defined for all t ≥ 0. From (4.5) it can be deduced that the set Λ is positively invariant and it is a global attractor for the system (2.1).

Theorem 4.3. Assume that the matrix Θ = (θ ij ) n×n (or equivalently, the connectivity matrix Θ 0 ) is irreducible. The model (2.1) has a unique positive equilibrium point which is GAS in the positive cone R n + \ {0}.

Proof. Consider the system (2.1). According to a result of Lu and Takeuchi [29,

Theorem 3] for the nonlinear diffusion system with logistic growth rate functions, the system (2.1) possesses a globally stable positive equilibrium if the matrix A given by (2.7) is stable, (i.e., there exists a positive diagonal matrix C such that CA + A T C is negative definite), which equivalent s(A) < 0 by Lemma 3 of [START_REF] Lu | Global asymptotic behavior in single-species discrete diffusion systems[END_REF]. According to Corollary 3.2, the model (2.1) has a unique positive equilibrium point which is GAS in the positive cone R n + \ {0}.

Remark 1. The matrix Θ being irreducible means that the set of patches cannot be partitioned into two nonempty disjoint subsets, I and J, such that there is no migrations between a patch in subset I and a patch in subset J. The matrix Θ is assumed to be irreducible throughout the rest of the paper. Therefore species can reach any patch from any patch either directly or through other patches.

In all of this work, the GAS equilibrium of the system (2.1), whose existence is shown

in Theorem 4.3, is denoted by E * n (β) = (x * 1 (β), . . . , x * n (β))
, and by T * n (β) the total equilibrium population (4.7)

T * n (β) = n i=1
x * i (β).

Large diffusion rate.

In this section our aim is to study the behavior of E * n (β) and T * n (β), defined by (4.7), for large migration rate, i.e when β → ∞.

5.1. The fast dispersal limit. We have the following result:

Theorem 5.1. We have:

(5.1) lim β→+∞ E * n (β) = n i=1 √ δ i r i n i=1 δ i α i ( δ 1 , . . . , δ n )
where α i = r i /K i . Moreover, if the matrix Θ is symmetric, then the equilibrium

E * n (β) converges to an element of ker Θ (5.2) lim β→+∞ E * n (β) = n i=1 r i n i=1 α i (1, . . . , 1).
Proof. The equilibrium E * n is a solution of the algebraic system:

(5.3) 0 = r i x i 1 - x i K i + β n j=1,j̸ =i (θ ij x 2 j -θ ji x 2 i ), i = 1, . . . , n.
The sum of these equations shows that E * n (β) satisfies the following equation

(5.4) n i=1 r i x i 1 - x i K i = 0. 7
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Therefore E * n (β) belongs to the ellipsoid:

(5.5)

E n-1 := x ∈ R n : Υ(x) := n i=1 r i x i 1 - x i K i = 0 .
Note that, this ellipsoid is independent of the migration terms β and θ ij . The ellipsoid E n-1 passes through the points O, and A = (K 1 , . . . , K n ).

So, the equilibrium E * n is the solution in the positive cone, of the equation F β = 0, where (5.6)

F β (x) := F β 1 (x), . . . , F β n-1 (x), n i=1 r i x i 1 - x i K i , with (5.7) F β i (x) = 1 β r i x i 1 - x i K i + n j=1,j̸ =i (θ ij x 2 j -θ ji x 2 i ), i = 1, . . . , n -1.
On the other hand, the limit equations (obtained when β → ∞) are given by:

(5.8)

F ∞ (x) := F ∞ 1 (x), . . . , F ∞ n-1 (x), n i=1 r i x i 1 - x i K i , with (5.9) F ∞ i (x) = n j=1,j̸ =i (θ ij x 2 j -θ ji x 2 i ), i = 1, . . . , n -1.
According to Lemma B.1 of [START_REF] Elbetch | The multi-patch logistic equation with asymmetric migration[END_REF], the system (5.9) admits unique solution given by:

x i = δi δn x n for all i = 1, • • • , n -1. So, the solutions of the equation F ∞ (x) = 0 is
given by the solutions of the following system:

(5.10)

         x i = δ i δ n x n , i = 1, • • • , n -1. n i=1 r i x i 1 - x i K i = 0, which admits x n = 0 and x n = n i=1 √ δiri n i=1 δiαi
√ δ n as solutions. So, the equation F ∞ = 0 admits two solutions, 0 and

E * n (∞) := n i=1 √ δ i r i n i=1 δ i α i ( δ 1 , . . . , δ n ).
The ellipsoid E n-1 is compact, so the equilibrium E * n (β) has at least one limit point in the ellipsoid, when β goes to infinity. By the same reason as in [15, Proof of Theorem 3.3], we prove the convergence of

E * n (β) to E * n (∞).
If the matrix Θ is symmetric, one has δ i = 1, for all i.

As a corollary of the previous theorem we obtain the following result which describes the total equilibrium population for large growth rate:

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Corollary 5.2. We have

(5.11) T * n (+∞) = n i=1 δ i n i=1 √ δ i r i n i=1 δ i α i .
Moreover, if the matrix Θ is symmetric, then:

(5.12)

T * n (+∞) = n n i=1 r i n i=1 α i .
Proof. The sum of the n components of the point E * n (∞) immediately gives the equation (5.11).

We remark that, the total equilibrium population for large migration rate (5.12) is equal the total equilibrium population for large migration rate of the multi-patch logistic model with linear diffusion obtained in [START_REF] Elbetch | The multi-patch logistic equation[END_REF]Equation 24 ]. This result show that, if the movement between the n patches is symmetric, the nonlinear diffusion has no influence on the total equilibrium population for large migration rate.

Two time scale dynamics.

We can use the theory of singular perturbations to obtain a better understanding of the behavior of the system in the case of perfect mixing. We have the following result:

Theorem 5.3. Let (x 1 (t, β), . . . , x n (t, β)) be the solution of the system (2.1) with initial condition (x 0 1 , • • • , x 0 n ) satisfying x 0 i ≥ 0 for i = 1, . . . , n.
Let Y (t) be the solution of the equation

(5.13) dX dt = rX 1 - X n i=1 √ δ i K ,
where

(5.14) r = n i=1 √ δ i r i n i=1 √ δ i , K = n i=1 √ δ i r i n i=1 δ i α i and α i = r i /K i .
Then, when β → ∞, we have

(5.15) n i=1 x i (t, β) = Y (t) + o β (1) uniformly for t ∈ [0, +∞)
and, for any t 0 > 0, we have (5.16)

x i (t, β) = √ δ i n i=1 √ δ i Y (t) + o β (1) i = 1, . . . , n, uniformly for t ∈ [t 0 , +∞).
Proof. Let X(t, β) = n i=1 x i (t, β). We rewrite the system (2.1) using the variables (X, x 1 , • • • , x n-1 ), and get:

(5.17)

         dX dt = n i=1 r i x i 1 - x i K i , dx i dt = r i x i 1 - x i K i + β n j=1,j̸ =i (θ ij x 2 j -θ ji x 2 i ), i = 1, . . . , n -1.

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This system is actually a system in the variables (X, x 1 , • • • , x n-1 ), since, whenever

x n appears in the right hand side of (5.17), it should be replaced by (5.18)

x n = X - n-1 i=1 x i .
When β → ∞, (5.17) is a slow-fast system, with one slow variable, X, and n -1 fast variables, x i for i = 1 . . . n -1. As suggested by Tikhonov's theorem [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 variables in the time scale τ = βt. We get (5. [START_REF] Freedman | Mathematical Models of Population Interactions with Dispersal I: Stability of two habitats with and without a predator[END_REF])

dx i dτ = 1 β r i x i 1 - x i K i + n j=1,j̸ =i (θ ij x 2 j -θ ji x 2 i ), i = 1, . . . , n -1.
where x n is given by (5.18). In the limit β → ∞, we find the fast dynamics

dx i dτ = n j=1,j̸ =i (γ ij x 2 j -γ ji x 2 i ), i = 1, • • • , n -1.
This is an (n-1)-dimensional non linear differential system. According to [15, Lemma B.1], we can deduced that, this system admits unique equilibrium GAS given by

√ δ 1 n i=1 √ δ i X, . . . , δ n-1 n i=1 √ δ i X T .
Thus, the slow manifold of System (5.17) is given by (5.20)

x i = √ δ i n i=1 √ δ i X, i = 1, . . . , n -1.
As this manifold is GAS, Tikhonov's theorem ensures that after a fast transition toward the slow manifold, the solutions of (5.17) are approximated by the solutions of the reduced model, which is obtained by replacing (5.20) into the dynamics of the slow variable, that is:

dX dt = n i=1 r i X n i=1 √ δ i δ i 1 - X n i=1 √ δ i K i δ i = rX 1 - X n i=1 √ δ i K ,
where r and K are defined in (5.14). Therefore, the reduced model is (5.13). The system (5.13) admits

X * = n i=1 δ i K = n i=1 δ i n i=1 √ δ i r i n i=1 δ i α i
as a positive equilibrium point, which is GAS in the positive axis, 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 variables, where t 0 is as small as we want. Therefore, letting Y (t) be the solution of the reduced model (5.13) with initial condition Y (0) = X(0, β) = n i=1 x 0 i , then, when β → ∞, we have the approximations (5.15) and (5.16).

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Note that, in the case of perfect mixing, the approximation (5.15) shows that, the total population behaves like the unique logistic equation (5.13) and then, when t and β tend to ∞, the total population x i (t, β) tends toward

n i=1 δ i K = n i=1 δ i n i=1 √ δ i r i δ i α i
as stated in Theorem 5.1. The approximation (5.16) shows that, with the exception of a thin initial boundary layer, where the density population x i (t, β) quickly jumps from its initial condition x 0 i to the average

√ δ i X 0 / n i=1
√ δ i , each patch of the model behaves like the single logistic equation

du dt = ru 1 - u √ δ i K , i = 1, . . . , n.
where r and K are given in (5.14). Hence, when t and β tend to ∞, the density

population x i (t, β) tends toward K = √ δ i n i=1
√ δiri δiαi as stated in Theorem 5.1.

As a corollary of the previous theorem we obtain the following result which describes the behavior of the system (2.1) for perfect mixing and nonlinear symmetrical dispersal:

Corollary 5.4. Assume that the matrix Θ is symmetric. Let (x 1 (t, β), . . . , x n (t, β)) be the solution of the system (2.1) with initial condition

(x 0 1 , • • • , x 0 n ) satisfying x 0 i ≥ 0 for i = 1 • • • n. Let Y (t)
be the solution of the equation

(5.21) dX dt = rX 1 - X nK ,
where

(5.22) r = n i=1 r i n , K = n i=1 r i n i=1 α i and α i = r i /K i .
Then, when β → ∞, we have

(5.23) n i=1 x i (t, β) = Y (t) + o β (1) uniformly for t ∈ [0, +∞)
and, for any t 0 > 0, we have

(5.24) x i (t, β) = Y (t) n + o β (1) i = 1, . . . , n, uniformly for t ∈ [t 0 , +∞).
Proof. If Θ is symmetric, one has δ i = 1 for all i. Therefore, the formulas (5.13),

(5.14), and the approximations (5.15), (5.16) for δ i = 1, give the proof of the corollary.

Comparison of

T * n (+∞) with i K i .
According to Formula (5.11), it is clear that the total equilibrium population at β = 0 and at β = +∞ are different in general.

In the remainder of this section, we give some conditions, in the space of parameters r i , K i , α i and δ i , for limit of the total equilibrium population when β → ∞ to be greater or smaller than the sum of the carrying capacities. We show that all three cases are possible, i.e T * n (+∞) can be greater than, smaller than, or equal to T * n (0).

First, we start by giving some particular values of the parameters for which equality holds.

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(+∞) = i K i .
Proof. Direct consequence of the equation (5.11).

Note that, if the matrix Θ is symmetric, then, Prop. 5.5 says that if all α i are equal, then T * n (∞) = i K i , which is [14, Prop 4.4] obtained for the multi-patch logistic model with linear diffusion.

In the next proposition, we give two cases which ensure that T * n (0) can be greater or smaller than T * n (+∞). This result can be stated as the following proposition:

Proposition 5.6. Consider the system (2.1).

1. If K 1 √ δ 1 ≤ . . . ≤ K n √ δ n and √ δ 1 α 1 ≤ . . . ≤ √ δ n α n , or if K 1 √ δ 1 ≥ . . . ≥ K n √ δ n and √ δ 1 α 1 ≥ . . . ≥ √ δ n α n , then T * n (+∞) ≥ T * n (0). 2. If K 1 √ δ 1 ≥ . . . ≥ K n √ δ n and √ δ 1 α 1 ≤ . . . ≤ √ δ n α n , or if K 1 √ δ 1 ≤ . . . ≤ K n √ δ n and √ δ 1 α 1 ≥ . . . ≥ √ δ n α n , then T * n (+∞) ≤ T * n (0).
In both items, if at least one of the inequalities in

K 1 √ δ 1 ≤ . . . ≤ K n √ δ n or K 1 √ δ 1 ≥ . . . ≥ K n √ δ n
is strict, then the inequality is strict in the conclusion.

Proof. Apply Lemma B.2 in [START_REF] Elbetch | The multi-patch logistic equation with asymmetric migration[END_REF] with the following choice:

w i = √ δ i , u i = K i √ δ i ,
and v i = √ δ i α i , for all i = 1, . . . , n.

If the matrix Θ is symmetric, one has δ i = 1, for all i, and Prop. 5.6 becomes Corollary 5.7. Consider the system (2.1). Assume that Θ is symmetric.

1. If K 1 ≤ . . . ≤ K n and α 1 ≤ . . . ≤ α n , or if K 1 ≥ . . . ≥ K n and α 1 ≥ . . . ≥ α n , then T * n (+∞) ≥ T * n (0). 2. If K 1 ≥ . . . ≥ K n and α 1 ≤ . . . ≤ α n , or if K 1 ≤ . . . ≤ K n and α 1 ≥ . . . ≥ α n , then T * n (+∞) ≤ T * n (0).
6. Effect of nonlinear dispersal on total population size. In this section, we will investigate how non linear dispersal affects the total population in all patches.

In particular, when is it detrimental or beneficial to the sum of carrying capacities?

Mathematically speaking, we will compare, in some particular cases of the system (2.1), the total equilibrium population T * n (β) , with the sum of carrying capacities

T * n (0) = i K i ,
when the rate of migration β varies from zero to infinity. We show that the total equilibrium population, T * n (β), can either be greater than, smaller than, or equal to the sum of the carrying capacities T * n (0) (non dispersal).

6.1. Nonlinear dispersal my be unfavorable to T * n . In the next proposition, we show that if the growth rates are equal in all patches, then the total equilibrium population is always smaller than the sum of the carrying capacities: Proof. The equation of the tangent space to the ellipsoid E n-1 , defined by (5.5),

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at point A = (K 1 , . . . , K n ) is given by (6.1)

n i=1 (x i -K i ) ∂Υ ∂x i (A) = 0,
where Υ is given by (5.5). Since ∂Υ ∂xi (A) = -r i , (6.1) can be written as follows:

(6.2)

n i=1 r i (x i -K i ) = 0. If we take r 1 = • • • = r n in (6.
2), we get that the equation of the tangent plane to

E n-1 at the point A is n i=1 x i = n i=1 K i .
By the convexity of E n-1 , any point of E n-1 lies in the half-space defined by the inequation

n i=1 x i ≤ n i=1 K i . Therefore E * (β) satisfies n i=1 x * i (β) ≤ n i=1 K i for all β ≥ 0.
Which completes the proof of the proposition.

The result of the previous proposition is the same as Prop. 3.1 and Prop. 6.2 in [START_REF] Elbetch | The multi-patch logistic equation[END_REF] for the linear diffusion.

Independence of T *

n with respect to nonlinear dispersal . In the next proposition we give sufficient and necessary conditions for the total equilibrium population not to depend on the migration rate.

Proposition 6.2. The equilibrium E * n (β) does not depend on β if and only if

(K 2 1 , . . . , K 2 n ) T ∈ ker Θ.
In this case we have E * n (β) = (K 1 , . . . , K n ) for all β > 0.

Proof. The equilibrium E * n (β) is the unique positive solution of the equation

(6.3) f (x) + βΘx 2 = 0,
where f (x) = (f 1 (x 1 ), . . . , f n (x n )) T , and

f i (x i ) = r i x i (1 -x i /K i ), i = 1, . . . n.
Suppose that the equilibrium E * n (β) does not depend on β, then we replace in Equation (6.3):

(6.4) f (E * (β)) + βΘ(E * n (β)) 2 = 0.
The derivative of (6.4) with respect to β gives

(6.5) Θ(E * n (β)) 2 = 0.
Replacing the equation (6.5) in the equation (6.4), we get f (E * n (β)) = 0, so E * (β) = (K 1 , . . . , K n ). From the equation (6.5), we conclude that (K 2 1 , . . . , K 2 n ) T ∈ ker Θ. Now, suppose that (K 2 1 , . . . , K 2 n ) T ∈ ker Θ, then (K 1 , . . . , K n ) satisfies the equation (6.3), for all β ≥ 0. So, E * n (β) = (K 1 , . . . , K n ), for all β ≥ 0, which proves that the total equilibrium population is independent of the migration rate β.

If the matrix Θ is symmetric, the previous proposition asserts that, the K i , for i = 1, . . . , n, are equal if and only if E * n = (K, . . . , K), where K is the common value of the K i . This is [START_REF] Elbetch | The multi-patch logistic equation[END_REF]Proposition 3.2].

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6.3. Nonlinear dispersal my be favorable to T * n . In this section, we give a situation where the non linear dispersal is favorable to the total equilibrium population. Mathematically speaking: Proposition 6.3. Assume that for all j < i, α i θ ij = α j θ ji . Then

T * n (β) ≥ n i=1 K i for all β ≥ 0.
Moreover, if there exist i 0 and j 0 ̸ = i 0 such that r i0 ̸ = r j0 , then

T * n (β) > n i=1 K i , for all β > 0.
Proof. The equilibrium point E * n (β) satisfies the system

(6.6) 0 = α i x * i (β) (K i -x * i (β)) + β n j=1,j̸ =i (θ ij (x * j ) 2 (β) -θ ji (x * i ) 2 (β)), i = 1 • • • n.
Dividing (6.6) by α i x * i , one obtains

x * i (β) = K i + β n j=1,j̸ =i θ ij (x * j (β)) 2 -θ ji (x * i (β)) 2 α i x * i (β)
.

Taking the sum of these expressions shows that the total equilibrium population T * n satisfies the following relation:

T * n (β) = n i=1 K i + β n i=1 n j=1,j̸ =i θ ij (x * j (β)) 2 -θ ji (x * i (β)) 2 α i x * i (β) (6.7) = n i=1 K i + β j<i θ ij (x * j (β)) 2 -θ ji (x * i (β)) 2 α i x * i (β) + θ ji (x * i (β)) 2 -θ ij (x * j (β)) 2 α j x * j (β) = n i=1 K i + β j<i θ ij x * j (β) + θ ji x * i (β) θ ij x * j (β) -θ ji x * i (β) α i x * i (β) + θ ji x * i (β) -θ ij x * j (β) α j x * j (β) = n i=1 K i + β j<i θ ij x * j (β) + θ ji x * i (β) θ ij x * j (β) -θ ji x * i (β) α j x * j (β) -α i x * i (β) α j α i x * j (β)x * i (β)
.

The conditions α i θ ij = α j θ ji can be written ω ij := α i / θ ji = α j / θ ij for all j < i, such that θ ij ̸ = 0 and θ ji ̸ = 0. Therefore, there exists ω ij > 0 such that α j = ω ij θ ij and α i = ω ij θ ji for all i, j with θ ij ̸ = 0 and θ ji ̸ = 0.

Replacing α i and α j in (6.7), one obtains (6.8)

T * n (β) = n i=1 K i + β j<i ω ij θ ij x * j (β) + θ ji x * i (β) θ ij x * j (β) -θ ji x * i (β) 2 α j α i x * j (β)x * i (β)
.

Therefore, T * n (β) ≥ n i=1 K i for all β ≥ 0. As the matrix Θ is assumed irreducible, then the equality holds if and only if β = 0 or θ ij x * j (β) -θ ji x * i (β) = 0, for all i and j. Let us prove that if at least two patches have different growth rates, then equality cannot hold for β > 0. Suppose that there exists β * > 0 such that the positive equilibrium satisfies (6.9) ∀i, j,

θ ij x * j (β * ) = θ ji x * i (β * ) ⇐⇒ ∀i, j, θ ij (x * j ) 2 (β * ) = θ ji (x * i ) 2 (β * ).
Replacing the equation (6.9) in the system (6.6), we get that x * i (β * ) = K i , for all i. Therefore, from (6.9), it is seen that, for all i and j, K 2 j θ ij = K 2 i θ ji . From these equations and the conditions α i θ ij = α j θ ji , we get r i = r j , for all i and j. This is a contradiction with the hypothesis that there exist two patches with different growth rates. Hence the equality in (6.8) holds if and only if β = 0.

As corollary of the previous proposition, we obtain the following result:

Corollary 6.4. Assume that Θ is symmetric. If α 1 = . . . = α n , then T * n (β) ≥ i K i for all β ≥ 0.
Proof. When the matrix Θ is irreducible and symmetric, the hypothesis of Prop.

6.3 implies that α i = α j for all i and j. Indeed if two patches i and j are connected (i.e θ ij = θ ji ̸ = 0), then we have α i = α j . As the matrix Θ is irreducible, for two arbitrary patches, there exists a finite sequence (i, . . . , j) which begins in i and ends in j, such that θ ab ̸ = 0 for all successive patches a and b in (i, . . . , j). Hence α a = α b for all a and b in (i, . . . , j). Hence,

α i = α j .
The result of the corollary 6.4 says that if all α i are equal, non linear dispersal enhances population growth, which is [START_REF] Elbetch | The multi-patch logistic equation[END_REF]Prop. 3.3], which has been proven for the multi-patch logistic model with linear diffusion.

Remark 2. For three patches or more, if the matrix Θ does not verify the condition (∀i, j, θ ij = 0 ⇐⇒ θ ji = 0), then the hypothesis of Prop. 6.3, that for all j < i, α i θ ij = α j θ ji cannot be satisfied. Note that the hypothesis α i θ ij = α j θ ji implies that α 2 i θ ij = α 2 j θ ji , which implies, for all i = 1, . . . , n, one has

n j=1 θ ij α 2 j = n j=1,j̸ =i θ ij α 2 j - n j=1,j̸ =i θ ji α 2 i = n j=1,j̸ =i α 2 i θ ij -α 2 j θ ji α 2 i α 2 j = 0.
Therefore, the hypothesis of Prop. 6.3 implies that ( 1

α 2 1 , . . . , 1 α 2 n ) T ∈ ker Θ. 6.4. Derivative of T * n at β = 0.
Proposition 6.5. The derivative of the total equilibrium population T * n (β) at β = 0 is given by:

(6.10) dT * n dβ (0) = n i=1   1 r i n j=1 θ ij K 2 j   .
Proof. By differentiating the equation (6.7) at β = 0, we get:

dT * n dβ (0) = n i=1 n j=1,j̸ =i θ ij (x * j (0)) 2 -θ ji (x * i (0)) 2 α i x * i (0)
, which gives (6.10), since x * i (0) = K i for all i = 1, . . . , n.

The derivative (6.10)can be written in matrix form as follows:

dT * n dβ (0) = R T ΘK 2 , with R = 1 r1 , . . . , 1 rn T and K 2 = (K 2 1 , . . . , K 2 n ) T .
Note that, if the growth rates are equal in all patches, or, (K 2 1 , . . . , K 2 n ) lies in the vector space ker Θ, then the derivative of the total equilibrium population T * n at β = 0 equal to zero.

7. Two-patch case. In this section, we concentrate on the two-patch model:

(7.1)        dx 1 dt = r 1 x 1 1 - x 1 K 1 + β θ 12 x 2 2 -θ 21 x 2 1 , dx 2 dt = r 2 x 2 1 - x 2 K 2 + β θ 21 x 2 1 -θ 12 x 2 2 .
The total equilibrium population of the model (7.1) in the perfect mixing case (i.e β → ∞) is given by the following formula:

(7.2) T * 2 (+∞) = θ 12 + θ 21 √ θ 12 r 1 + √ θ 21 r 2 θ 12 r 1 /K 1 + θ 21 r 2 /K 2 ,
and the derivative of the total equilibrium population T * 2 (β) at β = 0 becomes

(7.3) dT * 2 dβ (0) = θ 12 K 2 2 -θ 21 K 2 1 1 r 1 - 1 r 2 .
The equilibrium of the system (2.1) is the solutions of the following algebraic system:

(7.4)        0 = r 1 x 1 1 - x 1 K 1 + β θ 12 x 2 2 -θ 21 x 2 1 , 0 = r 2 x 2 1 - x 2 K 2 + β θ 21 x 2 1 -θ 12 x 2 2 .
The sum of two equations of (7.4) shows that the equilibrium points are in ellipse noted E, where its equation is given by: (7.5)

E : r 1 x 1 1 - x 1 K 1 + r 2 x 2 1 - x 2 K 2 = 0.
The ellipse E 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 β and θ ij (shown in red in Fig. 1).

Solving the first equation of system (7.4) for x 2 yields a hyperbola noted H β defined by

H β : h β (x 1 ) := 1 θ 12 θ 21 x 2 1 - r 1 β 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 β . The hyperbola H β ( shown in blue in Fig. 1) depend on the migration rate β. It always passes through the origin and the point B := K 1 , θ21 θ12 K 1 . Notice that, the hyperbola H β intersect the axis (Ox 1 ) at 0 and a second positive point

x 1 = r1/β
θ21+α1/β which always smaller than K 1 . So, the equilibrium points are the

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O

x 1

E * 2 (β) x 2 K 1 K 2 E • H β Fig. 1.
The ellipse E and the hyperbola H β . The equilibrium points are the intersection in the positive cone between E and H β , this intersections contains the origin and a second positive point E * (β).

non negative intersection between the ellipse E and H β . There are two equilibrium points. The first is the trivial point (0, 0) and the second is a non trivial point

E * 2 (β) = (x * 1 (β), x * 2 (β))
whose position depend on migration rate β ( see Fig. 1).

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

x 1 = K 1 . Moreover, E ∩ H 0 = {(K 1 , 0), (K 1 , K 2 )}.
In the case when β → ∞, the hyperbola H β tend to the oblique line

H ∞ : x 2 = θ21 θ12 x 1 . Moreover, E ∩ H ∞ = (0, 0), θ 12 √ θ 12 r 1 + √ θ 21 r 2 θ 12 r 1 /K 1 + θ 21 r 2 /K 2 , θ 21 √ θ 12 r 1 + √ θ 21 r 2 θ 12 r 1 /K 1 + θ 21 r 2 /K 2
It was shown by Arditi et al. [4, Proposition 2, page 54] that only three situations can occur: the case where the total equilibrium population is always greater than the sum of carrying capacities, the case where it is always smaller, and a third case, where the effect of migration is beneficial for lower values of the migration coefficient β and detrimental for the higher values. More precisely, it was shown in [START_REF] Arditi | In dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation[END_REF] that, if n = 2 in (1.7), the following trichotomy holds

• If T * 2 (+∞) > K 1 + K 2 then T * 2 (β) > K 1 + K 2 for all β > 0. • If dT * 2 dβ (0) > 0 and T * 2 (+∞) < K 1 + K 2 , then there exists β 0 > 0 such that T * 2 (β) > K 1 + K 2 for 0 < β < β 0 , T * 2 (β) < K 1 + K 2 for β > β 0 and T * 2 (β 0 ) = K 1 + K 2 . • If dT * 2 dβ (0) < 0, then T * 2 (β) < K 1 + K 2 for all β > 0.
Therefore, the condition T * 2 (β) = K 1 + K 2 holds only for β = 0 and at most for one positive value β 0 . The value β 0 exists if and only if

dT * 2 dβ (0) > 0 and T * 2 (+∞) < K 1 + K 2 .
In the remainder of this section, we analyze the effect of nonlinear dispersal on the total equilibrium population for the two-patch system (7.1). Using the method of Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF], we describe the position affects the equilibrium E * (β) of (7.1) when the migration rate varies from zero to infinity, we will give the condition whether T * 2 is greater or smaller than sum of carrying capacity T * 2 (0) = K 1 + K 2 . We prove there are only three cases as in the 2-patch logistic model with linear diffusion can occur.

We consider the regions in the set of the parameters θ 21 and θ 12 , denoted J 0 , J 1 and J 2 , depicted in Fig. 2 and defined by:

(7.6)                                If r 2 > r 1 then            J 1 = (θ 21 , θ 12 ) : θ12 θ21 > α 2 2 α 2 1 J 0 = (θ 21 , θ 12 ) : α 2 2 α 2 1 ≥ θ12 θ21 > K 2 1 K 2 2 J 2 = (θ 21 , θ 12 ) : K 2 1 K 2 2 > θ12 θ21 If r 2 < r 1 then            J 1 = (θ 21 , θ 12 ) : θ12 θ21 < α 2 2 α 2 1 J 0 = (θ 21 , θ 12 ) : α 2 2 α 2 1 ≤ θ12 θ21 < K 2 1 K 2 2 J 2 = (θ 21 , θ 12 ) : K 2 1 K 2 2 < θ12 θ21 Case r 2 > r 1 0 J 1 J 0 J 2 θ 21 θ 12 θ12 θ21 = K 2 1 K 2 2 θ12 θ21 = α 2 2 α 2 1 Case r 2 < r 1 0 J 2 J 0 J 1 θ 21 θ 12 θ12 θ21 = K 2 1 K 2 2 θ12 θ21 = α 2 2 α 2 1
Fig. 2. The domains J 0 , J 1 and J 2 . In the figure

α 1 = r 1 /K 1 and α 2 = r 2 /K 2 .
We have the following result which gives the conditions for which patchiness is beneficial or detrimental in model (7.1).

Theorem 7.1. The total equilibrium population of (7.1) satisfies the following properties

1. If r 1 = r 2 then T * 2 (β) ≤ K 1 + K 2 for all β ≥ 0.
2. If r 2 ̸ = r 1 , let J 0 , J 1 and J 2 , be defined by (7.6). Then we have:

• if (θ 21 , θ 12 ) ∈ J 0 then T * 2 (β) > K 1 + K 2 for any β > 0 • if (θ 21 , θ 12 ) ∈ J 1 then T * 2 (β) > K 1 + K 2 for 0 < β < β 0 and T * 2 (β) < K 1 + K 2 for β > β 0 , where β 0 = r 2 -r 1 √ θ12 α2 - √ θ21 α1 1 α 1 + α 2 . • if (θ 21 , θ 12 ) ∈ J 2 then T * 2 (β) < K 1 + K 2 for any β > 0 • If θ12 θ21 = K 2 1 K 2 2 , then x * 1 (β) = K 1 and x * 2 (β) = K 2 for all β ≥ 0. Therefore T * 2 (β) = K 1 + K 2 for all β ≥ 0.
Proof. To facilitate comparison of the total equilibrium population T * 2 (β) and

T * 2 (0) = K 1 + K 2 ,
we define a straight line ∆ : If we take r 1 = r 2 , we get that the equation of tangent space to the ellipse E at point A is the equation of ∆. By Prop. 6.1, we deduce T * 2 (β) ≤ T * 2 (0) for all β ≥ 0.

x 1 + x 2 = K 1 + K 2 .
In the case when r 1 ̸ = r 2 , direct calculation finds that the ellipse E and the line ∆ have two intersections:

A = (K 1 , K 2 ), and C = α 2 K 1 + K 2 α 1 + α 2 , α 1 K 1 + K 2 α 1 + α 2 .
We denote by Σ the straight line joint the origin and C. The slope of Σ is equal to

α 2 2 α 2 1
. We distinguish three cases relative position of the three points A, E * 2 (∞), and C, or equivalently, the three lines [OA), H ∞ and Σ whose slopes are

K 2 K 1 , θ 21 θ 12 and α 2 α 1 .
By the method graphic of Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF], we conclude the complete proof.

The result of the previous theorem show that the nonlinear dispersal can lead to an increased, a decreased the total population size in two patches. This prove that, the effect of nonlinear dispersal is the same as the effect of linear dispersal in two-patch logistic model [START_REF] Arditi | In 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]. Moreover, if the movement between two patches is symmetric, then we have the same results. In particular, Theorem 7.1, formula of equilibrium for large migration rate.

For the two-patch model (7.1), we plot in the figure 4.2, the curves of the total equilibrium population T * 2 and the line K 1 + K 2 in term of the migration rate β, for the sets of parameters choosing in Table 1.

Table 1 Parameters values of the three case used in Fig. 3-a,3-b and Fig 3-c. The perfect mixing abundance T * 2 (+∞) is computed with Eq. (7.2) and the derivative of the total equilibrium population at β = 0 is computed with Eq. (7.3). [START_REF] Elbetch | Effects of rapid population growth on total biomass in Multi-patch environment[END_REF], Elbetch is interested in some biological situations that can be found in the nature, that is, the case where several sub-populations grow with different speed. Mathematically speaking, he studied the system (1.7) under the hypothesis that some growth rates tend to infinity (i.e r i → ∞ for some i). In this section, we consider the two-patch model (2.1) and we assume that the growth rate

Figure r

1 r 2 K 1 K 2 θ 12 θ 21 dT * 2 dβ (0) T * 2 (+∞) 3-a 1 2 3.5 1
β β K 1 + K 2 K 1 + K 2 T * 2 T * 2 (a) (b) β K 1 + K 2 T * 2 (c)
Fig. 3. Total equilibrium population T * 2 of the system (7.1) as a function of the migration rate β for the sets of the parameter values given in Table 1. The horizontal line is K 1 + K 2 . . r 2 is much larger than r 1 . For simplicity we denote θ 2 := θ 12 > 0 the migration rate from patch 2 to patch 1 and θ 1 := θ 21 > 0 from patch 1 to patch 2. The model is written:

(8.1)        dx 1 dt = r 1 x 1 1 - x 1 K 1 + β θ 2 x 2 2 -θ 1 x 2 1 , dx 2 dt = r 2 ϵ x 2 1 - x 2 K 2 + β θ 1 x 2 1 -θ 2 x 2 2 ,
where ϵ is assumed to be a small positive number. Denote

E * 2 (β, ϵ) = (x * 1 (β, ϵ), x * 2 (β, ϵ))
the positive equilibrium of (8.1), and T * 2 (β, ϵ), the total equilibrium population. We recall that the derivative of T * 2 (β, ϵ) with respect to β at β = 0 is written as follow:

(8.2) dT * 2 dβ (0, ϵ) = (θ 2 K 2 2 -θ 1 K 2 1 ) 1 r 1 - ϵ r 2 .
The total equilibrium population of the model (8.1) for perfect mixing (i.e β → ∞) is given by the following formula:

(8.3) T * 2 (+∞, ϵ) = ( θ 1 + θ 2 ) ϵ √ θ 2 r 1 + √ θ 1 r 2 ϵθ 2 α 1 + θ 1 α 2 ,
where α i = r i /K i . First, we have the result:

Theorem 8.1. Let (x 1 (t, ϵ), x 2 (t, ϵ)) be the solution of the system (8.1) 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 differential equation

(8.4) dx 1 dt = r 1 x 1 1 - x 1 K 1 + β(θ 2 K 2 2 -θ 1 x 2 1 ) =: φ(x 1 ),
with initial condition u(0) = x 0 1 . Then, when ϵ → 0, we have

(8.5) x 1 (t, ϵ) = u(t) + o ϵ (1), uniformly for t ∈ [0, +∞)
and, for any t 0 > 0, we have

(8.6) x 2 (t, ϵ) = K 2 + o ϵ (1), uniformly for t ∈ [t 0 , +∞).
Proof. When ϵ → 0, the system (8.1) is a slow-fast system, with one slow variable,

x 1 , and one fast variable, x 2 . Tikhonov's theorem [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 ϵ t. One obtains (8.7)

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

(8.8) dx 2 dτ = r 2 x 2 1 - x 2 K 2 .
The slow manifold is given by the positive equilibrium of the system (8.8), i.e x 2 = K 2 , which is GAS in the positive axis. When ϵ goes to zero, Tikhonov's theorem ensures that after a fast transition toward the slow manifold, the solutions of (8.1) converge to the solutions of the reduced model (8.4), obtained by replacing x 2 = K 2 into the dynamics of the slow variable.

The differential equation (8.4) admits as a positive equilibrium

x * 1 (β, 0 + ) := 1/2

r 1 K 1 + r 1 2 K 1 2 + 4 β K 1 θ 2 K 2 2 r 1 + 4 β 2 K 1 2 θ 2 K 2 2 θ 1 r 1 + β K 1 θ 1 (8.9)
As φ(x 1 ) > 0 for all 0 ≤ x 1 < x * 1 (β, 0 + ) and φ(x 1 ) < 0 for all x 1 > x * 1 (β, 0 + ) then, the equilibrium x * 1 (β, 0 + ) is GAS 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 u(t) be the solution of the reduced model (8.4) of initial condition u(0) = x 0 1 , then, when ϵ → 0, we have the approximations (8.5) and (8.6).

As a corollary of the previous theorem, we have the following result which give the limit of the total equilibrium population T * 2 (β, ϵ) of the model (8.1) when ϵ goes to zero:

r 1 K 1 + r 1 2 K 1 2 + 4 β K 1 θ 2 K 2 2 r 1 + 4 β 2 K 1 2 θ 2 K 2 2 θ 1 r 1 + β K 1 θ 1 + K 2 . 21
This manuscript is for review purposes only.

Proof. According to the equations (8.5), (8.6) and (8.9), when ϵ goes to zero, the equilibrium E * 2 (β, ϵ) of the model (8.1) is converge to E * 2 (β, 0 + ) := (x * 1 (β, 0 + ), K 2 ), where x * 1 (β, 0 + ) is given in (8.9).The sum of the coordinates of E * 2 (β, 0 + ) gives the formula (8.10).

In the following proposition, we calculate the derivative and the formula of perfect mixing (i.e when β → ∞) of the total equilibrium population defined by (8.10). By taking the limit of (8.10) when β → ∞, we get that the total equilibrium population X * T (β, 0 + ) tend to (8.12).

Remark 3. We can deduce dT * 2 dβ (0, 0 + ) by using the equation (8.2). Indeed, the limit of (8.2) when ϵ goes to zero gives:

lim ϵ→0 dT * 2 dβ (0, ϵ) = lim ϵ→0 (θ 2 K 2 2 -θ 1 K 2 1 ) 1 r 1 - ϵ r 2 = -θ 1 K 2 1 + θ 2 K 2 2 r 1 = dT * 2 dβ (0, 0 + ).
We can also deduce T * 2 (+∞, 0 + ) by compute the limit of (8.3) when ϵ goes to zero:

lim ϵ→0 T * 2 (∞, ϵ) = lim ϵ→0 ( θ 1 + θ 2 ) ϵ √ θ 2 r 1 + √ θ 1 r 2 ϵθ 2 α 1 + θ 1 α 2 = √ θ 1 + √ θ 2 √ θ 1 K 2 = T * 2 (∞, 0 + ).
We consider the regions in the set of the parameters θ 1 and θ 2 , denoted J 0 and J 1 defined by: (8.14) J 0 = (θ 1 , θ 2 ) :

θ 2 θ 1 > K 2 1 K 2 2 , J 1 = (θ 1 , θ 2 ) : θ 2 θ 1 < K 2 1 K 2 2 .
We have the following result which gives the conditions for which patchiness is beneficial or detrimental in model (8.1) when ϵ goes to zero.

Theorem 8.4. Let J 0 and J 1 be the domains defined in (8.14). Consider the total equilibrium population T * 2 (β, 0 + ) given by (8.10). Then, we have: Therefore, we conclude that the first and second items of the theorem are hold.

9. Concluding remarks. In summary, this work was aimed to find out whether the total equilibrium population of the n patches connected by nonlinear migration asymmetrical migrations can be greater than the sum of the carrying capacities of the n isolated patches, i.e. at equilibrium. Is there a way to make connections between patches that increases the total population? we have assumed that the population obey a logistic type growth at each patches.

We have shown that if the growth rates are equal in all patches, then the total equilibrium population on the n connected patches is always less than or equal to the sum of carrying capacity of the isolated patches.This result can be seen as negative but constitutes in our eyes an interesting result. We then addressed this issue in the general case when the growth rates are different, we have determined criteria for which the total population size of the n connected patches at equilibrium may be greater than the sum of carrying capacities of n isolated patches. we have shown that for high movement rates between the patches, the total equilibrium population cloud exceed total carrying capacity of the n patches for certain combinations of the parameters r i , K i and δ i . For two-patch model, the parameter space is fully classified as to whether nonlinear dispersal is beneficial or detrimental to the sum of two carrying capacities.

Proposition 6 . 1 .

 61 If r 1 = • • • = r n , then the total equilibrium population, defined by (4.7) satisfies T * n (β) ≤ n i=1 K i , for all β ∈ [0, ∞[.

  If the intersection of the ellipse E and the hyperbola H β , i.e., the equilibrium (x * 1 (β), x * 2 (β)), is on or below the line ∆, then T * 2 (β) ≤ T * 2 (0), whereas if the intersection is above the line, then T * 2 (β) ≥ T * 2 (0). The equilibrium point E * 2 (β) is always in ellipse , then, for β = 0, the equilibrium point states at A , and when β increases, E * 2 (β) describes an arc of the ellipse and ends at point E * 2 (∞).

Corollary 8 . 2 .

 82 We have:

Proposition 8 . 3 .

 83 Consider the total equilibrium population (8.10). Then,

2 dβ

 2 (0, 0 + ) ̸ = 0 then β = 0 and the curve of the total equilibrium population intersects the straight line β → K 1 + K 2 in a unique point which is (0, K 1 + K 2 ).

  = θ 12 θ 13 + θ 12 θ 23 + θ 32 θ 13 , δ 2 = θ 21 θ 13 + θ 21 θ 23 + θ 31 θ 23 , δ 3 = θ 21 θ 32 + θ 31 θ 12 + θ 31 θ 32 .

	1 of Cosner et al. [9], Lemma 4.1 and Lemma 1 of Elbetch et al. [14, 15]. Note
	that, if the matrix Θ is symmetric, then ker Θ is generated by δ = (1, . . . , 1) T . On
	the other hand, it is shown in Guo et al. [23, Lemma 2.1] and Gao and Dong [22,
	Lemma 3.1] that the vector (Θ * 11 , . . . , Θ * nn ) T is a right eigenvector of Θ associated with the zero eigenvalue. Here, Θ * ii is the cofactor of the i-th diagonal entry of Θ, and sgn(Θ * ii ) = (-1) n-1 . As in our work, the matrix Θ is assumed to be irreducible, then (-1) n-1 (Θ * 11 , . . . , Θ * nn ) T is strictly positive, i.e δ i = (-1) n-1 Θ * ii > 0 for all i.
	Therefore, we have explicit formula for the components of the vector δ, as functions
	of the coefficients of Θ, at our disposal. For two patches we have δ = (θ 12 , θ 21 ) T , and
	for three patches we have δ = (δ 1 , δ 2 , δ 3 ) T , where
	(3.1)	 	δ 1
		

  Proposition 5.5. Consider the system (2.1). Let we denote

				√	ker Θ the vector
	space generated by the vector ( √ √ ker Θ, then T * n	δ 1 , . . . ,	√	δ n ) T . If the vector 1 α1 , . . . , 1 αn	T	lies in

This manuscript is for review purposes only.

ξ * 1 ,for all t ≥ 0.

This manuscript is for review purposes only.

Acknowledgments. We would like to acknowledge the assistance of volunteers in putting together this example manuscript and supplement.


Hamza Ammar
email: hamza.ammar@enit.utm.tn

Moncef Mahjoub

Néjib Zemzemi

Global Lipschitz stability of multiple space dependent ionic parameters in cardiac electrophysiology

Keywords: Parameter identification, Inverse problem, Reaction-diffusion equations, Carleman estimate, Uniqueness, Stability result

This paper is devoted to analyze the inverse problem of determining two space dependent ionic parameters of a nonlinear reaction diffusion system described by a parabolic-elliptic one, modeling the electrical activity in the heart. We consider the classical phenomenological model in cardiac electrophysiology of FitzHugh-Nagumo to describe the ionic exchanges at the microscopic level. Our main result is the uniqueness and a Lipschitz stability estimate for two ionic parameters (k, γ) of the model using sub-boundary observations over an interval of time. The key ingredients are global Carleman-type estimates with a suitable observations acting on a part of the boundary.

I INTRODUCTION

Let Ω ⊂ R 3 be a bounded connected open set whose boundary ∂Ω is regular enough and Γ 0 , Γ 1 and Γ 2 are nonempty sub-boundaries of ∂Ω. Given T > 0, in Ω we consider a modified version of the bidomain model, it decribes the electrical activity of human induced pluripotent stem cells-cardiomyocytes (hiPSC-CMs) based on multi-electrode array (MEA) experiments where the drug acts directly on the maximal conductance of the targeted ion channel:

                                                 c m (1 + σ -1 e σ i )∂ t v -div(σ i ∇v) = (1 + σ -1 e σ i )(I stim -I ion (v, w)) -A(x) • ∇u e in Q := Ω × (0, T ),
-div(σ e ∇u e ) = c m ∂ t v -I stim + I ion (v, w), in Q := Ω × (0, T ),

∂ t w + f (v, w) = 0 in Q := Ω × (0, T ), σ i ∇v.ν + σ i ∇(u e ).ν = 0 on Σ := ∂Ω × (0, T ), u e = 0 on Σ 0 := Γ 0 × (0, T ), σ e ∇(u e ).ν = 0 on Σ 2 := Γ 2 × (0, T ).
σ e ∇(u e ).ν =

9 k=1 I k el |e k | χ e k on Σ 1 := Γ 1 × (0, T ) , v(., 0) = v 0 , w(., 0) = w 0 in Ω a.e, (I.1)
where

A(x) := -∇(σ -1 e σ i )σ e = σ -1 e (σ i ∇σ e -σ e ∇σ i ). (I.2)
Functions u e and v are respectively the extracellular and transmembrane potentials. The constant c m is the transmembrane specific capacitance, σ i and σ e are respectively the intracellular and extracellular conductivities. |e k | denotes the surface of the electrode k, χ e k denotes its characteristic function, i.e. the function equal to one inside the electrode and zero outside and I k el is the electric current measured by the k th electrode. In an isolated heart, no current flows out of the heart, as expressed by the Neumann condition on Σ 2 . We use homogeneous Dirichlet boundary condition for the extracellular potential (u e = 0) on the three edges connected to the ground denoted by Γ 0 which has a "U" shape on each of our 6 wells and Γ 1 is the surface of our nine electrodes. For the remaining surface denoted by Γ 2 , we use a non flux boundary condition. σ e ∇u e .ν = 0. The transmembrane ionic current I ion is provided by the FitzHugh-Nagumo phenomenological model given by

I ion (v, w) = -kv(v -a)(1 -v) + w, f (v, w) = γv + βw (I.3)
where k, a, γ, β only depend on space, belong to L ∞ (Ω) and satisfy

0 ≤ a(x) ≤ 1, k(x) ≥ k 0 > 0, γ(x) ≥ 0, β(x) ≥ 0, ∀x ∈ Ω (I.4)
and that there exists M > 0 such that

∥k∥ L ∞ (Ω) + ∥γ∥ L ∞ (Ω) + ∥β∥ L ∞ (Ω) ≤ M (I.5)
The variable w denotes the ionic variable. In the first and second equation of system (I.1), the current I stim is an external stimulation, which can be applied at a certain location of the domain for a certain duration. The current I k el is computed using the electric model of Figure 1, by solving the following

dI k el dt + I k el τ = C el τ dU k dt With U k = 1 |e k | e k u e dx (I.6)
where τ = (R i + R el )C el , R i standing for the inner resistance, R el and C el for the resistance and the capacitance of the electrode. In Eq. (I.6) we have introduced U k as the mean value of the extracellular potential u e over the electrode k. We complete this model with initial data u e (•, 0) = u e,0 .

The anisotropic properties of the two media are modeled by an intracellular and extracellular conductivity tensors σ i (x) and σ e (x). Generally, the conductivities σ i and σ e are two matrices given by σ j (x) = σ t j (x)I + (σ l j (x) -σ t j (x))a l (x)a T l (x), where σ l j and σ t j , j ∈ {i, e} are the intra-and extracellular conductivities along and transversal to the direction of the fiber (parallel to a l (x)), respectively. In the case of equal anisotropy, i.e the so-called anisotropy ratios σ t i /σ l i = σ t e /σ l e = 1, the σ i and σ e are simplified as σ i (x) = σ i (x)I and σ e (x) = σ e (x)I with σ i (x) = σ t i (x), σ e (x) = σ t e (x), which is the case we discussed. The purpose of this work is to give an answer to the following question: Is it possible to identify simultaneously, the parameters k and γ, from observed data of the solution v, u e , w and I el on a sub-boundary of ∂Ω and some measurement at fixed times t 0 and t 1 . The proof of the results relies on Carleman estimates and a certain regularity of solutions for the MAE/ hiPSC-CM assays.

II INVERSE PROBLEM: MAIN RESULTS

Let

(v (n) , u (n)
e , w (n) ) be solutions to our problem (I.1) respectivly with initial conditions (v

(n) 0 , u (n) e,0 , w (n) 0 )
and ionic parameters (k (n) , γ (n) ) for n = 1, 2 and they are both under hypothesis of assumptions of property (I.4)-(I.5).

Our first main result in this work gives a Carleman estimate for the Problem (I.1). Let t 0 and t 1 two different times between 0 and T such that t 0 < t 1 and t 1 -t 0 < t 0 2 , without loss of generality we assume that t 0 + t 1 ≤ T (by changing the scale of t and modifying slightly the definition of the weight functions θ m in what follows). We put T 0 = t 0 + t 1 , δ := t 1 -t 0 4 and from now on, we denote by T := T 0 (for notation simplicity). This estimate involves a function ψ in Ω such that

ψ ∈ C 2 (Ω) , ψ > 0 in Ω , ψ = 0 on ∂Ω \ Γ 0 , |∇ψ(x)| > 0 in Ω .
Then we define, for all λ > 0 and s > 0 the weight functions:

φ m (x, t) = e λψ(x) θ m (t) , η m (x, t) = (e 2λ∥ψ∥ ∞ -e λψ(x) )θ m (t) , m = 1, 2 ,
the weighted integral:

I ψ (v, u e ) = Q sφ m (|∂ t v| 2 + |div(σ i ∇v)| 2 ) + (sφ m ) 3 λ 2 |∇v| 2 + (sφ m ) 5 λ 4 |v| 2 + |div(σ e ∇u e )| 2 +(sφ m ) 2 λ 2 |∇u e | 2 + (sφ m ) 4 λ 4 |u e | 2 e -2sηm dx dt,
and the weighted local function of observations: [START_REF] Buhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] , v [START_REF] Buhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] , w [START_REF] Buhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] ) -I ion (k [START_REF] Fursikov | Controllability of evolution equations[END_REF] , v [START_REF] Fursikov | Controllability of evolution equations[END_REF] , w [START_REF] Fursikov | Controllability of evolution equations[END_REF] )

N Γ 0 ,ψ (v, u e , I el ) = Q (sφ m ) 2 I ion (k
2 e -2sηm dx dt

+ Σ 0 |∂ t v| 2 + |∇v| 2 + |v| 2 + |∇u e | 2 dSdt + I el H 1 (0,T ) .
Here v = v (1) -v [START_REF] Fursikov | Controllability of evolution equations[END_REF] , u e = u

e -u

e and I el := (I el 1 , . . . , I el 9 ) = (I ) ∈ (C 1 (0, T )) 9 and θ 0 , θ 1 ∈ C 1 (0, T ) defined by:

θ 0 (t) =          1 (t(2t 0 -t)) 2 , 0 < t ≤ t 0 + δ t 2 0 +δ 2 -2( T 2 -t) 2 (t 2 0 -δ 2 ) 3 t 0 + δ < t ≤ T 2 θ 0 (T -t), T 2 < t < T and θ 1 (t) =          θ 0 (t), 0 < t ≤ t 0 constant, t 0 < t ≤ t 1 θ 0 (t), t 1 < t < T .
The Carleman inequality is then the following:

Theorem II.1: Then there exists a number λ 0 > 0 such that for an arbitrary λ ≥ λ 0 , we can choose a constant s 0 (λ) ≥ 1 satisfying: there exists a constant C = C(s 0 , λ 0 ) > 0 such that, for all s > s 0 ,

I ψ (v, u e ) ≤ CN Γ 0 ,ψ (v, u e , I el ). (II.1)
The constant C > 0 depends continuously on λ 0 but it is independent of s, and λ 0 depends continuously on γ.

Proof The proof of Theorem II.1 is very much technical, follwing Fursikov-Immanuvilov [2] and based on integrations by parts and bounded functions.

The next second main result gives a positive answer to the question made above. Then, using the classical Bukhgeim-Klibanov method [1], we show Lipschitz stability inequalities on the two reaction parameters k and γ simultaneously which appears in the first three equations of (I.1).

Theorem II.2:

Let (v (n) , u (n)
e , w (n) ) be solutions to our problem (I.1) respectivly with initial conditions (v

(n) 0 , u (n)
e,0 , w (n) 0 ) and ionic parameters (k (n) , γ (n) ) for n = 1, 2 and they are both under hypothesis of assumptions of property (I.4)-(I.5). In addition to that we assume that there exists t 0 ∈ (0, T ) such that: g(v [START_REF] Fursikov | Controllability of evolution equations[END_REF] (x, t 0 )) ̸ = 0 ∀x ∈ Ω., then there exists a constant C > 0 which depends only on Ω, T , M, M , k 0 ,I stim and initial data of solutions (v (n) , u (n) e , w (n) ), n = 1, 2 to our problem (I.1), such that:

γ (1) -γ (2) L 2 (Ω) + k (1) -k (2) L 2 (Ω) ≤ CN T,Γ 0 (v, u e , w, I el ) ,
where

N T,Γ 0 (v, u e , w, I el ) = v (1) -v (2) X + u (1) e -u (2) e X + I (1)
el -I

(2) el H 2 (0,T )

+ (u (1) e -u (2) e )(•, t 1 ) H 2 (Ω) + 1 i=0 (w (1) -w (2) )(•, t i ) L 2 (Ω) .
Here X is the vector space H 2 (0, T ; H

1 2 (Γ 0 )) ∩ H 1 (0, T ; H 3 2 (Γ 0 )) equipped with the norm ∥u∥ X = ∥u∥ 2 H 2 (0,T ;L 2 (Γ 0 )) + ∥∇u∥ 2 H 1 (0,T ;L 2 (Γ 0 )) 1/2
and t 1 is sufficiently close to t 0 .

By Theorem II.2, we can readily derive the uniqueness in the inverse problem.

Corollary II.3:

Under the same assumptions as in Theorem II.2 and if w (1) (x, t 0 ) = w (2) (x, t 0 ) for all x ∈ Ω (II.2) (u [START_REF] Buhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] e (x, t i ), w (1) (x, t i )) = (u (2) e (x, t i ), w (2) (x, t i )) for all x ∈ Ω, i = 0, 1 (II.3) v [START_REF] Buhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] = v [START_REF] Fursikov | Controllability of evolution equations[END_REF] , ∇u [START_REF] Buhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] e = ∇u [START_REF] Fursikov | Controllability of evolution equations[END_REF] e in Γ 0 × [0, T ] (II.4)

I (1) el = I (2) el in x ∈ (0, T ) (II.5)
then γ [START_REF] Buhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] = γ [START_REF] Fursikov | Controllability of evolution equations[END_REF] and k [START_REF] Buhgeim | Uniqueness in the large of a class of multidimensional inverse problems[END_REF] = k [START_REF] Fursikov | Controllability of evolution equations[END_REF] in Ω.

III DISCUSSION AND CONCLUSION

In this work, we use a mathematical model of the field potential recorded by the MEA device which results in a system of partial differential equations with boundary conditions acting on the four edges of our multi-electrode array device. We proved that the parameter identification inverse problem is stable under certain conditions. These conditions have to be satisfied in order to proceed to the computational estimation. Our approach is based on a new Carleman inequality for the MEA/hiPSC-CM assays. We established a new global Carleman estimate for solutions to a boundary value problem of a linear parabolic equation with Robin boundary condition. This Carleman inequality is crucial for solving the parameter identification stability problem. In order to prove the stability estimate for our ionic parameters k and γ at the same time, we began by establishing the first stability estimate of parameter k only, then we used both types of weight functions e -sη 0 and e -sη 1 to establish the last estimate of both k and γ using that the second weight function is constant with respect to time between t 0 and t 1 . This result is a very important step in order to numerically solve the parameters identification problem in cardiac electrophysiology because it provides the condition in which this problem is stable.

Figure 1 :

 1 Figure 1: The electrical circuit for the electrodes description

IV ACKNOWLEDGEMENTS

This work has been supported by EPICARD cooperative research program, funded by INRIA international laboratory LIRIMA. The LAMSIN researcher's work is supported on a regular basis by the Tunisian Ministry of Higher Education, Scientific Research and Technology. This work was also supported by the Agence Nationale de la Recherche (Grant IHU LIRYC ANR-10-IAHU-04).