Leif Arkeryd
email: arkeryd@chalmers.se

The paper proves existence of stationary solutions to the Boltzmann equation in a bounded set of R 2 for given indata, hard forces and truncation in the collision kernel for small velocities and close to parallel colliding velocities. It does not use any averaging in velocity lemma and uses techniques from the discrete velocity stationary case recently developped in [7]-[8]-[9], where the averaging in velocity lemmas are not valid.

Stationary solutions to the Boltzmann equation in the plane.

1 Introduction.

Consider the stationary Boltzmann equation in Ω

⊂ R 2 , v • ∇ z f (z, v) = Q(f, f ), z ∈ Ω, v ∈ R 2 , (1.1)
where Ω is a strictly convex domain with C 1 boundary. The nonnegative function f represents the density of a rarefied gas with z the position and v the velocity. The operator Q is the nonlinear Boltzmann collision operator with angular cut-off and a truncation for small velocities and close to parallel colliding velocities, From now on denote χ η B by B. The inward and outward boundaries in phase space are

Q(f, f )(z, v) = R 2 S 1 χ η (v, v * , ω) B(v -v * , ω) f (z, v )f (z, v * ) -f (z, v)f (z, v * ) dv * dω. (1.2) S 1 is the unit circle in R 2 , v = v+v * 2 + |v-v * | 2 ω, v * = v+v * 2 -|v-v * | 2 ω. For η ∈]0,
∂Ω + = {(z, v) ∈ ∂Ω × R 2 ; v • n(z) > 0}, ∂Ω -= {(z, v) ∈ ∂Ω × R 2 ; v • n(z) < 0},
where n(z) denotes the inward normal on ∂Ω. Given a function f b defined on ∂Ω + , solutions f to (1.1) are sought with f (z, v) = f b (z, v), (z, v) ∈ ∂Ω + .

(1.5)

For any (z, v) ∈ Ω × R 2 , denote by

s + (z, v) = inf{s > 0; (z -sv, v) ∈ ∂Ω + }, s -(z, v) = inf{s > 0; (z + sv, v) ∈ ∂Ω -}, z + (z, v) = z -s + (z, v)v, z -(z, v) = z + s -(z, v)v. (1.6)
Solutions are understood in mild form, i.e.

f (z, v) = f b (z + (z, v), v) + s + (z,v) 0 Q(f, f )(z + (z, v) + sv, v)ds, a.a. (z, v) ∈ Ω × R 2 . (1.7)
The main result of the paper is the following.

Theorem 1.1 Let f b be a non negative measurable function such that

∂Ω + v • n(z) 1 + v 2 + ln f b f b (z, v)dσ(z)dv < ∞. (1.8)
Then equation (1.1) has a non negative solution satisfying the boundary condition (1.5).

For the stationary Boltzmann equation the control of mass and entropy is not straightforward, contray to the evolutionary case. Existence results in the slab, i.e. one-dimensional spatial and three-dimensional velocity frame, together with an invariance with respect to the two remaining space variables of the distribution function were first established for the nonlinear Boltzmann equation. Stationary integrable solutions to the Boltzmann equation in a slab have been proven in [START_REF] Arkeryd | L 1 solutions to the stationary Boltzmann equation in a slab[END_REF], [START_REF] Arkeryd | The stationary Boltzmann equation in a slab, with given weighted mass for hard and soft forces[END_REF] and [START_REF] Arkeryd | A compactness result related to the stationary Boltzmann equation in a slab, with applications to the existence theory[END_REF], for different boundary conditions, bounds on the entropy production term and a weighted moment of the distribution function giving control of the entropy. For higher space dimension, stationary unsigned solutions close to Maxwellians were constructed in convex domains [START_REF] Guiraud | Problème aux limites intérieur pour l'équation de Boltzmann linéaire[END_REF], [START_REF] Guiraud | Problème aux limites intérieur pour l'équation de Boltzmann en régime stationnaire, faiblement non linéaire[END_REF]. In [START_REF] Esposito | Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law[END_REF], the existence and uniqueness of the stationary solution to the Boltzmann equation close to a uniform Maxwellian have been proven in a bounded domain of R n , 1 ≤ n ≤ 3, for diffuse reflection boundary conditions. Its hydrodynamic limit to a solution to the steady incompressible Navier-Stokes-Fourier system has been performed in [START_REF] Esposito | Stationary solutions to the Boltzmann equation in the hydrodynamic limit[END_REF]. Existence of stationary solutions to the Boltzmann equation in a bounded domain of R n , n ≥ 1, and given indata has been proven in [START_REF] Arkeryd | The stationary Boltzmann equation in R n with given indata[END_REF]. There, scaling arguments from [START_REF] Arkeryd | On the stationary Boltzmann equation in R n[END_REF] were used. In this paper, we prove existence of solutions to the stationary Boltzmann equation in the plane with the help of the entropy production term and the construction of 'good' characteristics where the distribution function is bounded and 'bad' characteristics of arbitrarily small measure. This is inspired by recent results for discrete velocity models for the Boltzmann equation where averaging lemmas do not hold and new arguments are required. In [START_REF] Arkeryd | Stationary solutions to the two-dimensional Broadwell model[END_REF], [START_REF] Arkeryd | On stationary solutions to normal, coplanar discrete Boltzmann equation models[END_REF], [START_REF] Arkeryd | Discrete velocity Boltzmann equations in the plane: Stationary solutions[END_REF] a weaker property than L 1 compactness of averages in velocity, i.e. the L 1 compactness of the integrated collision frequencies of a sequence of approximations is proven. It strongly depends on the two-dimensional spatial dimension. In this paper we use the tools developped in [START_REF] Arkeryd | Stationary solutions to the two-dimensional Broadwell model[END_REF], [START_REF] Arkeryd | On stationary solutions to normal, coplanar discrete Boltzmann equation models[END_REF], [START_REF] Arkeryd | Discrete velocity Boltzmann equations in the plane: Stationary solutions[END_REF] and do not use any averaging lemma. Work is in progress to fill a gap in the proof of Lemma 4.1 of [START_REF] Arkeryd | On the evolutionary velocity-discrete Boltzmann equation[END_REF] that uses these techniques in the discrete velocity evolutionary case. The construction of a first sequence of approximations with damping and convolutions is performed in Section 2. In Section 3, the damping and convolutions are removed, leading to a more involved sequence of approximations. In Section 3, the phase space is split into 'good' characteristics where the approximations are uniformly bounded and 'bad' characteristics of arbitrarily small measure. In Section 4, the L 1 compactness of the integrated collision frequency sequence is proven. The passage to the limit in the mild form satisfied by the approximations is performed in Section 5.

2 First approximations.

In the paper we denote by c constants that do not depend on approximations.

To emphasize their dependence on the indatum f b , we sometimes denote them by c b . We use the following approximation scheme. Let (B α ) α∈]0,1[ , be a family of

C ∞ regularizations of B. Let (ϕ α ) α∈]0,1[ be mollifiers defined from ϕ ∈ C ∞ 0 (R 4 ) such that ϕ(z, v) = 0 for |z| ≥ 1 or |v| ≥ 1, ϕ(z, v)dzdv = 1, by ϕ α (z, v) = 1 α 4 ϕ( z α , v α ).
Outside the boundary the function to be convolved with µ α , is continued in the normal direction by its boundary value. Let μk be a smooth mollifier on ∂Ω × R 2 in a ball of radius 1 k . Denote by

f bk = min{f b , k} * μk . Lemma 2.1 For every (α, k) ∈]0, 1[×N * , there is a non negative solution F to αF + v • ∇ z F = B α F 1 + F k (z, v ) F * ϕ α 1 + F * ϕα k (z, v * ) - F 1 + F k (z, v) F * ϕ α 1 + F * ϕα k (z, v * ) dv * dω, (z, v) ∈ Ω × R 2 , (2.1)
F (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + . (2.2)
Proof of Lemma 2.1. It follows the lines of the proofs in Section 2 in [START_REF] Arkeryd | On stationary solutions to normal, coplanar discrete Boltzmann equation models[END_REF] that we refer to for details. Let (α, k) ∈]0, 1[×N * be given. Let K be the closed and convex subset of L 1 (Ω × R 2 ) defined by

K = {f ∈ L 1 + (Ω × R 2 ); f (z, v)dzdv ≤ 1 α ∂Ω + v • n(z)f b (z, v)dσ(z)dv}.
Define the map T from K into K by T (f ) = F , where F is the solution to

αF (z, v) + v • ∇ z F (z, v) = R 2 ×S 1 B α F 1 + F k (z, v ) f * ϕ α 1 + f * ϕα k (z, v * ) - F 1 + F k (z, v) f * ϕ α 1 + f * ϕα k (z, v * ) dv * dω, (z, v) ∈ Ω × R 2 , (2.3) F (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + .
(2.4) F = T (f ) can be obtained as the limit in L 1 + (Ω × R 2 ) of the sequence (F q ) q∈N defined by F 0 = 0 and

αF q+1 + v • ∇ z F q+1 = R 2 ×S 1 B α F q 1 + F q k (z, v ) f * µ α 1 + f * µα k (z, v * ) - F q+1 1 + F q k (z, v) f * µ α 1 + f * µα k (z, v * ) dv * dω , (2.5)
F q+1 (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + , q ∈ N. (2.6)
In exponential form F q+1 can be written as

F q+1 (z, v) = f bk (z + (z, v), v)e -αs + (z,v) e -0 -s + (z,v) 1 1+ F q k (z+sv,v) Bα f * µα 1+ f * µα k (z+sv,u * )du * dγ ds + 0 -s + (z,v) B α F q 1 + F q k (z + sv, v ) f * µ α 1 + f * µα k (z + sv, v * )e αs e s 0 1 1+ F q k (z+rv,v) Bα f * µα 1+ f * µα k (z+rv,u * )du * dγ dr . (2.7)
The sequence (F q ) q∈N is monotone. Indeed, F 0 ≤ F 1 , by the exponential form of F 1 . If F q ≤ F q+1 , then it follows from the exponential forms of F q+1 and F q+2 that F q+1 ≤ F q+2 . Moreover,

α F q+1 (z, v)dzdv ≤ ∂Ω + v • n(z)f bk (z, v)dσ(z)dv + B α F q -F q+1 1 + F q k (z, v) f * µ α 1 + f * µα k (z, v * )dzdvdv * dω, so that F q+1 (z, v)dzdv ≤ 1 α ∂Ω + v • n(z)f b (z, v)dσ(z)dv, q ∈ N. (2.8)
By the monotone convergence theorem, (F q ) q∈N converges in L 1 (Ω × R 2 ) to a solution F of (2.3)-(2.4). The solution of (2.3)-(2.4) is unique in the set of non negative functions. Indeed, let G be a non negative solution of (2.3)-(2.4). It follows by induction that

F q ≤ G, q ∈ N.
(2.9) Indeed, (2.9) holds for q = 0, since G ≥ 0. Assume (2.9) holds for q. Using the exponential form of F q+1 implies F q+1 ≤ G. Consequently,

F ≤ G. (2.10)
Moreover, subtracting the equation satisfied by G from the equation satisfied by F , and integrating the resulting equation on Ω × R 2 leads to

α Ω×R 2 (G -F )(z, v)dzdv + ∂Ω - |v • n(z)|(G -F )(z, v)dσ(z)dv = 0. ( 2
) q∈N in K converge in L 1 (Ω × R 2 ) to f ∈ K. Set F q = T (f q ).
Because of the uniqueness of the solution to (2.3)-(2.4), it is enough to prove that there is a subsequence of (F q ) converging to F = T (f ). Now there is a subsequence of (f q ), still denoted (f q ), such that decreasingly (resp. increasingly) (G q ) = (sup r≥q f r ) (resp. (g q ) = (inf r≥q f r )) converges to f in L 1 . Let (S q ) (resp. (s q )) be the sequence of solutions to

αS q + v • ∇ z S q = B α S q 1 + S q k (z, v ) G q * µ α 1 + G q * µα k (z, v * ) - S q 1 + S q k (z, v) g q * µ α 1 + g q * µα k (z, v * ) dv * dω , (2.12
)

S q (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + , (2.13
)

αs q + v • ∇ z s q = B α s q 1 + s q k (z, v ) g q * µ α 1 + g q * µα k (z, v * ) - s q 1 + s q k (z, v) G q * µ α 1 + G q * µα k (z, v * ) dv * dω , (2.14)
s q (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + . (2.15) (S q
) is a non-increasing sequence, since that holds for the successive iterates defining the sequence. Then (S q ) decreasingly converges in L 1 to some S. Similarly (s q ) increasingly converges in L 1 to some s. The limits S and s satisfy (2.3)-(2.4). It follows by uniqueness that s = F = S, hence that (F q ) converges in L 1 to F . The map T is compact in the L 1 -norm topology. In [START_REF] Arkeryd | On stationary solutions to normal, coplanar discrete Boltzmann equation models[END_REF] an averaging lemma was used. Here we replace it by the following argument. Let (f l ) l∈N be a bounded sequence of L 1 and

F l = T (f l ). Denote by Fl = F l 1+ F l k , fl = f l * ϕ j 1+ f l * ϕ j k
, by B(0, V ) the open ball in R 2 centered at the origin and of radius V > 0, and by I(α, β) the interval with end points (α, β) ∈ R 2 . (F l ) l∈N is the sum of

f bk (z + (z, v), v) + 0 -s + (z,v) B α Fl (z + sv, v ) fl (z + sv, v * )dv * dωds l∈N * , (2.16) and - 0 -s + (z,v) B α Fl (z + sv, v) fl (z + sv, v * )dv * dωds l∈N * .
(2.17)

In order to prove that (2.16) is compact in L 1 , it is sufficient to prove that for any V > 0 and µ ∈]0, 1[,

0 -s + (z,v) B(0,V )×Aµ B α Fl (z + sv, v ) fl (z + sv, v * )dv * dωds l∈N * , (2.18)
is compact in L 1 (Ω × B(0, V ))
where

A µ = {ω ∈ S 1 ; | (v, ω)| ≥ µ and |π -(v, ω)| ≥ µ}. (2.19)
Indeed, noticing that v is parallel to v if and only if ω = ± v |v| , and expressing ω by its angle with the vector v |v| , the integral over Ω × B(0, V ) of

0 -s + (z,v) B(0,V )×A c µ B j Fl (z + sv, v ) fl (z + sv, v * )dv * dωdsdzdv is smaller than ck 2 V 2 µ. The sequence (2.18) is uniformly bounded in L 1 (Ω×B(0, V )).
Let us prove that it is uniformly equiintegrable with respect to the z variable. By the restriction to

A µ in (2.18), any (v, v ) ∈ R 2 × R 2
considered when integrating the absolute value of (2.18) over Ω × B(0, V ) forms a basis in R 2 . For any h ∈ R 2 , denote by (a(h), b(h)) its coordinates in this basis. Split the difference of (2.18) between (z, v) and (z + h, v) into the three following terms,

I(-s + (z,v),-s + (z+h,v)) Aµ B α Fl (z + h + sv, v ) fl (z + h + sv, v * )dv * dωds, 0 -s + (z,v) Aµ B α Fl (z + h + sv, v ) fl (z + h + sv, v * )dv * dω - Aµ B α Fl (z + b(h)v + sv, v ) fl (z + b(h)v + sv, v * )dv * dω ds = I(-s + (z,v)-a(h),-s + (z,v))∪I(0,a(h)) Aµ B α Fl (z + h + sv, v ) fl (z + h + sv, v * )dv * dω ds, 0 -s + (z,v) Aµ B α Fl (z + b(h)v + sv, v ) fl (z + b(h)v + sv, v * ) -fl (z + sv, v * ) dv * dω ds,
that tend to zero h → 0 when integrated over Ω × {v ∈ R 2 ; |v| < V }, and

0 -s + (z,v) Aµ B α Fl (z + b(h)v + sv, v ) -Fl (z + sv, v ) fl (z + sv, v * )dv * dωds. (2.20)
Notice that the integrand in the first line of (2.20) is a directional derivative in the direction v . Consequently, (2.20) is equal to

0 -s + (z,v) Aµ B α Fl (z + sv, v ) fl (z + sv, v * )e -αb(h) exp(- b(h) 0 B α fl (z + sv + rv , u * )du * dγ (1 + F l k )(z + sv + rv , v ) dr) -1 dv * dωds + 0 -s + (z,v) Aµ B α fl (z + sv, v * ) b(h) 0 B α Fl (z + sv + rv , V ) fl (z + sv + rv , V * )du * dγ e α(r-b(h)) exp - b(h) r B α fl (z + sv + tv , u * )du * dγ (1 + F l k )(z + sv + tv , v )
dt dr dv * dωds.

(2.21)

Here, V (resp V * ) denotes v -(v -u * , γ)γ (resp. u * + (v -u * , γ)).
(2.21) tends to zero when h → 0 when integrated over Ω × B(0, V ) in absolute value since all integrands are uniformly bounded and the domains of integration are of order h. This ends the proof of the uniform equiintegrability of (2.18) w.r.t. the z variable. The proof of its uniform equiintegrability w.r.t. the v variable is analogous. The L 1 compactness of (2.17) can be proven analogously. Hence the sequence (F l ) l∈N * is compact in L 1 . And so, the Schauder fixed point theorem applies to T , leading to a solution F of (2.1)-(2.2).

3 Removal of the damping and convolutions.

For any k ∈ N * , denote by

Q + k (resp. Q k , resp. ν k (F ), resp. D k
) the approximate gain term (resp. collision operator, resp. collision frequency, resp. entropy production term) defined by

Q + k (F, F ) = R 2 ×S 1 B F (v ) 1 + F (v ) k F (v * ) 1 + F (v * ) k dv * dω, (3.1)
ν k (F )(v) = 1 1 + F (v) k B F (v * ) 1 + F (v * ) k dv * dω, (3.2)
Q k (F, F ) = Q + k (F, F ) -F ν k (F ), D k (z, v) = R 2 ×S 1 B F k 1 + F k k F k * 1 + F k * k - F k 1 + F k k F k * 1 + F k * k ln F k F k * (1 + F k k )(1 + F k * k ) F k F k * (1 + F k k )(1 + F k * k ) dv * dω . (3.3) For any (α, k) ∈]0, 1[×N * , denote by F α,k the solution to (2.1)-(2.2) obtained in the previous section. (F α,k ) α∈]0,1[ is weakly compact in L 1 loc (Ω × R 2
) since it is bounded by a multiple of k 2 . Denote by F k the limit for the weak topology in L 1 loc (Ω × R 2 ) of a converging subsequence when α → 0. In the next lemma we prove that for a subsequence, the convergence is strong in

L 1 (Ω × R 2 ). Lemma 3.1
There is a sequence (α q ) q∈N tending to zero when q → +∞, such that when

q → +∞, (F αq,k ) q∈N strongly converges in L 1 (Ω × R 2 ) to F k . Moreover, F k is a solution to v • ∇ z F k = Q k (F k , F k ), (z, v) ∈ Ω × R 2 , (3.4)
F k (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + , (3.5)
and

(1 + v 2 )F k (z, v)dzdv ≤ c b , (3.6)
D k (z, v)dzdv ≤ c b , k ∈ N * . (3.7) ∂Ω - F k (z, v) | v • n(z) | dσ(z)dv + ∂Ω -,F k ≤k F k ln F k (z, v) | v • n(z) | dσ(z)dv + ln k 2 ∂Ω -,F k ≥k F k | v • n(z) | dσ(z)dv ≤ c b . (3.8) Proof of Lemma 3.1 Consider the approximation scheme (f α,ρ ) ρ∈N of F α,k , f α,0 = 0, (3.9)
αf α,ρ+1 + v • ∇ z f α,ρ+1 = B α F α,k 1 + F α,k k (z, v ) F α,k * µ α 1 + F α,k * µα k (z, v * ) - f α,ρ+1 1 + f α,ρ+1 k (z, v) f α,ρ * µ α 1 + f α,ρ * µα k (z, v * ) dv * dω, (3.10) f α,ρ+1 (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + , ρ ∈ N. (3.11)
f α,1 is explicitly given in terms of F α,k . It follows from the exponential forms of F α,k and f α,1 that

F α,k ≤ f α,1 , α ∈]0, 1[.
The sequence (f α,ρ ) ρ≥2 is constructed as follows. Denote by S the map from

(L 1 (Ω × R 2 )) 2 mapping (X, Z) into W = S(X, Z) ∈ L 1 (Ω × R 2 ) solution to αW + v • ∇ z W = B α F α,k 1 + F α,k k (z, v ) F α,k * µ α 1 + F α,k * µα k (z, v * ) - W 1 + X k (z, v) Z * µ α 1 + Z * µα k (z, v * ) dv * dω , W (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + .
Denote by f α,1,0 = S(0, f α,1 ), f α,1,r = S(f α,1,r-1 , f α,1 ),

F α,k,0 = S(0, F α,k ), F α,k,r = S(F α,k,r-1 , F α,k ), r ∈ N * .
First,

f α,1,0 ≤ F α,k,0 .
Then the sequence (f α,1,r ) r∈N (resp. (F α,k,r ) r∈N ) is increasing with limit

f α,2 (resp. F α,k ). It follows from f α,1,r ≤ F α,k,r , r ∈ N, that f α,2 ≤ F α,k . (3.12) Let f α,2,0 := S(0, f α,2 ), f α,2,r := S(f α,2,r-1 , f α,2 ), r ∈ N * .
It follows from (3.12) that

f α,2,0 ≥ F α,k,0 .
The sequence (f α,2,r ) r∈N is also increasing with limit f α,3 and with f α,2,r ≥ F α,k,r . Hence

f α,3 ≥ F α,k .
From here by induction on ρ, it holds that

f α,2ρ ≤ f α,2ρ+2 ≤ F α,k ≤ f α,2ρ+3 ≤ f α,2ρ+1 , α ∈]0, 1[, ρ ∈ N. (3.13)
By induction on r, for each r the sequence (f α,1,r ) α∈]0,1[ is translationally equicontinuous in α. The limit sequence (f α,2 ) α∈]0,1[ is also translationally equicontinuous. This is so, since given > 0, r and then h 0 can be taken so that

(f α,2 -f α,1,r )(z, v)dzdv < and |f α,1,r (z + h, v + h) -f α,1,r (z, v)|dzdv < , |h| < h 0 , | h| < h 0 .
It can analogously be proven that for each ρ ∈ N, (f α,ρ ) α∈]0,1[ is translationally equicontinuous in α. Let (α q ) q∈N be a sequence tending to zero. Take a subsequence in (α q ) q∈N , still denoted by (α q ) q∈N , such that (f αq,2 ) q∈N converges in L 1 to some f 0,2 when q → +∞.

Continuing by induction gives a sequence (f 0,ρ ) ρ∈N satisfying

f 0,2ρ ≤ f 0,2ρ+2 ≤ F k ≤ f 0,2ρ+3 ≤ f 0,2ρ+1 , ρ ∈ N, (3.14)
v • ∇ z f 0,ρ+1 = G -B f 0,ρ+1 1 + f 0,ρ+1 k (z, v) f 0,ρ 1 + f 0ρ k (z, v * )dv * dω, f 0,ρ+1 (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + .
Here, G is the weak L 1 limit when α → 0 of the gain term

B α F α,k 1 + F α,k k (z, v ) F α,k * µ α 1 + F α,k * µα k (z, v * )dv * dω .
In particular, (f 0,2ρ ) ρ∈N (resp. (f 0,2ρ+1 ) ρ∈N ) non decreasingly (resp. non increasingly) converges in L 1 to some g (resp. h) when ρ → +∞. The limits satisfy

0 ≤ g ≤ F k ≤ h, (3.15) v • ∇ z h = G -B h 1 + h k (z, v) g 1 + g k (z, v * )dv * dω, (3.16) v • ∇ z g = G -B g 1 + g k (z, v) h 1 + h k (z, v * )dv * dω, (3.17)
(h -g)(z, v) = 0, (z, v) ∈ ∂Ω + .
Subtracting (3.17) from (3.16) and integrating the resulting equation on Ω × R 2 gives that

∂Ω - | v • n(Z) | (h -g)(Z, v)dσ(Z)dv = 0, so that h -g = 0 also on ∂Ω -. Hence, s -(z,v) -s + (z,v) h(z + sv, v)e s 0
Bh(z+rv,u)dudγdr

B(h -g)(z + sv, v * )dv * dω ds = 0, (z, v) ∈ Ω × R 2 . (3.18)
(3.15) and (3.18) imply that g = h and is equal to

F k . (F αq,k ) q∈N converges to F k in L 1 (Ω × R 2
) when q → +∞. Indeed, given η > 0, choose ρ 0 big enough so that

f 0,2ρ 0 +1 -f 0,2ρ 0 L 1 < η and f 0,2ρ 0 -F k L 1 < η,
then q 0 big enough, so that

f αq,2ρ 0 +1 -f 0,2ρ 0 +1 L 1 ≤ η and f αq,2ρ 0 -f 0,2ρ 0 L 1 ≤ η, q ≥ q 0 .
Then split F αq,k -F k L 1 as follows,

F αq,k -F k L 1 ≤ F αq,k -f α,2ρ 0 L 1 + f α,2ρ 0 -f 0,2ρ 0 L 1 + f 0,2ρ 0 -F k L 1 ≤ f α,2ρ 0 +1 -f α,2ρ 0 L 1 +2η by (3.13) ≤ f α,2ρ 0 +1 -f 0,2ρ 0 +1 L 1 + f 0,2ρ 0 +1 -f 0,2ρ 0 L 1 + f 0,2ρ 0 -f α,2ρ 0 L 1 +2η ≤ 5η, q ≥ q 0 .
It remains to prove (3.6)-(3.8). Multiplying (3.4) by 1 + v 2 and integrating over Ω × R 2 , leads to

∂Ω - |v • n(z)|(1 + v 2 )F k (z, v)dσ(z)dv ≤ ∂Ω + v • n(z)(1 + v 2 )f b (z, v)dσ(z)dv. (3.19)
Denote by (v 1 , v 2 ) the components of v. Multiply (3.4) by v 1 and integrate it over Ω a × R 2 , where Ω a is the part of Ω with z 1 < a. Set S a = Ω ∩ {z 1 = a} and ∂Ω a = ∂Ω ∩ Ωa .

This gives

Sa×R 2 v 2 1 F k (a, z 2 , v)dz 2 dv = - ∂Ωa×R 2 v 1 v • n(z)F k (z, v)dzdv.
Ω×R 2 v 2 1 F k (z, v)dzdv ≤ c b . (3.21) ( v 2 2 F k (z, v)dzdv)
k∈N * is analogously bounded from above. Thus the boundedness of energy holds. Recalling the small velocity cut-off χ η , this in turn implies the boundedness of mass. The boundedness of the mass outflow in (3.8) follows from an integration of (3.4) on Ω×R 2 . Finally, Green's formula for

F k ln F k 1+ F k k implies that for some c b > 0, ∂Ω - |v • n(z)| F k ln F k -(k + F k ) ln(1 + F k k ) (z, v)dσ(z)dv + Ω×R 2 D k (z, v)dzdv ≤ c b , k ∈ N * . Moreover, k ∂Ω - ln(1 + F k k )(z, v) | v • n(z) | dσ(z)dv ≤ ∂Ω - F k (z, v) | v • n(z) | dσ(z)dv ≤ c b .
Hence

∂Ω - F k ln F k 1 + F k k (z, v) | v • n(z) | dσ(z)dv + Ω×R 2 D k (z, v)dzdv ≤ c b . (3.22)
It holds that

∂Ω - F k ln(1 + F k k )(z, v) | v • n(z) | dσ(z)dv ≤ ∂Ω -,F k ≤k F k ln(1 + F k k )(z, v) | v • n(z) | dσ(z)dv + ∂Ω -,F k ≥k F k ln(1 + F k k )(z, v) | v • n(z) | dσ(z)dv ≤ ln 2 ∂Ω - F k (z, v) | v • n(z) | dσ(z)dv + ∂Ω -,F k ≥k F k ln 2F k k (z, v) | v • n(z) | dσ(z)dv ≤ c b + ∂Ω -,F k ≥k F k ln F k (z, v) | v • n(z) | dσ(z)dv -ln k 2 ∂Ω -,F k ≥k F k (z, v) | v • n(z) | dσ(z)dv.
Together with (3.22), this implies that

∂Ω -,F k ≤k F k ln F k (z, v) | v • n(z) | dσ(z)dv + ln k 2 ∂Ω -,F k ≥k F k | v • n(z) | dσ(z)dv ≤ c b .
This ends the proof of Lemma 3.1.

Compactness of the integrated collision frequency.

Denote by Q + (f, f ) (resp. ν(f )) the gain term (resp. the collision frequency) of the nonlinear Boltzmann collision operator,

Q + (f, f )(z, v) = Bf (z, v )f (z, v * )dv * dω, ν(f )(z, v) = Bf (z, v * )dv * dω, (4.1)
so that

Q(f, f ) = Q + (f, f ) -f ν(f ).

Lemma 4.1

For any V > 1, the sequence

s -(z,v) -s + (z,v) ν(F k )(z + sv, v)ds k∈N * is uniformly bounded by cV 2 on Ω × {v ∈ R 2 ; η < |v| < V }.
Proof of Lemma 4.1.

For any (Z, v) ∈ ∂Ω + with η < |v| < V , the truncation χ η in B implies that

ν(F k )(Z + sv, v) = |v * |>η B(v, v * , ω)F k (Z + sv, v * )dv * dω ≤ c χ η (v, v * , ω)(v 2 + v 2 * )F k (Z + sv, v * )dv * ≤ c V 2 + η 2 η 4 v 2 * sin 2 (v, v * )F k (Z + sv, v * )dv * .
(z, v * ) ∈ Ω Z,v × R 2 . This gives s -(Z,v) 0 (v * • v ⊥ ) 2 |v| 2 F k (Z + sv, v * )dv * ds ≤ ∂Ω Z,v ×R 2 v * • v ⊥ |v| v * • n(z) F k (z, v * )dσ(z)dv * ≤ c b , (Z, v) ∈ ∂Ω + . (4.3)
Together with (4.2) this ends the proof of the lemma.

Lemma 4.2

The sequence (F k ) k∈N * is weakly compact in L 1 .

Proof of Lemma 4.2. By (3.6) it is sufficient to prove that for any V > 1 the sequence

(F k /Ω×{v∈R 2 ;|v|≤V } ) k∈N * is weakly compact in L 1 (Ω × {v ∈ R 2 ; η ≤ |v| ≤ V }). It follows from the exponential form of F k (z, v) from the outgoing boundary ∂Ω -, F k (z, v) = F k (z -(z, v), v) exp s -(z,v) 0 ν(F k )(z + sv, v)ds - s -(z,v) 0 Q + (F k , F k )(z + sv, v) exp s 0 ν(F k )(z + rv, v)dr ds,
and Lemma 4.1 that

F k (z, v) ≤ ce cV 2 F k (z -(z, v), v), z ∈ Ω, η ≤ |v| ≤ V. (4.4) By (3.8), (F k / ∂Ω -) k∈N * is weakly compact in L 1 |v•n(Z)| (∂Ω -).
This completes the proof of the lemma.

Lemma 4.3 For k ∈ N * and ∈]0, 1[, there is a subset Ω k, of characteristics of Ω × {v ∈ R 2 ; η < |v| < 1 },
with measure smaller than c , such that

F k (z, v) ≤ c 3 e c 2 , (z, v) ∈ Ω × {v ∈ R 2 ; η < |v| < 1 } \ Ω k, . (4.5)
Proof of Lemma 4.3.

Let ∈]0, 1[ be given. By the strict convexity of Ω and its C 1 regularity, the set

ω 1 := {(Z, v) ∈ ∂Ω × R 2 ; η < |v| < 1 and -2 < v • n(Z) < 0} (4.6)
is of measure smaller than c for some constant c > 0 and small enough. It follows from (3.8) that the measure of the subset of ∂Ω -where

η < |v| < 1 , v • n(Z) < -2 and F k (Z, v) > 1 3 , is smaller than c b . Denote this set by ω k, 2 . Define Ω k, as Ω k, = {(Z + sv, v); (Z, v) ∈ ω 1 ∪ ω k, 2 , s ∈ [-s + (Z, v), 0]}. ( 4.7)
Together with (4.4), this ends the proof of the lemma.

For any (k, ) ∈ N * ×]0, +∞[, denote by χ k, the characteristic function of (Ω k, ) c .

Lemma 4.4

For any V > 0 the sequence (

s -(z,v) -s + (z,v) ν(F k )(z + sv, v)ds) k∈N * is compact in L 1 (Ω × {v ∈ R 2 ; |v| < V }).
Proof of Lemma 4.4. Let V > 0 be given. By (3.6) and Lemma 4.2, it is sufficient to prove that for any > 0 and W > 0, the sequence

s -(z,v) -s + (z,v) |v * |<W Bχ k, F k (z + sv, v * )dv * ds k∈N * (4.8) is compact in L 1 (Ω × {v ∈ R 2 ; |v| < V }). By (3.6) this sequence is bounded in L 1 .
Let us prove that it is uniformly equiintegrable w.r.t. the z variable.

For any (α, β) ∈ R 2 , denote by I(α, β) the interval with end points α and β. For any h ∈ R 2 , split

s -(z+h,v) -s + (z+h,v) Bχ k, F k (z + h + sv, v * )dv * ds - s -(z,v) -s + (z,v) Bχ k, F k (z + sv, v * )dv * ds (4.9)
into

I(-s + (z,v),-s + (z+h,v))∪I(s -(z,v),s -(z+h,v)) Bχ k, F k (z + h + sv, v * )dv * ds, (4.10)
which absolute value tends to zero when integrated over Ω × R 2 and h → 0 by the continuity of (s + , s -) on Ω × {v; |v| ≤ V }, and

s -(z,v) -s + (z,v) Bχ k, F k (z + h + sv, v * ) -F k (z + sv, v * ) dv * ds. (4.11)
Almost every (v, v * ) ∈ R 2 × R 2 considered when integrating the absolute value of (4.11) over Ω × {v ∈ R 2 ; |v| < V } forms a basis in R 2 . Denote by (a(h), b(h)) the coordinates of h in this basis. Split (4.11) into

s -(z,v) -s + (z,v) Bχ k, F k (z + h + sv, v * ) -F k (z + b(h)v * + sv, v * ) dv * ds = s -(z,v) -s + (z,v) Bχ k, F k (z + h + sv, v * ) -F k (z + h + (s -a(h))v, v * ) dv * ds = I(-s + (z,v)-a(h),-s + (z,v))∪I(s -(z,v),s -(z,v)-a(h)) Bχ k, F k (z + h + sv, v * )dv * ds, (4.12)
and

s -(z,v) -s + (z,v) Bχ k, F k (z + b(h)v * + sv, v * ) -F k (z + sv, v * ) dv * ds = s -(z,v) -s + (z,v) Bχ k, f bk (z + (z + b(h)v * + sv, v * ), v * ) -f bk (z + (z + sv, v * ), v * ) dv * ds + s -(z,v) -s + (z,v) B I(0,b(h)) χ k, Q k (F k , F k )(z + sv + rv * , v * )drdv * ds. (4.13)
When integrated over Ω × {v ∈ R 2 ; |v| < V }, the limit when h → 0 of the first term of (4.13) is zero by the integrability of

f b . Notice that (χ k, Q k (F k , F k )) k∈N * is weakly compact in L 1 (Ω × {v ∈ R 2 ; |v| < V }), since χ k, Q k (F k , F k ) ≤ 1 ln Λ D k + cΛ 3 e c 2 ν(F k ), Λ > 1.
When integrated over Ω×{v ∈ R 2 ; |v| < V }, the limit when h → 0 of the second term of (4.13) is zero by the weak

L 1 compactness of (χ k, Q k (F k , F k )) k∈N * and lim h→0 b(h) = 0, (4.14)
uniformly on Ω × {v ∈ R 2 ; |v| < V }.

The uniform equiintegrability w.r.t. the v variable of

s -(z,v) -s + (z,v) |v * |<V Bχ k, F k (z + sv, v * )dv * ds k∈N *
follows from similar arguments.

Passage to the limit in the approximations

For each > 0, let F be the weak L 1 limit of a subsequence of (χ k, F k ) k∈N * . (F ) ∈]0,1[ is non increasing with respect to decreasing and bounded in L 1 . Let f be its strong L 1 limit when → 0. Notice that f is also the weak L 1 limit of (F k ) k∈N when k → +∞.

For proving that f is a mild solution of (1.1)-(1.5), it is sufficient to prove that for any β > 0, there is a set X β of characteristics with complementary set of measure smaller than cβ, such that if χ β denotes the corresponding characteristic function,

(χ β f )(z, v) = (χ β f b )(z + (z, v), v) + 0 -s + (z,v) (χ β Q(f, f ))(z + sv, v)ds, (z, v) ∈ Ω × R 2 . (5.1)
This in turn is satisfied if for any test function ϕ ∈ L ∞ ( Ω×R 2 ), continuously differentiable along characteristics, with v

• ∇ z ϕ ∈ L ∞ (Ω × R 2 ), compact
support and vanishing on ∂Ω -,

Ω×R 2 ϕχ β f (z, v)dzdv = Ω×R 2 ϕχ β f b (z + (z, v), v)dzdv + Ω×R 2 0 -s + (z,v) χ β f v • ∇ z ϕ + ϕχ β Q(f, f ) (z + sv, v)ds dzdv.
(5.2)

Let 0 > 0 be such that the support of ϕ is included in Ω×{v ∈ R 2 ; |v| ≤ 1 0 }. Define the set X β as follows. Using the weak L 1 (Ω × R 2 ) compactness of (F k ) k∈N * and the weak L 1 (∂Ω -) compactness of (F k /∂Ω -) k∈N * , pass to the limit when k → +∞ in

F k (z, v) ≤ ce c 2 0 F k (z -(z, v), v), a.a. z ∈ Ω, η < |v| ≤ 1 0 , k ∈ N * . (5.3) It implies that f (z, v) ≤ ce c 2 0 f (z -(z, v), v), a.a. z ∈ Ω, η < |v| ≤ 1 0 .
From here the proof follows the lines of the proof of Lemma 4.3, so that given β > 0, X β can be defined as a set of characteristics, with complementary set of measure smaller than cβ, such that

f (z, v) ≤ c β 3 e c 1 2 0 + 1 β 2 , a.a. (z, v) ∈ X β . (5.4)
The following lemma is a preliminary step to pass to the limit when k → +∞ in quadratic terms along the 'good characteristics' (z + sv, v), (z, v) / ∈ Ω k, .

Lemma 5.1

For any test function ϕ ∈ L ∞ ( Ω × R 2 ), continuously differentiable along characteristics, with v • ∇ z ϕ ∈ L ∞ (Ω × R 2 )
, compact support and vanishing on ∂Ω -,

lim k→+∞ Ω×R 2 0 -s + (z,v) ϕχ β χ k, F k ν(F k )(z + sv, v)ds dzdv = Ω×R 2 0 -s + (z,v)
ϕχ β F ν(f )(z + sv, v)ds dzdv.

(5.5)

Proof of Lemma 5.1. Since ϕ has compact support, one can restrict to the passage to the limit when k → +∞ and V fixed of

Ω×{v∈R 2 ;|v|<V } 0 -s + (z,v) ϕχ β χ k, F k ν(F k )(z + sv, v)ds dzdv. (5.6)
By an integration by parts,

Ω×{v∈R 2 ;|v|<V } 0 -s + (z,v) ϕχ β χ k, F k ν(F k )(z + sv, v)ds dzdv = Ω×{v∈R 2 ;|v|<V } 0 -s + (z,v) ϕχ β χ k, f bk (z + (z, v), v) 0 -s + (z,v) ν(F k )(z + sv, v)ds(z + sv, v)ds dzdv + Ω×{v∈R 2 ;|v|<V } 0 -s + (z,v) χ β χ k, v • ∇ z ϕ F k + ϕQ k (F k , F k ) (z + sv, v) 0 s ν(F k )(z + rv, v)dr ds dzdv. ( 5

Hence,

Ω×{v∈R 2 ;|v|<V } 0 -s + (z,v) ϕχ β χ k, F k ν(F k )(z + sv, v)ds dzdv = Ω×{v∈R 2 ;|v|<V } 0 -s + (z,v) ϕχ β χ k, f bk (z + (z, v), v) 0 -s + (z,v) ν(F k )(z + sv, v)ds(z + sv, v)ds dzdv + Ω×{v∈R 2 ;|v|<V } χ β χ k, (v • ∇ z ϕ)F k + ϕQ k (F k , F k ) (Z, v) s -(Z,v) 0 (s -(Z, v) -r)ν(F k )(Z + rv, v)dr dZdv.
(5.9) Lemma 5.1 follows from the passage to the limit when k → +∞ in (5.9). It uses the weak

L 1 compactness of (χ k, F k ) and (χ k, Q k (F k , F k )), the strong L 1 compactness of s -(Z,v) 0 (s -(Z, v) -r)ν(F k )(Z + rv, v)dr , and the L ∞ boundedness of χ k, (Z, v) s -(Z,v) 0 (s -(Z, v) -r)ν(F k )(Z + rv, v)
dr . An integration by parts back in the s variable is finally performed, taking into account that

χ k, Q k (F k , F k )(z + sv, v) = d ds (χ k, F k )(z + sv, v) .
Lemma 5.2 f is a solution of (1.1)-(1.5), i.e. for test functions ϕ defined as in Lemma 5.1,

Ω×R 2 ϕχ β f (z, v)dzdv = Ω×R 2 ϕχ β f b (z + (z, v), v)dzdv + Ω×R 2 0 -s + (z,v) χ β f v • ∇ z ϕ + ϕχ β Q(f, f ) (z + sv, v)ds dzdv.
(5.10)

Proof of Lemma 5.2.

For ∈]0, 0 [, write the mild form of ϕχ β χ k, F k and integrate it on Ω × R 2 . It results

Ω×R 2 ϕχ β χ k, F k (z, v)dzdv = Ω×R 2 ϕχ β χ k, f bk (z + (z, v), v)dzdv + Ω×R 2 0 -s + (z,v) χ β χ k, F k v • ∇ z ϕ(z + sv, v)ds dzdv + Ω×R 2 0 -s + (z,v) ϕχ β χ k, Q + (F k , F k ) -F k ν(F k ) (z + sv, v)ds dzdv.
(5.11)

By the weak L 1 compactness of (F k ) k∈N * and the linearity with respect to χ k, F k of the first two lines of (5.11), their passage to the limit when k → +∞ is straightforward. The passage to the limit when k → +∞ in the last term of (5.11) follows from Lemma 5.1. Finally, using monotonicity arguments together with the L ∞ boundedness of χ β f allows to pass to the limit when → 0 in

Ω×R 2 ϕχ β F (z, v)dzdv - Ω×R 2 ϕχ β f b (z + (z, v), v)dzdv - Ω×R 2 0 -s + (z,v) χ β F v • ∇ z ϕ(z + sv, v)ds dzdv + Ω×R 2 0 -s + (z,v) ϕχ β F ν(f )(z + sv, v)ds dzdv, (5.12)
and obtain the limit (5.16)

Ω×R 2 ϕχ β f (z, v)dzdv - Ω×R 2 ϕχ β f b (z + (z, v), v)dzdv - Ω×R 2 0 -s + (z,v) χ β f v • ∇ϕ(z + sv, v)ds dzdv + Ω×R 2 0 -s + (z,v) ϕχ β f ν(f )(z + sv, v)ds dzdv. ( 5
One can restrict to the study of the limit of (5.21)

Ω×R 2 {v * ∈R 2 ;|sin( v,v * )|>µ}×S 1 ϕχ β χ k, s -(Z, v )B F k 1 + F k k (Z, v * )dv * dω F k 1 + F k k (Z, v)dZdv, µ ∈]0, 1[, (5.17
Passing to the limit when (α, 1 , µ) → (0, 0, 0) in (5.21) leads to (5.14).

  1[ and fixed,χ η (v, v * , ω) = 0 if |v| ≤ η or |v * | ≤ η or |v | ≤ η or |v * | ≤ η, χ η (v, v * , ω) = 1, else. (1.3)Here (u, v) denotes the angle between vectors u and v.The function B is the kernel of the classical nonlinear Boltzmann operator for hard forces,|v -v * | β b(ω) with β ∈ [0, 2[, b ∈ L 1 + (S 1), b(ω) ≥ c > 0 a.e.(1.4)

  20) on [l, L], where l = inf{a; S a = ∅}, L = sup{a; S a = ∅}, and using (3.19) leads to

(4. 2 )

 2 Let Ω Z,v be one of the two subsets of Ω split by the segment [Z, Z +s-(Z, v)v] and ∂Ω Z,v = ∂Ω ∩ ΩZ,v . Let v ⊥ beone of the vectors orthogonal to v such that |v ⊥ | = |v|. Multiply (2.1) written in the variables (z, v * ) by v * • v ⊥ |v| and integrate the resulting equation over

. 7 )

 7 The change of variables (z, s) → (Z, s) = (z + sv, s), (5.8) moves the domain Ω × ] -s + (z, v), 0[ into the domain {(Z, s) ; Z -sv ∈ Ω and s < 0}, i.e. (Z, -s) ∈ Ω×]0, s -(Z, v)[.

2 0 2 0Ω×R 2 0

 222 -s + (z,v)ϕχ β χ k, Q + (F k , F k )(z + sv, v)ds dzdv = Ω×R -s + (z,v) ϕχ β Q + (f, f )(z + sv, v)ds dzdv.(5.14)For any (v, v * , ω), the change of variables(z, s) → (Z, s) = (z + sv , s),(5.15)moves the domain Ω × ] -s + (z, v ), 0[ into the domain {(Z, s) ; Z -sv ∈ Ω and s < 0}, i.e. (Z, -s) ∈ Ω×]0, s -(Z, v )[. Hence, -s + (z,v) ϕχ β χ k, Q + (F k , F k )(z + sv, v)ds dzdv = Ω 0 -s + (z,v ) ϕχ β χ k, (z + sv , v )B sv , v * )dsdz dvdv * dω = Ω×R 2 ϕχ β χ k, s -(Z, v )B F k 1 + F k k (Z, v * )dv * dω F k 1 + F k k (Z, v)dZdv.

) since Ω×R 2 2 Λe c 2 F 2 s 2 s

 22222 {v * ∈R 2 ;|sin( v,v * )|<µ}×S 1ϕχ β χ k, s -(Z, v )B F k > 1, µ ∈]0, 1[. (5.18) The sequence(Z, v) → F k (Z, v) {v * ∈R 2 ;|sin( v,v * )|>µ}×S 1 ϕχ β χ k, s -(Z, v )BF k (Z, v * )dv * dω k∈N * is weakly compact in L 1 . Indeed, using the change of variables v * → v * for every (v, ω), which holds since |sin( v, v * )| > µ, F k (Z, v) ϕχ β χ k, s -(Z, v )BF k (Z, v * )dv * dω ≤ c ln Λ D k (Z, v) + c Λ 3 k (Z, v * )dv * , Λ > 1,where (D k ) k∈N * is defined in (3.1) and uniformly bounded in L 1 by (3.7).Consequently one can restrict to the passage to the limit when k → +∞ in(Z,v)∈Ω×R 2 ;s -(Z,v)>α {v * ∈R 2 ;|sin( v,v * )|>µ}×S 1 ϕχ β χ k, s -(Z, v )BF k (Z, v * )dv * dω χ k, 1 F k (Z, v)dZdv, (α, 1 ) ∈]0, 1[ 2 . (5.19) Moreover, (Z,v);s -(Z,v)>α {v * ∈R 2 ;|sin( v,v * )|<µ}×S 1 ϕχ β χ k, s -(Z, v )BF k (Z, v * )dv * dω χ k, 1 F k (Z, v)dZdv = {v * ∈R 2 ;|sin( v,v * )|<µ}×S 1 Z;s -(Z,v)>α s -(Z,v) 0 ϕχ β χ k, s -(Z, v ) χ k, 1 F k s -(Z, v) F k (Z, v * )dsdZ Bdvdv * dω = {v * ∈R 2 ;|sin( v,v * )|<µ}×S 1 Ω s -(z,v) max{0,α-s -(z,v)} ϕχ β χ k, s -(z + sv, v ) χ k, 1 F k s -(z + sv, v) F k (z + sv, v * )dsdz Bdvdv * dω = Ω×R -(z,v) max{0,α-s -(z,v)} χ k, 1 F k s -(z + sv, v) Bϕχ β χ k, s -(z + sv, v )F k (z + sv, v * )dv * ds dzdv.(5.20)As in the proof of Lemma 5.1,lim k→+∞ Ω×R 2 s -(z,v) max{0,α-s -(z,v)} χ k, 1 F k s -(z + sv, v) Bϕχ β χ k, s -(z + sv, v )F k (z + sv, v * )dv * ds dzdv = Ω×R -(z,v) max{0,α-s -(z,v)} F 1 s -(z + sv, v) {v * ∈R 2 ;|sin( v,v * )|<µ}×S 1Bϕχ β s -(z + sv, v )f (z + sv, v * )dv * ds dzdv = (Z,v);s -(Z,v)>α {v * ∈R 2 ;|sin( v,v * )|<µ}×S 1 ϕχ β s -(Z, v )Bf (Z, v * )dv * dω F 1 (Z, v)dZdv.


domain: math phys

text:
Gourab Bhattacharya

A METRIC THAT SATISFIES THE GRAVITATIONAL MONOPOLE EQUATIONS-III

We showed the existence of a Gravitational metric that necessarily satisfies generalised Calabi inequalities.

Introduction

The following solutions of the Gravitational Monopole Equations behave as if the scalar curvature is zero (but they are non-zero and strictly negative). The Einstein-Maxwell equations are used to achieve this goal [cf.3], [cf.4]. This is the main difference between our previous works [cf.1] and [cf.2]. Actually, the idea came into authors mind when he encountered for the first came time Edward's work [cf.3], later the work of LeBrun's work simplified much of the effort [cf.4]. We showed the existence of a Gravitational metric that necessarily satisfies generalised Calabi inequalities.

The Einstein-Maxwell equations

Let X be a 4-manifold and L be a line-bundle (possibly trivial) over X. Let A be an affine U (1)-connection and F := F A is the corresponding curvature, then the Einstein-Maxwell equations with the cosmological constant are given as follows [cf.4] dF = 0,

d * F = 0, [R + F • F ] 0 = 0, with (F • F ) jk = F l j F lk (2.1)
are the Euler-Lagrange equations of the functional

(2.2) (g, F ) → M (s g + |F | 2 g )dµ g .
When n = 4 that the Einstein-Maxwell equations imply that the scalar curvature s of g is constant. The

(2.3) C = [ω] ∈ H 2 (M, R) : [ω] 2 > 0 For any [ω] ∈ C, let G [ω] ⊂ G is a Fréchet submanifold of finite codimension b ( M ). G [ω] ̸ = ∅ forall [ω]

4 ]

 4 following observation is due to Yamabe as acknowledged by LeBrun in a work of his [cf.Proposition 1. A Riemannian metric has constant scalar curvature if and only if it is a critical point of the restriction of the Einstein-Hilbert action sdµ to the space of volume-V metrics in a fixed conformal class. Therefore, when n = 4, the conformal invariance of |F | 2 dµ implies that critical points of the functional (2.2) have constant scalar curvature [cf.4]. Thus [cf.4], Proposition 2. Einstein-Maxwell equations on a 4-manifold imply that the scalar curvature is constant. The following is true [cf.4] Proposition 3. Any constant-scalar curvature Kähler metric on a 4-manifold may be interpreted as a solution of the Einstein-Maxwell equations. Proposition 4. A Gravitational Monopole always satisfies c 1 • [ω] < 0. Proposition 5. Any constant strictly negative scalar curvature Kähler metric on a 4-manifold can be interpreted as a solution of the Einstein-Maxwell Gravitational Monopole equations. Remark 2.1. A Gravitational Monopole can not be rational or ruled, as there will always be Kähler classes for which c 1 • [ω] > 0. The following proposition is due to Donaldson [cf.6] and Gay-Kirby [cf.5]. Proposition 6. Let M be any smooth compact 4-manifold with b + (M ) ̸ = 0. Let

  in an open-dense subset of C.Theorem 2.2. A Kähler Gravitational Monopole (M 4 , ω, g, J) with g ∈ G [ω] necessarily satisfies generalised Calabi inequalities M s 2 dµ g > 32π 2 (c 1 • [ω]) 2 [ω]2 , Proof. A Gravitational Monopole always satisfies c 1 •[ω] < 0. For the rest of the proof one consults [cf.4]. □

	(2.4)	M	|R| 2 dµ g > 8π 2 2	[ω] 2 (c 1 • [ω]) 2	1 -c 2


text:
Gourab Bhattacharya

MUMFORD-GIESEKER STABILITY AND THE GRAVITATIONAL MONOPOLE METRIC

The existence of the Gravitational Monopole equation implies the ω-stability of Mumford-Gieseker for a rank-2 holomorphic bundle over a complex algebraic surface. The main argument is briefly described in §(5.1).

Introduction

In this note, we sketched an argument how the Gravitational Monopole equation may be used to prove the ω-stability of Mumford-Gieseker for a rank-2 holomorphic bundle over a complex algebraic surface. The main argument is briefly described in §(5.1). The details will appear elsewhere.

The Gravitational Monopole Equations

In [cf.1] the Gravitational Monopole equations were introduced in the following sense. Let (X 4 , g) be a Riemannian spin 4-manifold. Then the Clifford algebra bundle Cl(X 4 ) is a vector bundle over X 4 with fiber at x ∈ X 4 is the Clifford algebra Cl(T x X). With respect to the metric g, one identifies (isomorphism) Cl(T x X) with Cl(T * x X). Therefore, as a vector space, this is isomorphic to ∧T * x X. Let us also assume E → X is a Clifford module bundle with a covariant derivative ∇ E . Then for each x ∈ X there is a Clifford action c :

T * x X ⊗ E x → E x via c(α ⊗ s) = c(α)s. Definition 2.1. The twisted Dirac operator associated to (E, ∇ E ) is the operator, (2.1) / ∇ := c • ∇ E : C ∞ (X, E) → C ∞ (X, E).
The equations we wish to consider are (sometimes we omit the mapping c and the dimension 4 for the convenience of computations),

/ ∇ψ = (d + d * )ψ = 0, c(W + g ) = 1 4 ⟨e i • e j ψ, ψ⟩e i ∧ e j , or, c (W + g ) ijkl e i ∧ e j = 1 4 ⟨e k • e l ψ, ψ⟩.
(2.

2)

The Weitzenböck's formula: the decomposition of the Laplace-Beltrami operator as a generalized Laplacian is, with the left-Clifford multiplication ϵ l = ext l -int l , and corresponding right Clifford action ϵ r ,

(2.3) (d + d * ) 2 = ∆ ∧T * X - ijkl R ijkl ϵ i l ϵ j l ϵ k r ϵ l r + 1 4 s.
Proposition 1. For k ̸ = l, ⟨e k e l ψ, ψ⟩ is purely imaginary.

Proposition 2.

(2.4) |⟨e k e l ψ, ψ⟩| 2 = 2|ψ| 4 .

We keep in mind the Seiberg-Witten analysis, and analogously define and get the following definition of Gravitational-Monopole functional, Definition 2.2. The Gravitational-Monopole functional of a pair (ψ, g) is given by,

S(g, ψ) = X | / ∇ψ| 2 + |W + - 1 4 ⟨e i • e j ψ, ψ⟩e i ∧ e j | 2 d(vol) g i.e., S(g, ψ) = X |(d + d * )ψ| 2 + |W + - 1 4 ⟨e i • e j ψ, ψ⟩e i ∧ e j | 2 d(vol) g .
(2.5)

Proposition 3. (2.6) S(g, ψ) = X |∇ψ| 2 + |W + | 2 + s 4 |ψ| 2 + 1 8 |ψ| 4 d(vol) g Proposition 4.
As a direct consequence of (3), if the scalar curvature of X is non-negative, all solutions of (2.2) have ψ ≡ 0.

Action of Special Orthogonal Group

Let X be an oriented Riemannian manifold of even dimension 2l and we also assume X is a spin manifold, that is the first and second Stiefel-Whitney classes vanish. We denote by ∧ p the bundle of exterior p-forms with A p = Γ(∧ p ) its space of smooth sections. The Hodge star operator * ∧ p → ∧ 2l-p is defined by,

(3.1) α ∧ * β = (α, β)ω ∈ ∧ 2l
where α, β ∈ ∧ p , (α, β) is the induced inner product on p-forms and ω is the volume form. From now everything will be 4-dimensional unless otherwise stated. We start with the symmetry of the equations, namely the Lie algebras. The Lie algebra so(4) of the special orthogonal group SO( 4) is not simple. It can be decomposed into the direct sum of two copies of the Lie algebra so(3) of the group SO(3):

(3.2) so(4) ∼ = so(3) ⊕ so(3).

In terms of the group theory, one understands the above decomposition corresponds to the fact that the universal covering group of SO( 4) is the product of the two copies of SU (2). This fact in quantum mechanics corresponds to ± 1 2 spins of an electron for each factor SU (2). In terms of the geometry of the vector bundles, the decomposition so(4) ∼ = so(3) ⊕ so(3) induces the following decomposition (for a choice of g on X 4 ) for the vector bundle 2 T * X → X,

(3.3) ∧ 2 T * X ∼ = ∧ + ⊕ ∧ -,
as a Whitney sum of two oriented 3-plane bundles. One can choose an oriented orthonormal frame for T * U X for an open set U ⊂ X. One therefore has, (3.4)

∧ ± = Span (e 1 ∧ e 2 ± e 3 ∧ e 4 ), (e 2 ∧ e 3 ± e 1 ∧ e 4 ), (e 3 ∧ e 1 ± e 2 ∧ e 4 ) .

We now use the unique Levi-Civita connection ∇ on ∧ 2 T * X to find a suitable decomposition of the curvature tensor under the action of O(4). The first step towards it is to note that ∇g = 0, this however means that ∇ is covariantly constant, that is ∇ maps sections of ∧ ± into ∧ ± ⊗ T * X; there is no mixed term mapping ∧ + into ∧ -⊗ T * X. The curvature of the Levi-Civita connection defines a section of ∧ 2 T * X ⊗ 2 T * X, correspondingly a decomposition of ∧ 2 T * X ⊗ 2 T * X into four matrix-blocks of size 3 × 3, more precisely, the Riemann curvature tensor defines, in general, a self-adjoint linear transformation R :

∧ 2 → ∧ 2 such that, (3.5) R(e i ∧ e j ) = 1 2 k,l R ijkl e k ∧ e l ,
relative to the decomposition ∧ 2 = ∧ + ⊗ ∧ -, the operator R has the following form,

(3.6) R = A B B t C
where, B ∈ Hom(∧ -, ∧ + ) (is the traceless Ricci curvature) 0 Ric, and A ∈ End(∧ + ), that is A is symmetric about its diagonal, that is A t = A, similarly for C ∈ End(∧ -) we have C t = C. This representation of the curvature tensor R gives us a complete decomposition of it into irreducible components, namely

(3.7) R → (Tr A, B, A - 1 3 Tr A W + , C - 1 3 Tr C W - ) T rA = Tr C = 1 4
s where s is the scalar curvature. More elaborately, if the basis of ∧ ± in (3.4) is denoted by {x i ± } 3 i=1 , then the curvature tensor R has the following expansion

(3.8) R = W + ij x i + ⊗ x j + + W - ij x i -⊗ x j -+ B ij x i + ⊗ x j -+ B t ij x i -⊗ x j + - s 12 (x i + ⊗ x j -+ x i -⊗ x j + ).
If we denote the projection operator by, (3.9)

P ± := 1 2 (1 ± * ) : ∧ → ∧ ± , then, (3.10)
W ± = P ± • Rm • P ± - s 12 Id ±

The Kobayashi-Hitchin Correspondence

Let (M 4 , ω) a Káhler 4-manifold Consider a rank-2 holomorphic vector bundle V on a surface X with c 1 (V ) = 0. For general Kähler manifolds, the Kobayashi-Hitchin Correspondence is equivalent to saying that the holomorphic vector bundle structure on V is ω-stable. , this condition amounts to the following: if L is a line bundle and there is a nonzero holomorphic map L → V (not necessarily an inclusion on each fiber), then c 1 (L) • ω < 0. If ω corresponds to the Kähler form of a Hodge metric corresponding to an ample divisor H, then ω-stability exactly corresponds to stability with respect to the line bundle [H].

Stability via the Gravitational Monopole Equations

Consider the moduli problem of polarized varieties, so we consider moduli functor (5.1) F : Schemes/C → Sets with respect to the objects below

F(C) = {(Γ, H) : Γ is a projective variety, H is ample line bundle on Γ } (5.2)
H is called a polarization of Γ, and (Γ, H) is a polarized variety. If Γ is smooth, then the pair (Γ, H) is known as a polarized variety. The Euler-Poincaré characteristic χ(Γ, H m ) is a polynomial of in the variable m. Let p n (x) ∈ Q[x] is a polynomial of degree n. Consider the following moduli problem

(5.3) F pn(x) (C) = {(Γ, H) : (Γ, H) ∈ F, χ(Γ, H m ) = p n (m), ∀m ≥ 1} It is proved in [cf.4] if (M, L) ∈ F pn(x) (
C) a polarized manifold with a a Hermitian metric h such that for a large k the pointwise constant norm

n i=1 ∥s i (z)∥ 2 h of a frame s 0 , • • • , s n ∈ H 0 (M, L k ) with induced ω = Ric(h) the k-th Hilbert point of (M, L) is (GIT) stable with respect to SL(n + 1, C).
5.1. The main argument. The converse statement follows if we assume the existence of the Gravitational Monopole equation. We choose a holomorphic line bundle from the Spin C -structure defining the Gravitational Monopoles such that the Gravitational Monopole invariant is non-zero, that is G(L) ̸ = 0, this amounts to a nonzero holomorphic map L → V , with the rank of V is 2. Now the if ψ ̸ = 0 on M if and only if the scalar curvature is strictly negative, that is s < 0, this amounts to the condition c 1 (L) • ω as ω = Ric(h), and Ric(h) < 0 by the Gravitational Monopole equation (see the author's work in [cf.3]). Thus the existence of h. The holomorphic assumption on L and on the map L → V implies h is hermitian.


Bilel Elbetch

On the effect of density-dependent dispersal on total biomass in multi-patch environment

Keywords: Nonlinear diffusion, Migration rate, Logistic growth, Total biomass, Global stability

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, we compute the limit of the total equilibrium population, which in general is different from the sum of the n carrying capacities, and depends on the migration terms, carrying capacities and growth rates. Second, we determine 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.

We ends by considering the case of two patches. We give the explicit formula of the total equilibrium population, and we compare with the sum of two carrying capacities.

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 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],

Hanski [START_REF] Hanski | Metapopulation Ecology[END_REF], Holt [START_REF] Holt | Population dynamics in two patch environments: some anomalous consequences of an optimal habitat distribution[END_REF]and Levin [START_REF] Levin | Dispersion and population interactions[END_REF][START_REF] Levin | Spatial patterning and the structure of ecological communities[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:

dx i dt = x i (a i -b i x i ) + Υ i (x), i = 1, . . . , n, (1)
where a i and b 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:

Υ i (x) = n j=1,j =i D ij (x j -α ij x i ) , , i = 1, . . . , n, (2)
where D ij and α 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:

Υ i (x) = n j=1,j =i D ij x i (x j -α ij x i ) , , i = 1, . . . , n, (3)
.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:

Υ i (x) = n j=1,j =i D ij x 2 j -α ij x 2 i , , i = 1, . . . , n, (4)
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].

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 [START_REF] Allen | Persistence and extinction in Lotka-Volterra reaction-diffusion equations[END_REF](2) for n = 2 ( see Theorem 1 in [START_REF] Allen | Persistence and extinction in Lotka-Volterra reaction-diffusion equations[END_REF]). For model [START_REF] Allen | Persistence and extinction in Lotka-Volterra reaction-diffusion equations[END_REF](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)(3); Theorem 4 of [START_REF] Allen | Persistence and extinction in Lotka-Volterra reaction-diffusion equations[END_REF] for the n-patch model

(1)(4) and Proposition 1 of [START_REF] Allen | Persistence and extinction in Lotka-Volterra reaction-diffusion equations[END_REF] for 2-patch case.

In 1977, Freedman and Waltman [START_REF] Freedman | Mathematical Models of Population Interactions with Dispersal II: Differential Survival in a Change of Habitat[END_REF] considered a two-patch model with a single species in logistic population growth as follows:

       dx 1 dt = r 1 x 1 1 - x 1 K 1 + (x 2 -x 1 ), dx 2 dt = r 2 x 2 1 - x 2 K 2 + (x 1 -x 2 ), (5)
where x i represents the population density in patch i, the parameter r i is the intrinsic growth rate, K i is carrying capacity and m is the dispersal rate. Freedman and Waltman show that under certain conditions, the total population abundance can be larger than the total carrying capacities K 1 + K 2 . For more details and information on the existence and stability of [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF], see [START_REF] Freedman | Mathematical Models of Population Interactions with Dispersal I: Stability of two habitats with and without a predator[END_REF]. Holt [START_REF] Holt | Population dynamics in two patch environments: some anomalous consequences of an optimal habitat distribution[END_REF] generalized these results to a source-sink system. In 2015, Arditi et al. [START_REF] Arditi | Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation[END_REF] gave a full mathematical analysis of the model ( 5) of Freedman and Waltman with symmetric dispersal.

In 2018, Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF] extended the model ( 5) by considering asymmetric dispersal, i.e. the model:

       dx 1 dt = r 1 x 1 1 - x 1 K 1 + ( 12 x 2 -21 x 1 ), dx 2 dt = r 2 x 2 1 - x 2 K 2 + ( 21 x 1 -12 x 2 ), (6)
where 12 and 21 with ij > 0, i = j and ≥ 0, are the migration terms which describe the flows of individuals from the patch 2 to the patch 1, and from the patch 1 to the patch 2 respectively. These flows can for example depend on the distance between the patches. By noting that the positive equilibrium (x * 1 , x * 2 ) of model ( 6) is the unique positive solution to

       r 1 x 1 1 - x 1 K 1 + r 2 x 2 1 - x 2 K 2 = 0, x 2 = 1 12 21 x 1 - r 1 x 1 1 - x 1 K 1 ,
i.e., the intersection of an ellipse and a parabola, they used a graphical method to completely analyze model [START_REF] Arino | Number of Source Patches Required for Population Persistence in a Source?Sink Metapopulation with Explicit Movement[END_REF] in order to determine when dispersal is either favorable or unfavorable to total population abundance. In [START_REF] Elbetch | Effect of dispersal in Two-patch environment with Richards growth on population dynamics[END_REF], Elbetch generalizes the results 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] to a Two-patch environment with Richards growth.

Recently, , Elbetch et al. [START_REF] Elbetch | The multi-patch logistic equation[END_REF][START_REF] Elbetch | The multi-patch logistic equation with asymmetric migration[END_REF] considered the following model:

dx i dt = r i x i 1 - x i K i + n j=1,j =i ( ij x j -ji x i ), i = 1, • • • , n. (7)
They studied the total population at equilibrium, as a function of the migration rate . They gave conditions on the system parameters that ensure that symmetric and asymmetric migration is beneficial or detrimental.

Our aim in this work is to study the model of n patches coupled by nonlinear migration terms ( same type of dispersion (3)). In particular, we are interested in studying the effect of nonlinear dispersion on the dynamics of population and coexistence of species.

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 prove the global stability of the model [START_REF] Cvetković | Stabilizing the Metzler matrices with applications to dynamical systems[END_REF]. In Section 4, we study the behavior of the system [START_REF] Cvetković | Stabilizing the Metzler matrices with applications to dynamical systems[END_REF] in the case when the migration rate goes to infinity. In Section 5, we compare the total equilibrium population with the sum of the n carrying capacities for all parameter space of the model. In Section 6, using the graphical method of 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 Appendix A, we recall some inequalities for stability modulus of a cooperative matrix and In Appendix B, we prove the global stability of (43) according to [1, Proposition 1].

Throughout this text, the following notations and names are used for matrices. Let A, B be two n×n-matrices, with A = (a ij ) and B = (b ij ). Then, A ≥ 0 is a nonnegative matrix if a ij ≥ 0 for all i, j, and A ≥ B if A -B ≥ 0. The matrix A > 0 is a positive matrix if additionally, there exists i, j such that a ij > 0, and

A > B if A -B > 0.
A 0 is strongly positive if a ij > 0 for all i, j, and A B if A -B 0. We recall that, the matrix A is reducible if we can partition {1, . . . , n} into two nonempty subsets E and F such that a ij = 0 for all i ∈ E and j ∈ F . Otherwise A is irreducible.

The mathematical model

We consider the general model of multi-patch logistic growth, coupled by asymmetric migration terms given by:

dx i dt = r i x i 1 - x i K i + x i n j=1,j =i ( ij x j -ji x i ), i = 1, • • • , n, (8)
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 in the system. The parameter represents the dispersal rate of the population; ij ≥ 0 denotes the flux between patches j and i, for j = i. Note that, if ij > 0 there is a flux of migration between patches i and j and, if ij = 0 there is no migration. The system (8) of differential equations can be written:

dx i dt = r i x i 1 - x i K i + x i n j=1 ij x j , i = 1, • • • , n, (9)
where

ii = - n j=1,j =i ji , i = 1, • • • , n (10)
denotes the outgoing flux of patch i. We denote by E the following matrix

E =      -k =1 k1 12 . . . 1n 21 -k =2 k2 . . . 2n • . . . . . . . . . n1 . . . n,n-1 -k =n kn      . ( 11
)
to 0. if E is irreducible, then 0 is a simple eigenvalue of E and all non-zero eigenvalues of E have negative real part. Moreover, the kernel of the matrix E is generated by a strongly 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 strongly positive vector. For the existence , uniqueness, and strongly positivity of δ see Lemma

* ii ) = (-1) n-1 .
As in our work, the matrix E is assumed to be irreducible,

then (-1) n-1 (E * 11 , . . . , E * nn ) T is strongly positive, i.e δ i = (-1) n-1 E * ii > 0 for all i.
Therefore, we have explicit formula for the components of the vector δ, as functions of the coefficients of E, at our disposal. For two patches we have δ = ( 12 , 21 ) T , and

for three patches we have δ = (δ 1 , δ 2 , δ 3 ) T , where

   δ 1 =
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 as follow:

δ k = T ∈Tk (i,j)∈E(T ) ij , k = 1, . . . , n, (13)
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 . The matrix

E 0 := E -diag( 11 , . . . , nn ) (14)
which is the same as the matrix E, 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 ij > 0, with weight ij assigned to the arrow.

The system (8) can be also rewritten in matrix form as follow:

dx dt = f (x) + diag(x 1 , . . . , x n )Ex, (15)
where

x = (x 1 , . . . , x n ) T and f (x) = (f 1 (x 1 ), . . . , f n (x n )) T with f i (x i ) = r i x i (1 - x i /K i ).
Note that, the system (8) is cooperative, and without diffusion (i.e for = 0) he has a strongly positive globally stable equilibrium point (K 1 , . . . , K n ). The main result in the next section shows that the strongly positive and globally stable equilibrium point of model [START_REF] Cvetković | Stabilizing the Metzler matrices with applications to dynamical systems[END_REF] with diffusion term continues to exist for any diffusion rate .

Global stability

Our aim in this part is to study the global stability of [START_REF] Cvetković | Stabilizing the Metzler matrices with applications to dynamical systems[END_REF]. We prove the result:

Theorem 3.1. Assume that the matrix E = ( ij ) n×n (or equivalently, the connectivity matrix E 0 ) is irreducible. The model ( 8) has a unique positive equilibrium point which is globally asymptotically stable (GAS), in the positive cone R n + \ {0}.

Proof. The equilibrium points of the system (8) are given by the solutions of the following algebraic system:

0 = r i x i 1 - x i K i + x i n j=1,j =i ( ij x j -ji x i ), i = 1, • • • , n, (16)
which can be written in the form:

0 = x i   r i 1 - x i K i + n j=1,j =i ( ij x j -ji x i )   , i = 1, • • • , n, (17)
So, the interior equilibrium points of ( 8), if they exist, are the solutions of the following linear system:

0 = r i 1 - x i K i + n j=1,j =i ( ij x j -ji x i ) := F i (x), i = 1, • • • , n, (18)
where x = (x 1 , . . . , x n ) T . The system [START_REF] Gantmacher | The Theory of Matrices[END_REF] is writes in matrix form as follows

(E -diag(α 1 , . . . , α n ))x = -(r 1 , . . . , r n ) T . ( 19
)
Let we prove that the stability modulus of the matrix E -diag(α 1 , . . . , α n ) is negative.

Indeed, let u = (1, • • • , 1) T . We have (E -diag(α 1 , . . . , α n )) T u = -(α 1 , . . . , α n ) T ≤ λu,
where λ = -max {α 1 , . . . , α n } < 0. Therefore, since the matrix E -diag(α 1 , . . . , α n ) is a cooperative matrix, according to Lemma A.2, we have

s(E -diag(α 1 , . . . , α n )) = s((E -diag(α 1 , . . . , α n )) T ) ≤ λ < 0.
Therefore, based on Lemma A.3, the matrix E -diag(α 1 , . . . , α n ) is stable and invertible. We conclude also according to [26,Theorem 6.

1] that -(E -diag(α 1 , . . . , α n )) -1
0. Then, the system (8) admits unique strongly positive equilibrium, locally asymptotically stable (LAS), given by:

E * n ( ) := -(E -diag(α 1 , . . . , α n )) -1 (r 1 , . . . , r n ) T . Let
we prove the global stability of this equilibrium by applying Lemma A.4. Let ξ > 0 and we denote E * n ( ) = (x * 1 ( ), . . . , x * n ( )). We have:

F i (ξE * n ( )) = r i 1 - ξx * i ( ) K i + ξ n j=1,j =i ( ij x * j ( ) -ji x * i ( )), i = 1, • • • , n, (20)
where F i is given by the equation [START_REF] Gantmacher | The Theory of Matrices[END_REF]. At the equilibrium, we have n j=1,j =i

( ij x * j ( ) -ji x * i ( )) = -r i 1 - x * i ( ) K i , i = 1, • • • , n. (21)
We replace [START_REF] Holt | Population dynamics in two patch environments: some anomalous consequences of an optimal habitat distribution[END_REF] in [START_REF] Hofbauer | Evolutionary Games and Population Dynamics[END_REF] we get:

F i (ξE * n ( )) = r i (1 -ξ) > 0 if ξ < 1, < 0 if ξ > 1. ( 22
)
According to Lemma A.4 , the equilibrium

E * n ( ) is GAS in the positive cone R n + \ {0}.
In all of this work, the GAS equilibrium of the system (8), whose existence is shown

in Theorem 3.1, is denoted by E * n ( ) = (x * 1 ( ), . . . , x * n ( ))
, and by T * n ( ) the total equilibrium population

T * n ( ) = n i=1 x * i ( ). ( 23
)

The limit of fast dispersal

We have the following result which prove that the equilibrium E * n ( ) converges to an element of ker E:

Theorem 4.1. We have:

lim →+∞ E * n ( ) = n i=1 r i n i=1 δ i α i (δ 1 , . . . , δ n ) ( 24
)
where

α i = r i /K i . Moreover, if the matrix E is symmetric, then lim →+∞ E * n ( ) = n i=1 r i n i=1 α i (1, . . . , 1). ( 25
)
Proof. The equilibrium E * n is a solution of the algebraic system:

0 = r i 1 - x i K i + n j=1,j =i ( ij x j -ji x i ), i = 1, . . . , n. (26)
The sum of these equations shows that E * ( ) satisfies the following equation

n i=1 r i 1 - x i K i = 0. ( 27
)
Therefore E * ( ) belongs to the hyper plane:

H n-1 := x ∈ R n + : n i=1 α i x i - n i=1 r i = 0 . (28)
Note that, this hyper plane is independent of the migration terms and ij . The hyper plane H n-1 passes through the point A = (K 1 , . . . , K n ).

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

F (x) := F 1 (x), . . . , F n-1 (x), n i=1 r i 1 - x i K i , (29)
with

F i (x) = 1 r i i 1 - x i K i + n j=1,j =i ( ij x j -ji x i ), i = 1, . . . , n -1. ( 30
)
On the other hand, the limit equations (obtained when → ∞) are given by:

F ∞ (x) := F ∞ 1 (x), . . . , F ∞ n-1 (x), n i=1 r i 1 - x i K i , (31)
with

F ∞ i (x) = n j=1,j =i ( ij x j -ji x i ), i = 1, . . . , n -1. ( 32
)
According to Lemma B.1 of [START_REF] Elbetch | The multi-patch logistic equation with asymmetric migration[END_REF], the system (32) 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:

       x i = δ i δ n x n , i = 1, • • • , n -1. n i=1 r i 1 - x i K i = 0, (33)
which admits x n = n i=1 ri n i=1 δiαi δ n as solution. So, the equation F ∞ = 0 admits unique solution

E * n (∞) := n i=1 r i n i=1 δ i α i (δ 1 , . . . , δ n ).
Since the hyper plan is compact, the equilibrium E * n ( ) converge to E * n (∞) when tend to infinity.

If the matrix E 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:

Corollary 4.2. We have

T * n (+∞) = n i=1 δ i n i=1 r i n i=1 δ i α i . ( 34
)
Moreover, if the matrix E is symmetric, then:

T * n (+∞) = n n i=1 r i n i=1 α i . ( 35
)
Proof. The sum of the n components of the point E * n (∞) immediately gives the equation (34).

We remark that, the total equilibrium population for large migration rate (35) is equal the total equilibrium population for large migration rate of the multi-patch logistic model with linear diffusion obtained in [9, Equation 24]. This result show that, if the movement between the n patches is symmetric, the biased diffusion has no influence on the total equilibrium population for large migration rate compared to the linear diffusion.

Total equilibrium population

In the next proposition we give sufficient and necessary conditions for the equilibrium not to depend on the migration rate.

Proposition 5.1. The equilibrium E * n ( ) does not depend on if and only if (K 1 , . . . , K n ) T ∈ ker E. In this case we have E * n ( ) = (K 1 , . . . , K n ) for all > 0.

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

diag(r i -α i x i ) + Ex = 0. ( 36
)
Suppose that the equilibrium E * n ( ) does not depend on , then we replace in Equation (36):

diag(r i -α i x * i ( )) + E(E * n ( )) = 0. ( 37
)
The derivative of (37) with respect to gives

E(E * n ( )) = 0. ( 38
)
Replacing the equation (38) in the equation (37), we get E * ( ) = (K 1 , . . . , K n ). From the equation (38), we conclude that (K 1 , . . . , K n ) T ∈ ker E. Now, suppose that (K 1 , . . . , K n ) T ∈ ker E, then (K 1 , . . . , K n ) satisfies the equation (36), for all ≥ 0. So, E * n ( ) = (K 1 , . . . , K n ), for all ≥ 0, which proves that the total equilibrium population is independent of the migration rate .

= . . . = α n , then T * n ( ) = n i=1 K n , for all ≥ 0.
Proof. The equilibrium is always in the hyper plane

n i=1 α i x * i ( ) = n i=1 r i .
If we suppose in the previous equation that all α i are equals, then we get that

n i=1 x * i ( ) = n i=1 K i , for all ≥ 0.
Lemma 5.1. The derivative of the total equilibrium population at = 0 is given by:

dT * n d (0) = i j =i ij K j -ji K i α i . ( 39
)
Proof. The equilibrium E * n is a solution of the algebraic system:

0 = r i 1 - x i K i + n j=1,j =i ( ij x j -ji x i ), i = 1, . . . , n, (40)
which implies

x i = K i + 1 α i n j=1,j =i ( ij x j -ji x i ), i = 1, . . . , n. (41)
The derivative of x * i with respect to , at = 0 gives

dx * i d (0) = i j =i ij x * j (0) -ji x * i (0) α i . ( 42
)
The sum of the previous equations give the formula of the derivative since x * i (0) = K i for all i.

We have the following result:

Theorem 5.1. Consider the total equilibrium population of [START_REF] Cvetković | Stabilizing the Metzler matrices with applications to dynamical systems[END_REF]. Let dT * n d (0) be the derivative of the total equilibrium without dispersal. If there exists i and j = i such that α i = α j , then

• If dT * n d (
0) > 0, then T * n ( ) ≥ i K i for all ≥ 0, and

• if dT * n d (0) < 0, then T * n ( ) ≤ i K i for all ≥ 0.
Proof. The equilibrium E * ( ) belongs to the hyper plane H n-1 . Consider the hyper 10 plan P which passes by (K 1 , . . . , K n ) and has the following cartesian equation:

P : i x i = i K i .
The two hyper planes are identical if and only if, α i are equals. So, if there exists α i = α j , the this two hyper plans intersect in a straight line passing through the point (K 1 , . . . , K n ), and withe respect the sign of the derivative of the total equilibrium without dispersal, we conclude the complete proof.

Two-patch case

In this section, we consider the two patches model coupled by migration terms and

with assumption that the diffusion rate is a function of population density. The model can be written as follows:

       dx 1 dt = r 1 x 1 1 - x 1 K 1 + x 1 ( 12 x 2 -21 x 1 ) , dx 2 dt = r 2 x 2 1 - x 2 K 2 + x 2 ( 21 x 1 -12 x 2 ) . (43)
where x i represents the population density in the i-th patch, the parameters r i and K i are respectively the intrinsic growth rate and the carrying capacity of patch i. The parameter is the migration rate, 12 and 21 are positive and denotes the flux of migration between patches. If 12 = 21 , the migration is symmetric.

In this part, we study the effect of the variation of migration rate in [0, ∞[ on the total population at the equilibrium E ++ ( ). Denote T * 2 ( ), the total equilibrium population, which is the sum of the coordinates of the equilibrium E ++ ( ), i.e

T * 2 ( ) = (K 1 r 1 K 2 12 + K 1 K 2 12 r 2 + K 2 r 1 K 1 21 + K 2 K 1 21 r 2 ) + K 1 r 2 r 1 + r 1 K 2 r 2 (r 1 K 2 12 + K 1 21 r 2 ) + r 2 r 1 . (44) It's clear that T * 2 (0) = K 1 + K 2 , and
T * 2 → T * 2 (∞) = ( 12 + 21 ) r 1 + r 2 12 α 1 + 21 α 2 , as → ∞. (45)
In the remainder of this section, we analyze the effect of biased dispersal on the total equilibrium population for the two-patch system (43). Using the method of Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF] ( see also [START_REF] Arditi | Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation[END_REF]), we describe the position affects the equilibrium E ++ of (43) 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 consider the regions in the set of the parameters 21 and 12 , denoted J 1 and

                     If α 2 > α 1 then    J 1 = ( 21 , 12 ) : 21 12 < K2 K1 J 2 = ( 21 , 12 ) : 21 12 > K2 K1 If α 2 < α 1 then    J 1 = ( 21 , 12 ) : 21 12 > K2 K1 J 2 = ( 21 , 12 ) : 21 12 < K2 K1 (46) Case α 2 > α 1 0 J 1 J 2 21 12 12 21 = K1 K2 Case α 2 < α 1 0 J 2 J 1 21 12 12 21 = K1 K2 Figure 1.
The domains 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 (43).

Theorem 6.1. The total equilibrium population (44) of (43) satisfies the following properties

(1) If α 1 = α 2 then T * 2 ( ) = K 1 + K 2 for all ≥ 0.
(2) If α 1 = α 2 , let J 1 and J 2 , be defined by (46). Then we have:

• if ( 21 , 12 ) ∈ J 1 then T * 2 ( ) > K 1 + K 2 for any > 0, • if ( 21 , 12 ) ∈ J 2 then T * 2 ( ) < K 1 + K 2 for any > 0, • if 12 21 = K1 K2 , then E ++ = (K 1 , K 2 ) for all ≥ 0. Therefore T * 2 ( ) = K 1 + K 2 for all ≥ 0.
Proof. The equilibrium E ++ is given by the solution of the following system

       0 = r 1 1 - x 1 K 1 + ( 12 x 2 -21 x 1 ) , 0 = r 2 1 - x 2 K 2 + ( 21 x 1 -12 x 2 ) . (47)
The sum of these equations show that E ++ is in a straight line

L : α 1 x 1 + α 2 x 2 = r 1 + r 2 .
To facilitate comparison of the total equilibrium population T * 2 ( ) and T * 2 (0) = K 1 + below the line ∆, then T * 2 ( ) ≤ T * 2 (0), whereas if E ++ is on or above the line, then

T * 2 ( ) ≥ T * 2 (0). The equilibrium point E * 2 (
) is always in the line L, then, for = 0, the equilibrium point states at A, and when increases, E ++ describes an arc of the line L and ends at point E ++ (∞) := lim →∞ E ++ ( ).

If we take α 1 = α 2 , we get that the equations of L and ∆ are the same, which gives the prove of the first item. Assume that α 1 = α 2 , then the two lines intersect at a single point A. Therefore, if ( 21 , 12 ) ∈ J 1 then T * 2 ( ) > K 1 + K 2 for any > 0, and if ( 21 , 12 ) ∈ J 2 then T * 2 ( ) < K 1 + K 2 for any > 0 ( see figure 2).

Under the hypothesis 12 21 = K1 K2 , the equilibrium simplify to

E ++ = (K 1 , K 2 ). 0 L ∆ • A x 1 x 2 0 L ∆ • A x 1 x 2 Figure 2.
The two possibles configurations of the lines L (shown in red) and ∆ ( shown in blue) in the case when α 1 = α 2 . When varies from 0 to infinity, the equilibrium E ++ varies in the line L which starts at point A = (K 1 , K 2 ) and has two possible directions as shown by the arrows.

Remark 6.1. In the case when α 1 = α 2 =: α, the equilibrium E ++ ( ) becomes as:

E ++ ( ) = (r 1 + r 2 ) 12 + αr 1 ( 21 + 12 )α + α 2 , (r 1 + r 2 ) 21 + αr 2 ( 21 + 12 )α + α 2 ,
and the sum of the coordinates of E ++ ( ) equal to

K 1 + K 2 .
Let we denote by x * 1 ( ) and x * 2 ( ) the coordinates of E ++ ( ) given by (B3), we have

dx * 1 d ( ) = K 1 r 1 r 2 2 (K 2 12 -K 1 21 ) (r 2 r 1 + r 1 K 2 12 + K 1 21 r 2 ) 2 , and dx * 2 d ( ) = K 2 r 2 r 2 1 (K 1 21 -K 2 12 ) (r 2 r 1 + r 1 K 2 12 + K 1 21 r 2 ) 2 . The sum of dx * 1 d and dx * 2 d
gives the derivative of T * 2 with respect to :

dT * 2 d ( ) = α 1 α 2 (α 2 -α 1 )(K 2 12 -K 1 21 ) K 2 1 K 2 2 (r 2 r 1 + r 1 K 2 12 + K 1 21 r 2 ) 2 . ( 48
)
In particular, the derivative of T * 2 ( ) at = 0 is given by:

dT * 2 d (0) = ( 12 K 2 -21 K 1 ) 1 α 1 - 1 α 2 . ( 49
)
We have the following result which describes the variation of the population in each patch as function of the rate of migration as well as the variation of the total equilibrium population in two patches: Proposition 6.1. Consider the equilibrium population x * 1 ( ) and x * 2 ( ) of (43). Assume that α 1 = α 2 , let J 1 and J 2 , be defined by ( 46). Then we have:

• if ( 21 , 12 ) ∈ J 1 then x * 1 is increasing, x *

Conclusion

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.

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. For npatch model, 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 . We have shown that if α i are equal in all patches, then the total equilibrium population on the n connected patches is independent to the sum of capacities, and if α i are different, then , the total equilibrium population take two case: either greater or smaller than the sum of carrying capacities.

Finally, comparisons the present results with previous work [START_REF] Elbetch | The multi-patch logistic equation[END_REF][START_REF] Elbetch | The multi-patch logistic equation with asymmetric migration[END_REF] indicate that this simple density-dependent dispersal mechanism has an effect on the dynamics of population. Some questions remain open, for example, for susceptible-infected-susceptible (SIS) patch-model ( see [START_REF] Gao | How does dispersal affect the infection size?[END_REF]), how does the nonlinear dispersal affect the infection size?

Appendix A. Some inequalities for stability modulus of a cooperative matrix In this section, our goal is to recall some inequalities for stability modulus of a cooperative matrix. First, we start by giving some definitions.

Definition A.1. A matrix A = (a ij ) is called cooperative if a ij ≥ 0 for all i = j.

Definition A.2. The stability modulus of a matrix A is given by s(A) = max {Re(λ) : λ is an eigenvalue of A} , (A1)

The trace and the determinant of the matrix J(E ++ ) are given by: Hence by the Routh-Hurwitz criteria for stability, the real parts of the eigenvalues value of the Jacobian matrix J(E ++ ) are negative, proving that the equilibrium of coexistence is locally asymptotically stable (LAS) for the system (43).

     trJ(E ++ ) = - (r 2 r 1 +
Our aim in the remainder of this section, is to prove the global stability of the equilibrium of coexistence Since, the system (43) is cooperative ( see the jacobian matrix (B4)). For general cooperative systems, we have the following result, which guarantees that all solutions converge to equilibria/infinity without periodic oscillation ( see page 28 in Hofbauer and Sigmund [START_REF] Hofbauer | Evolutionary Games and Population Dynamics[END_REF]). As the solutions are bounded and E ++ is LAS, hence the stability must be global.

  E ++ . According to[1, Proposition 1] for n = 2, every solution x(t) of System 1 considered by Allen, is unbounded, limt→∞ x i (t) = ∞, i = 1, 2, if and only if b1 b2 -D 12 D 21 ≤ 0, where bi = b i + j =i D ij α ij , i = 1, 2. (B5) If inequality (B5) does not hold, then lim t→∞ x(t) = x * > 0. For the following choice of the parameters: D 12 = 12 , α 12 = 21 / 12 , D 21 = 21 , α 21 = 12 / 21 , a i = r i and b i = r i /K i , the system 1 of Allen transform to our model (43), and the condition (B5) becomes as: every solution x(t) of System (43) is unbounded, lim t→∞ x i (t) = ∞, i = 1, 2, if and only if b1 b2 -2 12 21 ≤ 0, where b1 = α 1 + 21 and b2 = α 2 + 12 . (B6) We have b1 b2 -2 12 21 = α 1 α 2 + (α 1 12 + α 2 21 ) > 0 for all ≥ 0. Therefore, from [1, Proposition 1], the solutions x(t) of (43) are bounded.

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

  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 [9, Proposition 3.2].

	Proposition 5.2. If α 1

  r 1 K 2 12 + K 2 12 r 2 ) (r 1 + K 1 21 ) r 2 r 1 + r 1 K 2 12 + K 1 21 r 2 -(r 1 K 1 21 + r 2 r 1 + K 1 21 r 2 ) (r 2 + K 2 12 ) r 2 r 1 + r 1 K 2 12 + K 1 21 r 2 < 0, det J(E ++ ) = (r 2 r 1 + r 1 K 2 12 + K 2 12 r 2 ) (r 1 K 1 21 + r 2 r 1 + K 1 21 r 2 ) r 2 r 1 + r 1 K 2 12 + K 1 21 r 2 > 0.

B Existence of equilibria and stability of (43)

Email : elbetchbilel@gmail.com

ρ(A) = max{| λ |: λ is an eigenvalue of A}.

(A2)

We have the following result [24, Lemma 8]:

Lemma A.1. Let A be a non negative matrix. Let u ∈ R n be a non-zero vector and λ ∈ R be a real number. If Au ≥ λu then ρ(A) ≥ λ. If for a strictly positive vector u we have Au ≤ λu then ρ(A) ≤ λ.

Proof. If Au ≥ λu then, since A is non negative, A k u ≥ λ k u for all k. Therefore

we obtain that ρ(A) ≥ λ. The second statement is a simple consequence of the representation (2) in [START_REF] Nesterov | Computing closest stable nonnegative matrix[END_REF].

We have also the following result [8, Lemma 8]:

If for a strictly positive vector u we have Au ≤ λu then s(A) ≤ λ.

Proof. Let A be a cooperative matrix, there exists h > 0 such that A + hI, where I is the identity matrix, is non negative. Let u and λ be such that Au ≥ λu. Since s(A+hI)u ≥ (λ+h)u, using Lemma A.1, we deduce that ρ(A+hI) ≥ λ+h. According to the Perron-Frobenius Theorem [18, Theorem 3, page 66], we have

Therefore we have s(

By the same method, we prove the second statement.

Recall that the differential system

is said to be cooperative, if its jacobian matrix is cooperative, i.e., for all i = j, i, j = 1, . . . , n, (∂F i /∂x i ) ≥ 0, for all x positive. To prove local stability in this paper, we used the following Lemma [START_REF] Hofbauer | Evolutionary Games and Population Dynamics[END_REF][START_REF] Smith | Cooperative systems of differential equations with concave nonlinearities[END_REF]:

Lemma A.3. For a cooperative matrix A = (a ij ), we have s(A) < 0 ⇐⇒ There exists a vector u positive such that Au < 0.

To prove global stability in this paper, we used the following Lemma [START_REF] Freedman | Global stability and predator dynamics in a model of prey dispersal in a patchy environment[END_REF][START_REF] Takeuchi | Cooperative system theory and global stability of diffusion models[END_REF]:

Lemma A.4. Consider the system (A3). Assume that the jacobian matrix of (A3) is irreducible. If system (A3) possesses a positive equilibrium point x * satisfying

then x * is globally stable.

Appendix B. Existence of equilibria and stability of (43)

The equilibria points of (43) are given by the solutions of the following system:

which can written as follows:

Therefore, the system (B2) admits four solutions given by:

, 0 , and

In the next, we will study the local stability of the equilibrium of (43). Let J be denote the Jacobian matrix of the system (43) evaluated at x, given by

(B4) The matrix (B4) evaluated at the equilibrium E 00 gives J(E 00 ) = diag(r 1 , r 2 ), which prove that the origin is unstable. The Jacobian matrix evaluated at E 0+ (resp. at E +0 ) is given by:

< 0 ), then, Routh-Hurwitz criteria for stability assure that the both equilibrium E 0+ and E +0 are unstable.

The Jacobian matrix evaluated at E ++ is given by: