Influence of a spatial structure on the long time behavior of a competitive Lotka-Volterra type system

To describe population dynamics, it is crucial to take into account jointly evolution mechanisms and spatial motion. However, the models which include these both aspects, are not still well-understood. Can we extend the existing results on type structured populations, to models of populations structured by type and space, considering diffusion and nonlocal competition between individuals? We study a nonlocal competitive Lotka-Volterra type system, describing a spatially structured population which can be either monomorphic or dimorphic. Considering spatial diffusion, intrinsic death and birth rates, together with death rates due to intraspecific and interspecific competition between the individuals, leading to some integral terms, we analyze the long time behavior of the solutions. We first prove existence of steady states and next determine the long time limits, depending on the competition rates and the principal eigenvalues of some operators, corresponding somehow to the strength of traits. Numerical computations illustrate that the introduction of a new mutant population can lead to the long time evolution of the spatial niche.


Introduction
The spatial aspect of populations is an important ecological issue which has been extensively studied (see [17], [18], [23], [25], [28]). The interplay between space and evolution is particularly crucial in the emergence of polymorphism and spatial patterns and the heterogeneity of the environment is considered as essential ( [21], [22]). The combination of spatial motion and mutation-selection processes is also known for a long time to have important effects on population dynamics ( [19], [24]). Recently biological studies observed that classical models could underestimate the invasion speed and suggested that invasion and evolution are closely related. The ecological parameters can have a strong effect on the expansion of invading species and conversely, the evolution can be conditioned by the spatial behavior of individuals related to the resources available. The paper by Philipps and Co [26] shows the strong impact of the morphologic parameters of the cane toads on the expansion of their invasion.
In this context, the study of space-related traits, such as dispersal speed or sensibility to heterogeneously distributed resources, is fundamental and has been object of mathematical developments. In Champagnat-Méléard [12], a stochastic individual-based model is introduced where individuals are characterized both by their location and one or several phenotypic and heritable traits. The individuals move, reproduce with possible mutation and die of natural death or because of competition for resources. The spatial motion is modeled as a diffusion and the spatial interaction between individuals is modeled by a convolution kernel in some spatial range. In a large population scale, it is shown that this microscopic stochastic model can be approximated by a nonlinear nonlocal reaction-diffusion equation defined on the space of traits and space. The latter has been studied in Ferrière-Desvillettes-Prévost [16] and Arnold-Desvillettes-Prévost [2] and existence and uniqueness of the solution, numerical simulations and steady states are studied. Propagation phenomena and existence of traveling waves are explored numerically and theoretically for different variants of such models in [1], [3], [4], [6]. This problem has also been studied from an asymptotic point of view using Hamilton-Jacobi equations [7], [8].
Despite several recent attempts to study such models, dynamics of populations structured by trait and space are not completely understood and several interesting and challenging questions remain to be resolved in this field (see for instance [7]). In particular, the works quoted above concentrate on the case where the mutations are frequent such that the diffusion in space and the mutations are modeled in the same time scale. Our objective is to understand the framework of adaptive dynamics where the mutations are rare enough such that between two mutations the dynamics is driven by a system of nonlocal reaction-diffusion equations, each of them describing the dynamics and the spatial distribution of one trait. We study the steady states and the long time behavior of such systems. Note that although the existence of steady states for a model with continuous trait and space is provided in [2], the long time behavior of solutions is not known, to our knowledge, for discrete or continuous traits. However, in the case of a single trait and considering only homogeneous environments, [5] provides a study of steady states and traveling waves.
In this paper we focus on this problem, for the simplest cases where the population is either monomorphic (a single type is involved) or dimorphic (the population is composed of two-type subpopulations). The space state is an open bounded subset X of R d with a boundary of class C 3 . In the dimorphic case, the spatial density of the population is modeled by the system of nonlinear partial differential equations of parabolic type x) + a 1 (x) − X I 11 (y)g 1 (t, y)dy − X I 12 (y)g 2 (t, y)dy g 1 (t, x), (0.1) where g 1 (t, x) (respectively g 2 (t, x)) denotes the density of individuals of type 1 (resp. of type 2), in position x at time t. The density dynamics is driven by growth rates a 1 and a 2 which depend on the spatial position of individuals and on their type. Further the competition is modeled by nonlocal death rates depending on the environment heterogeneity through kernels I ij , i, j = 1, 2. We will show that this system admits 4 non-negative steady states depending on the ecological parameters. The stability of these states is based on the signs of the principal eigenvalues Here H 1 is the Sobolev space of order 1 on X . The first steady state is the trivial one (0, 0) describing an extinct population, two of them describe long term specialization on a single type and only one subpopulation has a non trivial long time behavior. The last one describes a coexistence case where individuals with two types exist in a long time scale.
The main results of the paper (Theorems 2.3 and 2.4) give assumptions based on spectral parameters and competitive kernels under which the solution of the equation converges, as time goes to infinity, to one of these steady states. This result is new and interesting by itself but it will also be the first step in an adaptive dynamics framework, if we want to understand how mutant individuals invade the population at the evolutive scale (see [11], [13]).
Section 1 deals with the case of a monomorphic population where all individuals have the same type. The density dynamics is led by a nonlinear partial differential equation of parabolic type with a non-local competition term. In this case, we show existence of steady states for a more general competition term: We then explore the long-time behavior of the solution, for the particular case I(x, y) = I(y). Note that, the long-time behavior of the solution of (0.2) with general competition kernel I(x, y) is not yet understood to our knowledge. In [14] the steady states and the long time behavior of the solution of a similar model are studied using different techniques, for the monomorphic case. However, we provide a shorter result for the convergence in long time of solutions which is easily generalizable to dimension 2.
Section 2 is devoted to the two-type case of dimorphic population. In this section, we present our main results on the steady states and the long-time behavior of the solution for the system of partial differential equations (0.1). We also present some numerical results which confirm that the introduction of a new mutant population can lead to a coexistence or an invasion of the previous population, that is linked with an evolution of spatial niches.
Finally in Section 3 we provide the proofs of our main Theorems 2.3 and 2.4 on the long time behavior and the stability of steady states for a dimorphic population.
The mathematical analysis rely on the spectral decomposition of compact operators, fixed point arguments and the study of perturbed Lotka-Volterra type systems.

Notation:
The space set X is an open bounded subset of R d with a boundary of class C 3 . We will denote by L k the Lebesgue space on X of order k ∈ N * and by H k the Sobolev space on X of order k ∈ N * . We denote by C 0,1 the space of Lipschitz continuous functions on X .
For all x ∈ ∂X , we denote by n(x) the outward normal to the boundary ∂X at point x. For a sufficiently smooth function u and x ∈ ∂X , we denote by ∂ n u(x) the scalar product ∇u(x).n(x).

Monomorphic population
In this section, we study the case of a monomorphic population. The model is written as below (1.1) Here g(t, x) denotes the density of individuals in position x and at time t. The Laplace term corresponds to the diffusion of individuals in space and the positive constant m is the rate of this diffusion. We represent the intrinsic growth rate by a, which depends on the position of individuals. Finally, the last term corresponds to the mortality due to competition. We denote indeed by I(x, y) the competition rate between individuals at position x and individuals at position y.
We first give a necessary and sufficient condition for (1.1) to have a steady solution. We then prove, in a particular case where I(x, y) = I(y), that the solution converges in long time to the unique steady solution of (1.1).

Existence of steady state
In this section, we give a necessary and sufficient condition for (1.1) to have a steady solution. To this end, we make the following assumptions on the coefficients: a ∈ C 0,1 (X ), and |a(x)| ≤ a ∞ , for all x ∈ X , (1.2)      I(·, ·) ∈ C(X ×X ) is nonnegative and Lipschitz continuous with respect to the first variable, if d = 1 : 3) Consequently, there exists a positive constant I + such that I(x, y) ≤ I + , for all x, y ∈ X . (1.4) Here, we note that these assumptions are not necessarily optimal. In particular, the regularity of the coefficients may be relaxed but this is sufficient for our purpose and allows us to avoid some technical details.
To state our result we also need the following lemma which can be derived easily from Krein-Rutman's Theorem (see for instance the chapter 6 of [20]) and its proof is left to the reader. Lemma 1.1 (Eigenvalue problem). There exists a principal eigenvalue H to the following eigenvalue problem: This eigenvalue is simple and the corresponding eigenfunction u is the only eigenfunction which is strictly positive in X . Moreover, H can be computed from the following variational problem We are now ready to state our first result: (1.6) Proof. (i) Let H ≤ 0. We prove by contradiction that there is no nonnegative solution to (1.6).
To this end, we suppose that 0 ≤ g ∈ C 2 (X ) solves (1.6). Supposing that g is non-trivial, from the maximum principle we obtain that g is strictly positive and in particular I(x, y)g(y)g(x)u(x)dxdy > 0.
We now multiply (1.6) by u and integrate with respect to x to obtain from (1.5), This is in contradiction with the assumption H ≤ 0.
(ii) We now suppose that H > 0. To prove that (1.2) has a steady solution, we construct a mapping Υ : such that any fixed point of this mapping will be a steady state of our problem, as follows. Thanks to (1.2), we can choose δ > 0 small enough such that 1 − δa(x) > 0 for all x ∈ X . Let h ∈ L 2 . We define ψ(h) = h 1 − δ I(·, y)h(y)dy , and Υ(h) = g, where g ∈ H 1 (X ) is the unique solution of the following equation Notice that fixed points of the mapping Υ are steady solutions of our problem and conversely. So the last step is to show that Υ has a fixed point. We establish this result thanks to Schauder's fixed point Theorem (see for instance Theorem (4.1) in [15]). We first notice from the choice of δ that (−mδ∆ + (1 − δa)Id) −1 is a continuous and compact mapping. As ψ : L 2 → L 2 is a well-defined continuous mapping, we deduce the continuity and compactness of Υ.
We will split the rest of the proof into two cases depending on the dimension of the domain, denoted by d, as in the statement of the theorem.
• If d > 1, using Lemma 1.1, there exists a positive eigenfunction u associated to the positive eigenvalue H. We denoted by u + and u − its maximum and minimum values on X . Then we define and choose δ > 0 small enough such that Let us now introduce the convex closed subset of L 2 As h is a positive function, and using (1.9), We deduce that g is positive on X thanks to the maximum principle. Then we multiply (1.10) by u and integrate it over X , From an integration by parts, (1.5), (1.4) and (1.3), we find the following inequality: Thanks to (1.9), the two polynomial functions Finally we obtain from (1.8) and (1.
Spatially structured Lotka-Volterra systems We conclude from the Schauder's fixed point theorem that Υ has a positive fixed point.
• For d = 1, the previous proof is valid if I is strictly positive in X but we can relax this assumption to the one in (1.3) thanks to the following method. We first prove the following lemma Lemma 1.3. Assume (1.2) and (1.3). There exists R > 0 such that for all g ∈ L 2 , positive and t ∈ [0, 1[ satisfying g = tΥ(g), we have g L 2 < R.
Proof. We will use an argument which is similar to the one presented in [5]. Let g ∈ L 2 , positive and t ∈]0, 1[ such that g = tΥ(g). g attain its maximum value at a point x 0 ∈ X . As g satisfies Neumann boundary conditions, for all x 0 ∈ X , we have g (x 0 ) = 0 and g (x 0 ) ≤ 0. Using (1.7) at the point x 0 and since t < 1, we get (1.11) We then use Taylor-Lagrange's formula for the function g at point x 0 . For all y ∈ X , there exists Additionally, by using again (1.7), we obtain, for all ξ ∈ X , which is bounded since I(x 0 , .) is positive in a neighborhood of x 0 from (1.3), we conclude easily.
Thanks to this lemma, we choose δ satisfying Then we define the convex closed subset where λ − is defined as before by (1.8). For h ∈ Y, we have X I(x, y)h(y)dy ≤ I + R |X | < 1 δ which implies that ψ(h) and g = Υ(h) are positive functions. Moreover, following similar arguments as in the case d > 1, and noticing that the assumption (1.12) guarantees that , we obtain that X gu ≥ λ − . As we are not sure that Υ(h) L 2 ≤ R, we use the following method inspired by the proof of Schaefer's fixed point theorem: we introduce a new functionΥ : ThusΥ is a continuous and compact function that maps Y into itself. From the Schauder's fixed point theorem, we deduce that it has a fixed pointḡ. Suppose that Υ(ḡ)

Convergence to steady state solution
We now come back to our initial problem and we study the long time behavior of the solution to (1.1) for a particular case, where I satisfies the following assumptions: I ∈ L 2 (X ) and there exists I − > 0, for all y ∈ X , I(y) ≥ I − ., (1.13) We also assume the following assumption on the initial condition (1.14) We show that the positive solution of the following parabolic equation (1.15) tends to the unique steady state of the problem while t → +∞. (1.16) Remark 1. Notice that H > 0 is a necessary and sufficient condition to obtain a positive limit as t → ∞, that is, only the diffusion parameter and the growth rate a have an influence on the non-extinction of the population in long time. The competition rate affects only the total size of the population at the limit, but not its existence.
We also note that a simple example where the assumption H > 0 is satisfied is the case where the growth rate a is a positive function, that is, if the birth rate is greater than the death rate everywhere, the population will survive. Moreover, if a is a negative function, it is easy to deduce that H is negative and that the population goes extinct.
Proof. We first check that there exists only one positive steady state in the case H > 0. Let g,h ∈ H 1 be two positive solutions to (1.16). Henceḡ,h are positive eigenvectors of the compact, continuous operator L = m∆(.) + a.. As Lemma 1.1 implies the uniqueness of a positive eigenvector up to a multiplicative constant,ḡ = s ·h with s ∈ R + . Moreover, from (1.16), we deduce that the principal eigenvalue H is equal to X I(y)ḡ(y)dy and the same result holds for h. It follows thatḡ =h. We now denote byḡ the unique solution of (1.16).
The next step is to show the convergence in L ∞ towards the positive steady state if H > 0 and towards 0 if H ≤ 0. Let us make the following changes of variable function Thus v is a solution to the equation (1.17) It is well-known from the spectral decomposition of the operator L and the regularizing property of the Laplace operator that v(t, ·)e −Ht tends uniformly to βḡ, a principal eigenvector of the operator L, that is, for some positive constant β, We divide this limit by an integrated version of it to obtain where ρ(t) = X g(t, y)dy, and the r.h.s. is positive sinceḡ is a principal eigenvector of L. It remains to show that ρ(t) has a finite limit when t tends to infinity. Integrating (1.15), we find that ρ is a solution to Moreover −m∆ḡ = aḡ − Hḡ using definitions of H andḡ, which leads to X a(x)ḡ( , that is sufficient to conclude that sup t∈R + ρ(t) < +∞, and D(t) −→ 0, as t → +∞. Next, we show that ρ(t) tends to H/µ if H > 0 and to 0 if H ≤ 0 thanks to the following lemma, proven below. Lemma 1.5. Let µ ∈ R + * , and ρ be a positive solution on R to d dt ρ(t) = ρ(t)(r + E(t) − µρ(t)), where E(t) −→ t→+∞ 0, then ρ(t) tends to r µ if r ≥ 0 or to 0 if r < 0 as t tends to +∞.
Finally, we conclude from (1.19) and the above lemma that, for H > 0, and for H ≤ 0, Proof of Lemma 1.5. We split this proof into two parts, depending on the value of r.
As this is true for all ε 0 > 0, we can conclude.

Dimorphic population
In this section we study the dynamics of a system describing a dimorphic population. The model is written (2.1) Here, g 1 (t, x) (respectively g 2 (t, x)) denotes the density of individuals of type 1 (respectively of type 2) at position x ∈ X and at time t ≥ 0. As before, the diffusion of individuals in space is modeled by Laplace terms and the diffusion rates for the populations of type 1 and 2 are denoted respectively by the positive constants m 1 and m 2 . We represent the intrinsic growth rate of individuals of type i, for i ∈ {1, 2}, by a i . The last terms correspond to the mortality due to intraspecific and interspecific competition. The functions I ij (y), for i, j ∈ {1, 2}, represent indeed the pressure applied by individuals of type j at position y on individuals of type i at any position. We first identify the steady solutions of (2.1). Next, we study the stability of steady states and the long time behavior of the solutions. To this end, we will make the following assumptions on the coefficients, for i, j ∈ {1, 2}, We also make the following assumptions on the initial conditions g 0 i ∈ L 2 (X ), for i = 1, 2.
is an orthogonal basis of L 2 (X ).We normalize them by X |A i k (x)|dx = 1.
(iv) The first eigenvalue H i is simple and is given by The first eigenfunction A i 1 is positive, the eigenfunctions corresponding to the other eigenvalues change sign. Those eigenfunctions are smooth.
Let us also introduce the following notation for i, j ∈ {1, 2}

This steady state exists if and only if
Proof. The conditions on the existence of the three first steady states are immediate from Theorem 1.2. Moreover, it follows from Lemma 2.1 that there is no nonnegative steady state other than the ones stated above. We only prove the last point corresponding to the steady state with two positive exponents. We first suppose that r 1 and r 2 given by (2.8) are positive. It is then easy to verify, from (2.5) and (2.6), that (r 1 A 1 1 , r 2 A 2 1 ) is a steady solution of (2.1). We next notice using (2.7) that the matrix in (2.8) is invertible, and r 1 and r 2 are positive if and only if has also the same sign if and only if H 1 > 0. We conclude easily.

Long time behavior of the system
Our main results concern the long time behavior of solutions of (2.1). We give explicit conditions determining whether or not the population goes extinct or whether or not there is co-existence of the two types at equilibrium. The first theorem shows the convergence of the solution when time goes to infinity and gives sufficient conditions for convergence to the globally asymptotically stable states. The second theorem explores the more delicate cases, where there are several stable equilibria and different basins of attraction. 1. For any initial condition, as t → ∞, the unique solution of the parabolic system (2.1) tends to one of the steady states described in Lemma 2.2.

If
then for any initial condition such that g 0 1 is not identically zero and as t → ∞, the solution converges to (ḡ 1 , 0). We thus have fixation of type 1 in the population.

If
then for any initial condition such that g 0 2 is not identically zero and as t → ∞, the solution converges to (0,ḡ 2 ). We thus have fixation of type 2 in the population.

If
then for any initial condition such that g 0 1 and g 0 2 are not identically zero and as t → ∞, the solution converges to (ĝ 1 ,ĝ 2 ), i.e. we have co-existence of types 1 and 2.
Before writing the second theorem which deals with the other values of H 1 and H 2 , we illustrate this first theorem with some numerical examples. The numerics are computed with an algorithm based on finite difference method. Our aim is to illustrate the behavior of mutant individuals that appear in a well established population. Fisrt of all, thanks to Theorem 1.4, we have a mean to compute the principal eigenvalues H i when they are positive. In fact, in this case, Theorem 1.4 guarantees that any positive solution to ∂ t u = m i ∆ x (u) + a i − X I ii u u with Neumann boundary conditions tends to the steady stateḡ i , and Let us now describe our numerical simulations. We consider that the growth rates of the two populations are maximal at two different spatial positions. For instance, the space state can represent a variation of resources, as seed size for some birds, and so the two populations are not best-adapted to same resources. Different values ofā 2 , the maximum of the growth rate of the mutant population, will be explored, while the other parameters are fixed, Notice that around the trait u i , the growth rate of the population i is positive but far from this position, it becomes negative. Thus positions around u i are favorable for population i, and we suppose that the intraspecific competition is greater around that position: Then, we define the interspecific competition from the previous kernels by I 12 = I 21 = min{I 11 , I 22 }. Finally, we suppose that all individuals move with the same diffusion constant m 1 = m 2 = 0.01.
As we want to illustrate the invasion of a mutant, we suppose that the initial condition is near (ḡ 1 , 0), as presented in figure 1(a). We resolve numerically the system of parabolic equations (2.1) and present the solution after a long time, that is, when the densities are almost stable, see figure 1. Whenā 2 is small, the mutant population is not able to survive (case (b)). But when a 2 is big enough, coexistence (case (c)) and even invasion (case (d)) can appear. On the two last cases, the new population that has invaded the space does not live on the same spatial position as the previous one. From an ecological viewpoint, such examples are very interesting because we observe a change of spatial niche due to a selection event.
Finally, to give a complete picture of the long time behavior of the solution, let us now study the last cases where several equilibria can be reached.

If
then the steady state (ḡ 1 , 0) is asymptotically stable and (0,ḡ 2 ) is unstable. Nevertheless some solutions will converge to the latter.

If
then the steady state (ḡ 2 , 0) is asymptotically stable and (0,ḡ 1 ) is unstable. Nevertheless some solutions will converge to the latter.

Stability of the steady states -The proofs of Theorems 2.and 2.4
This section is devoted to the proofs of Theorems 2.3 and 2.4.

The proof of Theorem 2.3
In this section, we prove Theorem 2.3. To this end, noticing that the total density of the population is not constant, we will first study the limit of population densities normalized by the masses. Then we will study the long time behavior of a system of differential equations which describes the dynamics of the two masses (see Lemma 3.1).
Similarly to the proof of Theorem (1.4), we make the following change of variables, Following similar arguments as in subsection 1.2, we find a similar limit as (1.18) for g i , which leads to g 1 (t, .) Let ρ i (t) = X g i (t, y)dy for i ∈ {1, 2}. It remains now to understand the behavior of (ρ 1 (t), ρ 2 (t)). We deduce the following limits from (3.1),

Spatially structured Lotka-Volterra systems
Integrating (2.1) on X and using the previous limits, we find that (ρ 1 , ρ 2 ) is a solution to Here we have used the fact that, in view of (2.2) and (2.3), ρ j is a positive solution to and hence ρ j is bounded, for j ∈ {1, 2}.
To go further we need the following lemma which is proven below.
Proof of Lemma 3.1. We split the proof into several cases depending on the values of H 1 and H 2 .
Case 1: First of all, we will consider that at least one of the two variables is non-positive. For example, let assume that . Thanks to the results on the logistic equation, we conclude easily that lim sup t→+∞ ρ 1 (t) ∈ [0, ε]. As this is true for all ε > 0, ρ 1 (t) tends toward 0 when t approaches infinity.
Therefore, ρ 2 solves We conclude that ρ 2 convergence and evaluate its limit thanks to Lemma 1.5.
Let ε > 0 be small enough to satisfy We split (R + ) 2 into five disjoint sets presented bellow and drawn on figure 2: There exists t ε > 0 such that for all t ≥ t ε , max{|E 1 (t)|, |E 2 (t)|} ≤ ε 2 . It is then easy to verify that, As ρ 1 satisfies d dt ρ 1 (t) ≥ ε 2 ρ 1 (t) in D 1 , for all t ≥ t ε , if (ρ 1 (t), ρ 2 (t)) belongs to D 1 for somē t ≥ t ε , it will quit this domain after a finite time t 0 and reach the set D = D 3 ∪ D 4 ∪ D 5 . Same kind of results holds in D 2 . Thus after a finite time t 0 ≥ t ε , the trajectory of the solution reaches D , moreover it cannot quit this domain according to the signs of derivatives of ρ 1 and ρ 2 at the boundaries of D . The next step is to study the dynamic in D . Suppose that the trajectory belongs to D 3 ∪ D 4 , it cannot stay in that area for all t ≥ t 0 , so there exists t 1 such that (ρ 1 (t 1 ), ρ 2 (t 1 )) ∈ D 5 . We denote the entry point in D 5 by x 1 , drawn in figure 3. According to the derivatives of ρ 1 and ρ 2 , the trajectory of the solution does not quit the set: where (ρ x1 1 , ρ x1 2 ) are the coordinates of x 1 , this set is represented by the hatched area on the left scheme of figure 3. Moreover, as long as the trajectory stays in D 3 ∪ D 5 , ρ 2 satisfies ∂ t ρ 2 (t) ≤ − ε 2 ρ 2 (t). So two cases can happen: (a) either ρ 2 (t) −→ t→+∞ 0, and thus ρ 1 tends to H1 µ11 from Lemma 1.5, (b) or there exists t 2 > t 1 where the trajectory reaches D 4 . Let denote x 2 = (ρ x1 1 , ).

As the trajectory stays in
Moreover, for all t ≥ t 2 , the trajectory stays in D x2 (see the hatched area on the right scheme of figure 3).
Iterating the previous step, we construct a decreasing sequence of areas (D xn ) n≥0 which will be necesseraly finite. Indeed, the choice of ε (3.3) implies that there exists m ∈ N such that D x2m ∩ D 4 = ∅. Then we conclude as in the case (a) above. Let k ≥ 1 and ε > 0 be such that max µ 11 µ 21 , µ 21 µ 11 < k and min We divide the plan (R + ) 2 as presented in figure 4, where D 1 , D 2 , D 3 , D 4 , D 5 and D are defined as follows The constant k is chosen such that D 1 ∩ D 5 = ∅ and D 2 ∩ D 4 = ∅ and t ε is defined as before. There exists t 0 ≥ t ε such that for all t ≥ t 0 , the trajectory is belonging to D . Then we construct a sequence of sets (D xn ) n≥1 as before, but this time, this sequence can be infinite. So let D ε be the set where the above constants satisfy the following compatibility conditions then the solution to the equation (2.1) tends to the steady state (ḡ 1 , 0).
Let us express g 1 in the basis , ∀x ∈ X , and denote κ(t) = Xḡ 1 (x)dx + α 1 (t), for all t ∈ R + . From (2.1) and the representation of g 1 and ∂ t g 1 with respect to the basis (A 1 k , k ∈ N * ), we find the following dynamical system (3.7) Here, we have used the fact that since, from Lemma 2.1, L N = Xḡ 1 dxA 1 1 + N k=1 α k A 1 k tends to g 1 in L 2 as N → ∞, and since the domain X is bounded, L N tends to g 1 in L 1 .
We will show that for all t ≥ 0, We notice that this is true for t = 0 thanks to (3.4): The second line is justified using the representation of I i1 with respect to the orthonormal basis We prove the result (3.8) by contradiction. Denote and suppose that t 0 is finite. Thus, for i ∈ {1, 2}, In fact, from (3.9), we will find a lower bound greater than −C for any t ≤ t 0 which is a contradiction with the fact that t 0 is finite.
Moreover, Theorem 2.3 guarantees the existence of a limit for (g 1 (t, .), g 2 (t, .)). Let us identify that limit. On the one hand, we note that (3.12) holds for all t ≥ 0, since t 0 = +∞, and hence X |g 2 (t, x)| 2 dx tends to 0, as t approaches infinity. On the other hand, thanks to the equation (3.13), which holds for all t ≥ 0, and (3.6), Thus, the limit of g 1 (t, .) L 2 is positive. The limit of the solution is hence the steady state (ḡ 1 , 0).
Thanks to point (1a), we already know that the steady state (ḡ 1 , 0) is stable. We prove that the steady state (0,ḡ 2 ) is unstable; more precisely, we show that in any neighborhood of (0,ḡ 2 ), there exists a solution to (2.1) that does not tend to (0,ḡ 2 ) as t → ∞, but there also exist some solutions that tend towards it.