Existence of nontrivial steady states for populations structured with respect to space and a continuous trait

We are interested in this work in the steady solutions to models of population dynamics in which the population is structured both w.r.t. the space variable Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8, A-1040 Wien, Austria, e-mail: anton.arnold@tu-wien.ac.at CMLA, ENS Cachan, CNRS & IUF, PRES UniverSud, 61 avenue du Président Wilson, F-94235 Cachan, France, e-mail: desville@cmla.ens-cachan.fr MAPMO, Université d’Orléans, 45067 Orléans Cedex, e-mail: celine.prevost@acorleans-tours.fr


1.
Introduction. In this work we are interested in the steady solutions to models of population dynamics in which the population is structured both w.r.t. the space variable x (here, x ∈ Ω, a bounded regular open set of IR N ) and a trait variable denoted by v (here, v ∈ [0, 1] for the sake of simplicity).
The distribution function f := f (t, x, v) ≥ 0 shall then denote the number density of individuals at time t ∈ IR + , position x ∈ Ω, and whose trait is v ∈ [0, 1]. We also denote by ρ(t, x) = 1 0 f (t, x, v) dv the total number of individuals at time t and position x.
This paper is concerned with an integro-PDE model of reaction-diffusion type in infinite (continuous) dimension in which selection, mutations, competition, and migrations are taken into account.
Our modeling assumptions are the following: migration is described by a diffusion (w.r.t. x) operator with a rate ν := ν(x, v) [that is, individuals with different traits or at different positions can have a different migration rate]; mutations are described by a linear kernel K * := K * (x, v, v ′ ) ≥ 0 which is related to the probability at point x that individuals with trait v ′ have offsprings with trait v; selection is implemented in the model thanks to a fitness function k * := k * (x, v) ≥ 0 which may depend on both point x and trait v; finally a logistic term involving a kernel C * := C * (x, v, v ′ ) ≥ 0 models the competition (felt by individuals of trait v) at point x due to individuals of trait v ′ .
Under those assumptions, the evolution of the population is governed by the following integro-PDE: For a mathematical study of eq. (1), we refer to [7].
Our goal in this paper is to investigate the existence of (non-trivial) steady states for eq. (1), that is (non-zero) solutions to the following non-linear, elliptic integro- where . This study will be carried out by moreover assuming that the population is confined to the region Ω, that is f := f (x, v) satisfies the homogeneous Neumann boundary condition: where n(x) is the outward unit normal vector to ∂Ω at point x.
We use in the sequel coefficients which satisfy the following assumption: Assumption A • The selection and mutation parameters k := k(x, v) and K := K(x, v, v ′ ) satisfy Note that the upper bounds in this assumption are quite natural (the rates have no reason to become infinite on biological grounds). The lower bound for C means that we assume that all individuals are in competition (no cooperation, or even neutrality, can be treated in our theorem). The lower bound on k (or k + K) is of a different nature: it can be imposed by restricting the domain in v to the individuals for which it holds (that is, individuals which have a tendency to decay exponentially even without competition are not considered).
Our main result reads EXISTENCE OF STEADY STATES FOR POPULATIONS 3 Theorem 1.1. Let Ω be a smooth bounded subset of IR N (N ∈ IN ), and K, C, k satisfy Assumption A. We suppose moreover that K, C, k are continuous onΩ 2.

The function
5. The function µ also satisfies (2), (3) in the sense of distributions (more precisely, the equation is integrated against C ∞ functions including the boundary conditions), i.e. for all Note that all terms in (9) and (10) are well-defined thanks to the estimates (6), (7), and (8).
This result can be improved in situations when mutations are somehow predominant (compare (11) to (4)), as shown by the following Let Ω be a smooth bounded subset of IR N (N ∈ IN ), and K, C, k satisfy Assumption A. We suppose moreover that K ∈ L ∞ (Ω × [0, 1] × [0, 1]) and that Then, there exists f : is nonnegative, and f is a weak solution to eq. (2), (3) in the following sense: for , We remark that the improvements achieved in Theorem 1.2 are based on an L pestimate on f which relies on the additional assumption (11).
We now put these results in perspective. On one hand, models of selection/mutation/competition for populations structured w.r.t. a continuous trait were studied (especially from the point of view of their large time behavior) in [9], [8], [12], [3], etc. On the other hand, the very rich subject of reaction-diffusion equations with a finite number of equations has been the subject of innumerable studies (cf. [13] and [14] and the references therein). Extensions to an infinite (enumerable) number of such equations in the context of coagulation-fragmentation models can be found for example in [10]. Models involving infinite "continuous" dimensional reaction-diffusion equations were studied in [11,4,5], and the large time behavior of such equations in the presence of a Lyapunov functional was established (cf. [4]).
The present work is a first step towards the extension [to infinite-dimensional (continuous) reaction-diffusion equations modeling selection/mutation/competition/ migration] of two of the above research directions: of results on the large time behavior of the spatially homogeneous models [9], [8], [12], [3] on one hand, and of models with Lyapunov functionals, as in [4], on the other hand. The absence of a Lyapunov functional in the considered model (1) makes our analysis much more difficult than the one performed in [4]. Hence, we present here only existence results for steady states of equation (1) (but we shall not study their stability).
As when looking for nontrivial steady states of finite-dimensional reaction-diffusion equations, one needs topological tools. Here, we shall use Schauder's fixed point theorem (cf. [14]) for an approximate problem, together with a (weak) compactness method for removing the approximation.

EXISTENCE OF STEADY STATES FOR POPULATIONS 5
The paper is organized as follows: In §2 we prove existence of nontrivial solutions to problem (2), (3) in the space of measures (that is, Theorem 1.1). In §3 we prove Theorem 1.2, that is, when mutations are predominant, the solutions to (2), ). In the numerical examples of §4 we illustrate the effect of the diffusion strength on the steady states.
2. Stationary Solutions. The proof of Theorem 1.1 uses a compactness method based on the following regularized equations (with ε > 0): These boundary value problems have solutions thanks to the following , and K, C, k satisfy Assumption A. Then for all ε > 0, there exists a strong solution (14).
Moreover, f ε is nonnegative and satisfies (for a.e. x ∈ Ω) Proof of Proposition 1: We establish this result thanks to Schauder's fixed point theorem. In order to do so, we introduce (for δ > 0) on one hand the linear operator L δ defined by the elliptic Neumann problem (with constant coefficients) And on the other hand we define the (nonlinear) operator N δ , δ > 0, by Then the boundary value problem (12)-(14) is equivalent to the fixed point problem Next, we introduce the bounded convex (nonempty) closed subset (for a, b defined in (6)) of the Banach space L 1 (Ω × [0, 1]). We now prove the Lemma 2.1. The operator L δ N δ maps Y into itself as soon as δ > 0 is small enough.
Proof of Lemma 2.1: First we choose δ > 0 sufficiently small for the following inequality to hold: Then, the functions h 1 and h 2 : and Now we note that (for f ≥ 0) we have (using (4)) and Now, if f ∈ Y , we know that a ≤ f dv ≤ b. From (25), (26), and the monotonicity of h 1 and h 2 we thus conclude Next we consider the BVP satisfied by L δ N δ (f ) dv on Ω: and for x ∈ ∂Ω, Then, by a standard maximum principle applied to the solution L δ N δ (f )(x, v) dv of a Neumann elliptic problem, we see that In order to conclude the proof of Lemma 2.1, it remains to show that L δ N δ (f ) ≥ 0 for δ > 0 small enough. Using again the maximum principle (but this time on the domain Ω×]0, 1[) for the elliptic operator Id − δ∆ x − δε∂ 2 v , we just have to show that N δ (f ) ≥ 0 (for δ > 0 small enough). But C+b . This concludes the proof of Lemma 2.1.
We now turn to the proof of It remains to prove that L δ N δ is compact. For this, we observe that according to the proof of Lemma 2.1, N δ (Y ) ⊂ Y when δ > 0 is small enough, and that L δ sends L 1 (Ω × We now turn to the Proof of Theorem 1.1: We start from a sequence of solutions f ε = f ε (x, v), ε > 0 to the regularized problem (12) - (14) given by Proposition 1. Hence (15) holds.

ANTON ARNOLD AND LAURENT DESVILLETTES
As a consequence, we can extract from f ε a subsequence (still denoted by f ε ) such that where µ is some function x → µ x of L ∞ (Ω x ) with values in the set of (nonnegative) bounded measures on [0, 1] (denoted here by M 1 v ). This means that for all functions , Note that x → µ x clearly satisfies estimate (6).
To prove (9), we now introduce our test function ϕ(x, v) = ψ(x)ξ(v) with ψ ∈ H 1 (Ω) and, initially, again ξ ∈ C 2 n ([0, 1]). Then the weak form of problem (12) -(14) reads: Then, it is easy to pass to the limit in the (linear) first four terms (using estimate (30) for the first one). So we only need to show that we can also pass to the limit in the (nonlinear) last term: The following result holds: Proof of Lemma 2.3: Due to the uniform continuity of C (and estimate (15)), is also continuous (uniformly w.r.t. x ∈ Ω) that is, (8) holds. As a consequence, the second part of the l.h.s. of (33) is well-defined. Now we estimate the l.h.s. of (33) by The second term tends to zero since ( ). Then, the first term of (34) is bounded by b ψ L 2 (Ω) ξ L ∞ K ε L 2 (Ω) , where The property K ε L 2 → 0 when ε → 0 can be shown by first approximating C(x, v, v ′ ) by a sequence such that C n → C uniformly. Then, we estimate where the last term is bounded by 2 C n − C L ∞ b|Ω| 1/2 . The first term tends to 0 as ε → 0 for all fixed (30), and therefore compact in L 2 (Ω) (strong).
3. The case when mutations are predominant. We develop in this section the Proof of Theorem 1.2: We once again pass to the limit when ε → 0 in the regularized problem (12) - (14) in order to get a solution to problem (2), (3). In order to do so, we multiply (12) by (f ε ) p (p ≥ 1) and we integrate with respect to x and v.
Moreover, for any p ≥ 1, . Hence, we can extract from the family (f ε ) ε>0 a subsequence still denoted by and (for all x ∈ Ω) ∂ v ϕ(x, 0) = ∂ v ϕ(x, 1) = 0, we can write the weak form of (12) - (14): Then, we can pass to the limit (as ε → 0) in the following terms: We now wish to prove that We write therefore and observe that since f ε ⇀ f in L ∞ weak*, we only need to show that (up to a subsequence) Cf ε ϕdv strongly converges in L 1 (Ω × [0, 1] v ′ ) to Cf ϕdv. To this end we use the following lemma (with s(x, v, v ′ ) := ϕ(x, v)C(x, v, v ′ )): Then, a subsequence (still denoted by f ε ) satisfies Proof of Lemma 3.1: We first observe that Then, we approximate s (in We see that Thanks to (40), the second term tends to 0, hence f ε s dv converges to f s dv in . This ends the proof of Lemma 3.1.
Finally, we can pass to the limit when ε → 0 for all the terms in (the weak form (38) of) problem (12) - (14), and get a weak solution f to eq. (2), (3), for test and (for all x ∈ Ω) ∂ v ϕ(x, 0) = ∂ v ϕ(x, 1) = 0. These last assumptions can easily be removed thanks to an approximation procedure. Note that these additional assumptions on ϕ were only used to treat the term ε ∂ 2 v f ε which disappears in the limit. As a last remark, note that since f ∈ L ∞ (Ω×[0, 1]), we also have ∆ x f ∈ L ∞ (Ω×[0, 1]).
Remark 2: Theorem 1.2 shows that smooth solutions of problem (2), (3) exist when mutations are predominant. Our feeling is that in many situations, all steady solutions of this equation are smooth when mutations are present (i.e. K > 0). This is supported by the study of the time-dependent equation (1) with suitable initial and boundary conditions, cf. [7].

4.
Some numerical examples. We present here some computations which illustrate the results obtained in §2. In particular we show the influence of the diffusion strength on the shape of the steady states.
More precisely, we chose to use the following parameters for the selection, mutation, and competition: Note that this choice of k does not satisfy the positivity assumption from (4) on the whole (x, v)-domain. It would, however, hold on an appropriate subdomain. Anyhow, the above fitness function k prefers one single trait (at v = 1 2 ). On the other hand, the competition kernel C favors a clear splitting of population into well separated traits. This example hence illustrates the balance between these two opposing effects.
The computation is performed on the square [0, 1] × [0, 1] by letting t → ∞ in the time-dependent equation (1), (3). We use 200 cells in the x-space and 200 cells in the v-space. The time step is adjusted in order to obtain a CFL parameter of 0.396. We use a semi-implicit finite difference scheme (the 0-th order part of the equation is discretized in an implicit way, but not the diffusion part).
We present results obtained for A = 10 7 , 10 6 , 10 5 , 10 4 where 1/A plays the role of the diffusion coefficient in (2). The surface that defines the (quasi) stationary solution is presented at two different angles so that the shape of the solution is clearer. The coordinate x corresponds to the vertical axis and the coordinate v to the horizontal axis in the figures on the left. The graduation from 0 to 200 corresponds to the numbering of cells. The first figure (A = 10 7 ) corresponds to a case in which the diffusion is very small, so that its solution is very close to the case without diffusion which can be computed explicitly (cf. [8]): for x small, the function of v is a Dirac mass at v = 0.5, for x bigger, it is the sum of two Dirac masses, and for x large, it is the sum of three Dirac masses (one of them sits at v = 0.5). Note the quite sharp transitions (in x) between the regions populated by individuals with one, two, or three traits.
In the other figures, the diffusion w.r.t. x entails the presence of individuals with various v in the whole domain x ∈ [0, 1]. This is particularly clear in the last figure, where the diffusion is strong enough to build a "five-modal" function of v at point x = 1.