Nonlocal refuge model with a partial control

In this paper, we analyse the structure of the set of positive solutions of an heterogeneous nonlocal equation of the form: $$ \int_{\Omega} K(x, y)u(y)\,dy -\int_ {\Omega}K(y, x)u(x)\, dy + a_0u+\lambda a_1(x)u -\beta(x)u^p=0 \quad \text{in}\quad \times \O$$ where $\Omega\subset \R^n$ is a bounded open set, $K\in C(\R^n\times \R^n) $ is nonnegative, $a_i,\beta \in C(\Omega)$ and $\lambda\in\R$. Such type of equation appears in some studies of population dynamics where the above solutions are the stationary states of the dynamic of a spatially structured population evolving in a heterogeneous partially controlled landscape and submitted to a long range dispersal. Under some fairly general assumptions on $K,a_i$ and $\beta$ we first establish a necessary and sufficient criterium for the existence of a unique positive solution. Then we analyse the structure of the set of positive solution $(\lambda,u_\lambda)$ with respect to the presence or absence of a refuge zone (i.e $\omega$ so that $\beta_{|\omega}\equiv 0$).


INTRODUCTION
In this article we are interested in the positive bounded solutions of the nonlinear nonlocal equation (1.1) Ω K(x, y)u(y) dy − k(x)u + a 0 (x)u + λa 1 (x)u − β(x)u p = 0 in Ω, where Ω ⊂ R n is a bounded open set, K ∈ C(R n × R n ) is non negative, k(x) := Ω K(y, x) dy λ ∈ R, and a i , β are continuous functions. Our aim is to describe the properties of the positive bounded solutions of (1.2), in terms of the properties of K, a i , β and λ. That is, we look for existence criteria of positive bounded solutions of (1.1) and we describe some bifurcation diagrams i.e. depending on a i and β we analyse the properties of the curve (λ, u λ ).
The study of these kind of problems finds its justification in the ecological problematics related to the erosion of Biodiversity. In particular, some recent studies have focused on a better understanding of the impact of some agricultural practises on non targeted species [1,7,21,27,28,29]. Such problematic can be addressed through the analysis of the asymptotic behaviour of the positive solution of a reaction diffusion equation : where u represents a population density evolving in a partial controlled heterogeneous . Here the parameter λ is a control related to the practise and a 1 represents the region where the control is exerted.
For nonlocal equations such as (1.1), less is known and the analysis of the existence, uniqueness and the bifurcation diagram have been only studied in particular situations [2,12,14,15,19,22,25,31]. A large part of the literature is devoted to the existence of positive solution to (1.1) in situations where no refuge zone exists and for a fixed λ [2,12,14,15,22,31]. To our knowledge [19] is the first paper which considers a nonlocal logistic equation with a refuge zone and analyses the curves (λ, u λ ). More precisely, the authors investigate the existence, uniqueness of a positive bounded solution of u ≡ 0 in R n \Ω, (1.8) where J is a symmetric density of probability. They prove that a positive solution of the above problem exists if and only if Moreover, they have showed that this solution is unique and have established the following asymptotic behaviours: These results have been recently extended to the more general equation (1.1) with a quadratic nonlinearity (s(a(x) − b(x)s)) and under some assumptions on the symmetry of the kernel K and some extra conditions on a and λ, see [25].
Here we address these questions of existence, uniqueness and the description of some bifurcation diagrams for a general kernel K and with no restriction on the coefficients a i , λ and β.
In what follows we will always assume that the functions a i and β satisfy: For the dispersal kernel, we will also require that K satisfies: K(x, y) > c 0 .
A typical example of such dispersal kernel is given by with J ∈ C(R n ) continuous, J(0) > 0 and 0 < α i ≤ g i ≤ β i and 0 ≤ h i ≤ β i . Such type of kernel have been recently introduced in [11] to model a nonlocal heterogeneous dispersal process. To simplify the presentation of our results, we also introduce the notation L Ω [u] for the continuous linear operator In [19,25] the analysis essentially relies on the existence of positive eigenfunction associated with a principal eigenvalue µ 1 and a L 2 variational characterisation of µ 1 . However, such properties ( existence of a positive eigenfunction and a L 2 variational characterisation of µ 1 ) does not hold for general kernels K and a i [13] and a new approach and characterisation of the principal eigenvalue has to be developed.
In the past few years, the spectral properties of nonlocal operators such as L Ω + a have been intensively studied [2,12,13,14,15,20,22,23,24]. In particular a notion of generalized principal eigenvalue µ p of a linear operator L Ω + a has been introduced in [12,15] and is defined by µ p is called a generalized principal eigenvalue because µ p is not necessarily associated with a L 1 positive eigenfunction [12,13,24,30]. Such notion has been successfully used to derive an optimal criterium for the existence of a unique positive solution of (1.1) in absence of a refuge zone [12,15].
Equipped with this notion of generalised eigenvalue, we can now state our results. We first present an optimal criterium for the existence of a unique positive bounded solution to (1.1). Namely, we show Moreover the solution is unique.
Next we analyse the partially controlled problem (1.1) i.e. we describe the set {λ, u λ } where u λ is a positive bounded continuous solution to (1.1). We start by describing {λ, u λ } in a case of the absence of a refuge zone. We prove the following Theorem 1.2. Assume that K, a i and β satisfy (1.9)-(1.10). Assume further that β > 0 inΩ then there exists λ * ∈ [−∞; ∞), so that for all λ > λ * there exists a unique positive continuous solution u λ to (1.1). When λ * ∈ R, there is no positive solution to (1.1) for all λ ≤ λ * . Moreover, we have the following trichotomy: In addition, the map λ → u λ is monotone increasing and we have Furthermore, u ∞ is non trivial when µ p (L Ω\Ω 1 + a 0 ) < 0.
Before going to the proofs of theses results we would like to make some additional comments. The assumption can be relaxed and we can get a full description of the curves when a 1 > 0 inΩ.
The paper is organised as follows. In a preliminary section we recall some known results on µ p and on the positive solution of a KPP equation. Then in Section 3 we prove the existence criterium of Theorem 1.1. The proof of Theorem 1.2 is done in Section 4. Finally, in Section 5 we analyse the bifurcation diagram of (1.1) in the presence of a refuge zone (Theorem 1.3).

PRELIMINARIES
In this section, we recall some results on the principal eigenvalue of a linear nonlocal operator and some known results about the KPP equation below and f (x, s) is satisfying ) and is differentiable with respect to s f u (·, 0) ∈ C(Ω) f (·, 0) ≡ 0 and f (x, s)/s is decreasing with respect to s there exists M > 0 such that f (x, s) ≤ 0 for all s ≥ M and all x. (2.2) The simplest example of such a nonlinearity is where a(x) ∈ C(Ω). It has been shown in [2,12] that the existence of a positive solution of (2.1) is conditioned to the sign of the principal eigenvalue µ p of the linear operator L Ω + f u (x, 0) where µ p is defined by the formula That is to say Also noted in [12] the principal eigenvalue is not always achieved. This means that there is not always a positive continuous eigenfunction associated with µ p . However as shown in [13], we can always associate a positive measure dµ with µ p . More precisely,

Theorem 2.2 ([13]).
Let Ω be an open bounded set and assume that K and f u (x, 0) satisfy the assumptions (1.9) and (2.2). Then there exists a positive measure dµ ∈ M + (Ω), so that for any φ ∈ C c (Ω) we have is a non negative singular measure with respect to the Lebesgue measure whose support lies in the set Σ := {y ∈Ω|f u (y, 0) − k(y) = sup x∈Ω (f u (x, 0) − k(x))}.
As proved in [12,24,30], when Ω is an open bounded set we can find a condition on the coefficients which guarantees that dµ s (x) ≡ 0 and the existence of a positive continuous eigenfunction. For example the existence of principal eigenfunction is guaranteed, if we assume that the function a(x) := f u (x, 0) − Ω K(y, x)dy satisfies For the existence of principal eigenfunction as remark in [15] we also have this useful criteria: Next we recall some properties of the principal eigenvalue µ p that we will constantly use along this paper:

then for the two operators
respectively defined on C(Ω 1 ) and C(Ω 2 ) we have (ii) Fix Ω and assume that a 1 (x) ≥ a 2 (x), then

then for the two operators
respectively defined on C(Ω 1 ) and C(Ω 2 ). Assume that the corresponding principal eigenvalue are associated to a positive continuous principal eigenfunction. Then we have where C 0 depends on K and φ 2 . (v) We always have the following estimate

Proof:
We refer to [12] for the proofs of (i) − (iii) and (v), so we will be concerned only with (iv). Let us introduce the following quantity: . We obtain the equality by arguing as follows. Assume by contradiction that µ p (L Ω + a) > µ ′ p (L Ω + a). Then there exists µ so that µ ′ p (L Ω + a) < µ < µ p (L Ω + a) and from the definition of µ p and µ ′ p there exists two positive continuous functions ψ and φ so that From the last inequalities we deduce that ψ > 0 inΩ and by setting w := φ ψ it follows that Thus w cannot achieve a maximum inΩ without being constant. w being continuous inΩ, it follows that φ = cψ for some positive constant c. Thus we get the contradiction We are now in position to prove (iv). Let φ 2 be the eigenfunction associated toµ p (L Ω 2 + a(x)) normalized by φ 2 ∞ = 1 and let us set C 0 := K(·,·) ∞ minΩ 2 φ2 . Now, let us show that (φ 2 , µ p (L Ω 2 +a)+C 0 |Ω 2 \Ω 1 |) is an adequate test function for µ ′ p (L Ω 1 +a). By a direct computation and by using the normalisation of φ 2 we have

OPTIMAL EXISTENCE CRITERIUM
In this section we establish an optimal criterium for the existence of a positive continuous bounded solution to when there exists ω ⊂ Ω so that β |ω ≡ 0. Note that (3.1) is a particular case of (2.1) with f (x, s) := a(x)s − β(x)s p . However, due to the presence of refuge zone (i.e. β |ω ≡ 0) the function f (x, s) does not satisfy the assumptions (2.2) and the Theorem 2.1 does not apply. But we still have a complete characterisation of the existence of a bounded positive solution. Namely we can show the following Theorem: where we set µ p (L ω + a) = − sup ω a when • ω= ∅.

Proof:
First let us assume that µ p (L ω +a) ≤ 0, we will show that there is no positive bounded solution to (3.1). Let us suppose by contradiction that there exists u, a positive bounded solution to (3.1). So in ω, u satisfies L Ω [u] + au = 0, which implies that maxω a < 0 and u is continuous onω. Furthermore, we have If • ω= ∅ then we obtain easily a contradiction. Indeed in such case, we have µ p (L ω + a) = − sup ω a which leads to the contradiction In the other situations, • ω = ∅ and to obtain our desired contradiction we argue as follows. Since µ p (L ω + a) ≤ 0 < − maxω a, by Proposition 2.3 there exists a positive continuous eigenfunction associated with µ p (L ω +a). As a consequence there exists also a positive continuous eigenfunction We can easily check that µ p (L ω + a) = µ p (L * ω + a). Let us denote by φ * the positive continuous principal eigenfunction associated with µ p (L * ω + a). Now by multiplying (3.2) by φ * and then integrating over ω, it follows By using Fubini's Theorem in the above inequality we get the contradiction Thus in both cases, there is no bounded solution to (3.1) when µ p (L ω + a) ≤ 0.
Next we see that there is no positive bounded solution for (3.1) when µ p (L Ω + a) ≥ 0. In this situation, with some modifications we can reproduce the argumentation developed in [12] (Subsection 6.2). Let us assume that a positive solution of (3.1) exists and let us denote u this solution. We first observe that by following the argument developed in [2] we can see that u is continuous in Ω and there exists positive constants δ and c 0 so that From the monotone behaviour with respect to the s of the function g(x, s) : . By construction, we have γ(x) ≤ a(x) and we see by (ii) of Proposition 2.4 that Moreover, since u is a solution of (3.1), we have with a strict inequality for any x ∈ Ω \ ω. We claim that Claim -3.1. There exists δ > 0 and a positive continuous function φ so that inf Ω φ > δ and Assume for the moment that the Claim holds true then we get our desired contradiction by arguing as follow. Since φ > δ we can define the following quantity Obviously, by proving that τ * = 0 we get the contradiction Assume by contradiction that τ * > 0 and let us denote w : Therefore, since K satisfies (1.10) we must have w(y) = 0 for almost every y ∈Ω. Thus, we end up with τ * φ ≡ u and we get the following contradiction
Lastly, let us construct a positive bounded solution to (3.1) when the condition (3.6) µ p (L ω + a) > 0 > µ p (L Ω + a) is satisfied. The uniqueness of this solution follows form a similar argumentation as in [2,12], so we will omit the proof here.
From the condition 0 > µ p (L Ω + a), by reproducing the argument in [12] we can find a positive bounded subsolution φ 0 of the problem (3.1) so that κφ 0 is still a subsolution for any κ small and positive. Here the main difficulty is to find a positive supersolution ψ. Indeed, due to the existence of a refuge zone, the large positive constants are not supersolutions of (3.1). We claim Claim -3.2. When the condition (3.6) is satisfied, then there exists ψ > 0, ψ ∈ C(Ω) supersolution of (3.1) Note that by proving the claim we end the construction of the solution to (3.1). Indeed, since for κ small we have κφ ≤ ψ, by the monotone iterative scheme there exists a solution u to (3.1) so that κφ ≤ u ≤ ψ. Now, let us turn our attention to the proof of the Claim.

Proof of the Claim:
Let us first assume that • ω = ∅. In this situation, by following the argument in [12] (Subsection 6.1) we can introduce a regularisation a ǫ ∈ C(Ω) of a − k so that the following operator has a positive continuous principal eigenfunction. By continuity of µ p (L ǫ,ω ) with respect to a ǫ ((iii) of Proposition 2.4) we can find ǫ small so that Let ǫ be fixed and let us denote ω δ the following set ω δ := {x ∈ Ω | d(x; ω) < δ}. By continuity of the function sup ω δ a ǫ with respect to δ, there exists δ 0 so that for all δ ≤ δ 0 we have So, by (i) of Proposition 2.4, we deduce from the above inequality that we have for all δ ≤ δ 0 , Therefore, thanks to Proposition 2.3 for all δ ≤ δ 0 there exists a positive continuous eigenfunction associated with µ p (L ǫ,ω δ ). By continuity of µ p (L ǫ,ω δ ) with respect to the domain ((iv) of Proposition 2.4) we achieve for δ small enough, say δ ≤ δ 1 , By constructionΩ \ ω δ andω δ 2 are two disjoints bounded closed set, so by the Urysohn Lemma there exists a nonnegative continuous function Let ψ 1 , ψ 2 be the following continuous functions where Ψ δ denotes the positive continuous eigenfunction associated with µ p (L ǫ,ω δ ) normalized by Ψ δ ∞ = 1 and C 1 and C 2 are positive constants to be specified later on. Consider now the function ψ := sup(ψ 1 , ψ 2 ), we will prove that for well chosen C 1 and C 2 , ψ is a supersolution of (3.1).

Now by choosing
Hence from (3.8) and (3.9) we see that the function ψ is a positive continuous supersolution of (3.1).

THE PARTIALLY CONTROLLED PROBLEM: THE KPP CASE
In this section we analyse the dependence in λ of the positive continuous solutions to (1.1) in absence of a refuge zone and we prove the Theorem 1.2 that we recall below. More precisely, we look for positive continuous solution of the partially controlled problem: In absence of a refuge zone, we can show that there exists a critical value λ * characterising completely the existence/non existence of a positive stationary solution. More precisely we have, Theorem 4.1. Assume that K, a i and β satisfy (1.9)-(1.10). Assume further that β > 0 inΩ then there exists λ * ∈ [−∞; ∞), so that for all λ > λ * there exists a unique positive continuous solution u λ to (4.1). When λ * ∈ R, there is no positive solution to (1.1) for all λ ≤ λ * . Moreover, we have the following trichotomy: • λ * = −∞ when µ p (L Ω\Ω 1 + a 0 ) < 0, • λ * ∈ [−∞, ∞) when µ p (L Ω\Ω 1 + a 0 ) = 0.

Proof:
In absence of a refuge zone, we observe that the problem (4.1) is a particular case of the KPP equation (2.1) where the nonlinearity f is given by f (x, s) := a 0 s + λa 1 s − βs p . Therefore by the Theorem 2.1, for each λ ∈ R the existence of a positive solution to (4.1) is conditioned by the sign of µ p (L Ω + a 0 + λa 1 ).
Before proving the trichotomy, let us look at the asymptotic behaviours with respect to λ of the unique solution u λ . First let us observe that the map λ → u λ is monotone non decreasing. Indeed, thanks to the nonnegativity of a 1 , for any λ ≥ λ ′ , the continuous bounded function u λ ′ is a subsolution of the problem Observe that any large constant M is a super-solution of (4.2). Therefore by taking M large enough we have u λ ′ ≤ M and by the monotone iteration scheme we can construct a positive bounded solution of (4.2) which satisfies u λ ′ ≤ u ≤ M . We conclude by using the uniqueness of the solution of problem (4.2). Hence, u λ ′ ≤ u λ ≡ u.
The asymptotic behaviour of u λ when λ → +∞ is obtained by establishing a bound from below for the solution u λ when λ → +∞. More precisely we show that for all x ∈ Ω 1 we have for λ large enough Indeed from (4.1) using that u λ is non negative we have sup Ω β So for x ∈ Ω \ Ω 1 so that |B ǫ0 (x) ∩ Ω 1 | > 0 where ǫ 0 is given by (1.10) we conclude that lim λ→+∞ Bǫ 0 (x)∩Ω1 u λ (y) dy = +∞.

The later implies that
By repeating the above argument with z∈Ω1 B ǫ0 (z) instead Ω 1 , we show that By a finite iteration of the above argumentation, we get lim λ→+∞ u λ (x) = +∞ for all x ∈Ω.
Let us now deal with the limit of u λ when λ → λ * ,+ . First let us assume that λ * ,+ ∈ R. In this situation by using the positivity of u λ and the monotonicity of u λ with respect to λ, we deduce that u λ converges pointwise to u λ * ,+ when λ → λ * ,+ . Moreover thanks to the Lebesgue dominated convergence Theorem by passing to the limit in (4.1), we see that u λ * ,+ is a non negative solution of (4.1) with λ = λ * ,+ . Therefore by Theorem 2.1 we deduce that u λ * ,+ ≡ 0 since µ p (L Ω + a 0 + λa 1 ) = 0. Thus in this case lim λ→λ * ,+ u λ (x) = 0 for all x ∈Ω.

Proof of the Claim:
The proof of this claim relies on the construction of an adequate test function. By arguing as in the proof of Claim 3.2 we can introduce a regularisation a ǫ of a 0 − k so that the following operator L ǫ,Ω\Ω 1 [u] := Ω\Ω1 K(x, y)u(y)dy + a ǫ (x)u has a positive continuous principal eigenfunction. By continuity of µ p (L ǫ,Ω\Ω 1 ) with respect to a ǫ ((iii) of Proposition 2.4) we can find ǫ small so that Let ǫ be fixed and let Ω δ be the set As in the proof of the Claim 3.2, by continuity of sup Ω\Ω δ and µ p (L ǫ,Ω\Ω δ ) with respect to the domain we achieve for δ small enough, say δ ≤ δ 0 , By constructionΩ \ Ω δ 2 andΩ δ are two disjoints bounded closed set, so by the Urysohn Lemma there exists a nonnegative continuous function Let ψ 1 , ψ 2 be the following continuous functions where Ψ δ is the positive continuous eigenfunction associated with µ p (L ǫ,Ω\Ω δ ) normalized by Ψ δ ∞ = 1 and c 0 is positive constant to be specified later on. Consider now the function ψ := sup(ψ 1 , ψ 2 ) and let γ be a positive constant to be fixed later on. We will prove that for γ, δ, λ and c 0 well chosen the function ψ is an adequate test function for L Ω + a 0 + λa 1 + γ. So let us compute L Ω [ψ] + aψ + λa 1 ψ + γψ.
Now on Ω δ 2 we have by construction Since we get Hence from (4.9) and (4.10) we see that for λ negative enough, the function (ψ, γ) is an adequate test function for the operator L Ω + a 0 + λa 1 . That is: ψ is a positive continuous function on Ω which satisfies So by definition of µ p (L Ω + a 0 + λa 1 ) we deduce that for λ negative enough we have µ p (L Ω + a 0 + λa 1 ) ≥ γ > 0.

THE PARTIALLY CONTROLLED PROBLEM: THE REFUGE CASE
In this Section, we analyse (1.1) in the presence of a refuge zone, i.e. when there exists ω ⊂ Ω so that β |ω ≡ 0. In a presence of a refuge zone, the analysis of (1.1) is more involved and the characterisation of the existence/non-existence of a positive solution of (1.1) cannot always be summarised to a single critical value λ * . In this situation, we prove the Theorem 1.3 that we recall below: Theorem 5.1. Assume that K, a i and β satisfy (1.9)-(1.10). Assume further that ω = ∅, then there exists two quantities λ * , λ * * ∈ [−∞, +∞] so that we have the following dichotomy : • Either λ * * ≤ λ * and there exists no positive bounded solution to (1.1).
Let us introduce the following quantities: We can see that the description of the set of positive bounded solutions of (1.1) is then equivalent to show whether or not we have λ * < λ * * . To answer this question, we analyse separately the three different situations : Let us start with the analysis the first situation.
Let us now look at the asymptotic behaviour of u λ with respect to λ. The monotone behaviour of u λ and its limit as λ → λ * (i.e (iii)) can be obtained by following the arguments in Section 4, so we drop the proof here and prove only (i) and (ii) i.e. we analyse the limits of u λ as λ → λ * * .
When λ * * = +∞, the behaviour of u λ can be obtained by reproducing the arguments of Section 4 and we get for all x ∈Ω lim λ→+∞ u λ (x) = +∞.
In the above arguments by replacingx by any x ∈ B(x, ǫ0 4 ) ∩ Ω, we achieve Since Ω is compact we achieve lim λ→λ * * u λ (x) = +∞ for all x ∈ Ω after a finite iteration of this argument.