Stabilization for a Periodic Predator-Prey System

A reaction-di ﬀ usion system modelling a predator-prey system in a periodic environment is considered. We are concerned in stabilization to zero of one of the components of the solution, via an internal control acting on a small subdomain, and in the preservation of the nonnegativity of both components. dis-tribution,


Introduction
This paper concerns the internal zero stabilization of the predator population of a predator-prey system in a periodic environment.Our starting point is the system describing the evolution of a predator population and a prey population distributed over the habitat Ω: where Ω ⊂ R N (N ≥ 2) is a bounded domain with a smooth enough boundary ∂Ω.Here h(x,t) is the density of preys at position x ∈ Ω and time t ≥ 0 and p(x,t) is the density of predators at position x ∈ Ω and time t ≥ 0; h and p are both nonnegative functions.d 1 , d 2 > 0 are the diffusivity constants of the two populations.r(t) is the intrinsic growth rate of preys in the absence of predators, at the moment t ≥ 0 (which can be positive, zero, or negative) and is T-periodic (T > 0).Usually, the period T is of one year.a(t) is the decay rate of predators in the absence of preys, at the moment t, and is also T-periodic.k is a T-periodic and positive function.k(t)h(x,t) represents an additional mortality rate of the preys due to the overpopulation.Homogeneous Neumann boundary conditions mean that there is no flux of species through the boundary ∂Ω (this corresponds to isolated populations).h 0 and p 0 are the initial densities of the two populations.
The following cases are well known in the literature.
Finally, in the case f 1 (t,h, p) = θ 1 /(1 + qh + qp) and f 2 (t,h, p) = θ 2 /(1 + qh + qp), for every h, p ≥ 0, and θ 1 , θ 2 , q, q positive constants, a Beddington-De Angelis functional response for predation is obtained.For a complete study of the solutions to this model we refer to [1].For a description of the predator-prey systems and some basic results we refer to [2,3].
Throughout this paper, the following assumptions will be considered: are continuous functions and locally Lipschitz continuous with respect to (h, p) and satisfy S. Anit ¸a and C. O. Tarniceriu 3 Condition T 0 r(t) dt > 0 is a persistence condition for the preys in the absence of predators.So, if p 0 ≡ 0 and h 0 (x) > 0 a.e. in Ω, then the necessary and sufficient condition for the persistence of preys is the above-mentioned one.
For basic results concerning the solutions of periodic predator-prey systems (without diffusion) we refer to [4].
Let ω ⊂ R N be a nonempty domain with a smooth-enough boundary ∂ω and satisfying ω ⊂ ⊂ Ω.We denote by m the characteristic function of ω.
The questions we want to investigate are the following.
Definition 1.1.Say that the predator population is p-zero stabilizable if for any h 0 , p 0 satisfying (H1), the answer to the first question is affirmative.p-zero stabilizable means that the zero stabilizability holds for controls acting only on the predator population.
Definition 1.2.Say that the predator population is h-zero stabilizable if for any h 0 , p 0 satisfying (H1), the answer to the second question is affirmative.h-zero stabilizable means that the zero stabilizability holds for controls acting only on the prey population.
We are dealing here with some results of zero stabilizability with state constraints.
First notice that, due to assumption (H3) and to the comparison principle for parabolic equations, the solution (h, p) to (1.1) satisfies 0 ≤ h(x,t) ≤ h(x,t) a.e.x ∈ Ω, ∀t ≥ 0, (1.8) where h is the solution to (1.9) Lemma 1.3.The solution h to (1.9) satisfies where h is the unique nontrivial nonnegative solution to the following problem: (1.11) Remark 1.4.In fact, we will show that (1.11) has exactly two nonnegative solutions, the trivial one and the unique nontrivial and nonnegative solution to (1.12) 12) has a unique nonnegative solution (the trivial one).This follows by a simple calculation and taking into account that the first equation in (1.12) is a Bernoulli equation.
Proof of Lemma 1.3.Since h 0 L ∞ (Ω) > 0, it follows that there exists a positive constant (this is a consequence of a result in [5]).Therefore, we can assert that where h ρ 1 (t) is the solution to corresponding to ρ := ρ 1 (h ρ 1 does not depend explicitly on x).
If we choose ρ 1 > 0 sufficiently small and taking into account that (1.16) By mathematical induction, we get that and consequently for any n ∈ N, where h for any n ∈ N * .In the same manner, taking ρ 2 > 0 sufficiently large, we can obtain a nonincreasing bounded sequence h Using the comparison result for parabolic equations, we have that where h 1 does not depend explicitly on x (because h ρ 1 n does not).We may extend h 1 by Tperiodicity to [0,+∞) and we deduce that h 1 is a positive solution to (1.11) and to (1.12).Since (1.12) has a unique nontrivial nonnegative solution, we may infer that this one is (1.23) In the same manner, it follows that Let us prove that there is only one nontrivial and nonnegative solution to (1.11).Let h 2 be a nontrivial and nonnegative solution to (1.11).It follows immediately that there exists ρ 0 > 0 (see [5]) such that h 2 (x,T) ≥ ρ 0 a.e.x ∈ Ω.If we choose ρ 1 and ρ 2 such that 0 < ρ 1 < ρ 0 ≤ h 2 (x,0) = h 2 (x,T) ≤ ρ 2 a.e.x ∈ Ω with ρ 1 small enough and ρ 2 large enough, then it follows as before that x ∈ Ω, for all t ∈ [0,T], for all n ∈ N) and so we get the conclusion of the lemma.

Let us consider now the corresponding equation in p for
(1.26) Having in mind (H5), we obtain that therefore, the solution p to (1.26) satisfies (using the comparison principle for parabolic equations) where p is a solution to (1.29) This may be rewritten as where Thus, the solution p can be written as with f solution to Let us denote by λ 1 the principal eigenvalue of the following eigenvalue problem (1.36) Remark that λ −1 = 0. Now, we notice that if λ 1 > α * , then (1.32) and (1.34) imply that the predator population goes to extinction without any control.Therefore, in the rest of this paper we will assume (H6) 0 < α * .
For basic results concerning the solutions to predator-prey systems we refer to [1,6].Stabilization of predator-prey systems with r, k, a constants has been investigated in [7,8].If in (1.1) the predator is an alien population, then our main goal is to eliminate this population.This problem and its importance have been discussed in [9].We will investigate next what happens in the cases when we act with a control with support in ω.
Section 2 is devoted to the study of p-zero stabilization, while Section 3 concerns the h-zero stabilization.Some remarks are given in Section 4. (2.1)

The p-zero stabilization of the predator population
Then, according to Rayleigh's principle (see [10]), λ ω,p Here is one of the main results of our paper.

.16)
where h is the unique nontrivial nonnegative solution to (1.11).
Proof.Since Let us denote now by h 1 and h 2 the solutions to the following problems, respectively: where ρ 1 > 0 is a small enough constant and ρ 2 is a large enough constant, such that (existence of such ρ 1 is a consequence of a result in [5]).Then, by the comparison principle for the parabolic equations, we obtain As in the proof of Lemma 1.3 we can prove that h 2 satisfies where h δ is the unique nontrivial nonnegative solution to (2.24) Since δ → h δ is a decreasing function, then we may pass to the limit in (2.24) and get that Proof of Theorem 2.1.Assume that p 0 (x) > 0 a.e.x ∈ Ω and let (h, p) be a nonnegative solution to (1.5) corresponding to the p-stabilizing control it follows by Lemma 2.3 that which implies, due to the continuity of the function f 2 , that, for any ε > 0, there exists for any t ≥ t ε .
Applying the comparison result for parabolic equations, we conclude that with respect to the location and geometry of ω and Ω.

The h-zero stabilization of the predator population
In this section, we are looking for a stabilizing control v acting indirectly (acting on the prey population).Let us consider (h, p) a solution to (1.7) corresponding to the feedback control v := −γh.The system becomes For any γ ≥ 0 we consider the following eigenvalue problem: and denote by λ h 1,γ its principal eigenvalue.Next, we denote by λ ω,h 1 the principal eigenvalue to It is a consequence of Rayleigh's principle that the mapping γ → λ h 1,γ is increasing and continuous, and In the same manner as in Section 2 it follows the next result.
Remark 3.4.In general, the habitat of preys is larger than Ω.The strategy to eradicate the predators via indirect control is the following one: we isolate the domain Ω (we do not permit migration through the boundary of it), then we eradicate firstly the preys in Ω and consequently the predators will extinct.Next, the preys are allowed to repopulate the domain Ω.

Final comments
The results in Sections 2 (and 3) show how important is to find the position and the geometry of ω and Ω in order to get a great value for λ ω,p 1 (and λ ω,h 1 ).This yields λ ω,p where λ 1 (ω,Ω) is the principal eigenvalue to  The following result has been proved in [8] using rearrangement techniques and can be used to obtain upper and lower bounds for λ 1 (ω,Ω).where B is a ball with the same measure as Ω and with the same center as ω.Moreover, we have equality only for Ω ≡ B and we conclude that the maximal value for λ 1 (ω,Ω), subject to all domains Ω ⊂ R N with smooth boundary and satisfying ω ⊂ ⊂ Ω and having a prescribed measure, is attained for the ball B of the same measure and with the same center as ω.