Stabilization strategies for some reaction–diffusion systems

Nonlinear Analysis: Real World Applications 10 (2009) 345–357 www.elsevier.com/locate/nonrwa

Stabilization strategies for some reaction–diffusion systems Sebastian Anit¸a a,b,∗ , Michel Langlais c a Faculty of Mathematics, University “Al.I. Cuza”, Ias¸i 700506, Romania b Institute of Mathematics “Octav Mayer”, Ias¸i 700506, Romania c UMR CNRS 5466 MAB and INRIA Futurs Anubis, case 26, Universit´e Victor Segalen Bordeaux 2, 33076 Bordeaux Cedex, France

Received 28 July 2007; accepted 13 September 2007

Abstract A two-component reaction–diffusion system is considered. The question of stabilizing to zero one of the components of the solution via an internal control acting on a small subdomain and preserving nonnegativity of both components is investigated. Our results apply to predator–prey systems. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Reaction–diffusion system; Internal stabilization; Comparison principle; Predator–prey system

1. Introduction and setting of the problem We consider a nonlinear two-component reaction–diffusion system describing interacting prey and predator populations distributed over a habitat Ω ⊂ Rn , n ≥ 1 (Ω is a bounded domain with a smooth boundary ∂Ω ). We denote by h(x, t) ≥ 0 the density of preys at position x ∈ Ω and time t ≥ 0 and by p(x, t) ≥ 0 the density of predators at position x ∈ Ω and time t ≥ 0. The dynamics of these two spatially distributed interacting populations are governed by a set of partial differential equations together with initial and boundary conditions, cf. [2] or [3]:  ∂ h − d1 ∆h = r h − kh 2 − f 1 (h, p)hp, x ∈ Ω, t > 0    t ∂t p − d2 ∆ p = −ap + f 2 (h, p)hp, x ∈ Ω, t > 0 (1) x ∈ ∂Ω , t > 0 ∂η h = ∂η p = 0,   h(x, 0) = h 0 (x) ≥ 0, p(x, 0) = p0 (x) ≥ 0, x ∈ Ω , where d1 , d2 , r, k, a are positive constants. We have denoted by ∂η the normal derivative. Homogeneous Neumann boundary conditions mean there is no flux of species through the boundary ∂Ω ; this corresponds to isolated populations. d1 and d2 are the diffusivity constants of the two populations; r is the intrinsic growth rate of preys. kh 2 is the logistic term for the prey dynamics; kh(t) is an additional mortality rate for the prey population (it is proportional to ∗ Corresponding author at: Faculty of Mathematics, University “Al.I. Cuza”, Ias¸i 700506, Romania.

E-mail addresses: [email protected] (S. Anit¸a), [email protected] (M. Langlais). c 2007 Elsevier Ltd. All rights reserved. 1468-1218/$ - see front matter doi:10.1016/j.nonrwa.2007.09.003

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its size) and is due to the overpopulation. rk is the carrying capacity of preys in the absence of predation and a is the decay rate of predators in the absence of preys. See [14] and [15] for details. When f 1 (h, p) = ρ1 and f 2 (h, p) = ρ2 , ∀h, p ≥ 0, where ρ1 , ρ2 are positive constants, we get a standard Lotka-Volterra system. ρ1 ρ2 The case f 1 (h, p) = 1+qh and f 2 (h, p) = 1+qh , ∀h, p ≥ 0, where ρ1 , ρ2 , q are positive constants, corresponds to a Holling II functional response to predation. ρ1 ρ2 Last f 1 (h, p) = 1+qh+ q˜ p and f 2 (h, p) = 1+qh+q˜ p , ∀h, p ≥ 0, ρ1 , ρ2 , q, q˜ being positive constants, yield a Beddington–De Angelis functional response to predation. For a qualitative analysis of (1) in this particular case we refer to [8]. We shall assume throughout this work that: (H1) h 0 , p0 ∈ L ∞ (Ω ), h 0 (x) ≥ 0, p0 (x) ≥ 0 a.e. x ∈ Ω ; (H2) f 1 , f 2 : R2 → R are locally Lipschitz continuous and satisfy ∃C > 0 such that : 0 ≤ f 1 (h, p), f 2 (h, p) ≤ C, ∀h ≥ 0, ∀ p ≥ 0, f 1 (h, p) = f 2 (h, p) = 0, ∀(h, p) ∈ R2 \ [0, +∞) × [0, +∞); (H3) the application h 7→ h f 2 (h, p) is nondecreasing on [0, +∞) for any p ≥ 0; (H4) the application p 7→ f 2 (h, p) is nonincreasing on [0, +∞) for any h ≥ 0. Let ω ⊂ Rn be a nonempty domain with a suitably smooth boundary ∂ω, satisfying ω ⊂⊂ Ω and denote by m the characteristic function of ω. The questions we are investigating are the following: 1o “Is there any control u ∈ L ∞ loc (ω ×[0, +∞)) such that the solution (h, p) to the following initial-boundary value problem:  ∂ h − d1 ∆h = r h − kh 2 − f 1 (h, p)hp, x ∈ Ω, t > 0    t ∂t p − d2 ∆ p = −ap + f 2 (h, p)hp + m(x)u(x, t), x ∈ Ω , t > 0 (2) ∂η h = ∂η p = 0, x ∈ ∂Ω , t > 0    h(x, 0) = h 0 (x), p(x, 0) = p0 (x), x ∈Ω satisfies both h(x, t) ≥ 0,

p(x, t) ≥ 0

a.e. x ∈ Ω , ∀t ≥ 0

(3)

and lim p(t) = 0

t→+∞

in L ∞ (Ω ) ?00

(4)

Definition 1. We say that the predator population is p-zero stabilizable if for any h 0 , p0 ∈ L ∞ (Ω ), h 0 (x), p0 (x) ≥ 0 a.e. x ∈ Ω , the answer to the above mentioned question is affirmative. “ p-zero stabilizable” means that the zero stabilizability holds for controls acting only on the predator population. 2o “Is there any control v ∈ L ∞ loc (ω × [0, +∞)) such that the solution (h, p) to the following initial-boundary value problem:  2  ∂t h − d1 ∆h = r h − kh − f 1 (h, p)hp + m(x)v(x, t), x ∈ Ω , t > 0  ∂t p − d2 ∆ p = −ap + f 2 (h, p)hp, x ∈ Ω, t > 0 (5) ∂ h = ∂ p = 0, x ∈ ∂Ω , t > 0  η η   h(x, 0) = h 0 (x), p(x, 0) = p0 (x), x ∈Ω satisfies both (3) and (4) ?” Definition 2. We say that the predator population is h-zero stabilizable if for any h 0 , p0 ∈ L ∞ (Ω ), h 0 (x), p0 (x) ≥ 0 a.e. x ∈ Ω , the answer to the above mentioned question is affirmative. “h-zero stabilizable” means that the zero stabilizability holds for controls acting only on the prey population. Typically prey are native species while predators are invading species. It is often desirable to get rid of unwelcome species, as the predators in our context. So, we are interested in driving predators to extinction.

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We can act directly, harvesting (removing) predators in the accessible subdomain ω of the habitat. Sometimes, controlling and reducing the size of more accessible native population (the prey) below some threshold may turn out to be easier to handle. If the prey is brought below this threshold, then the prey cannot sustain anymore the predator population, which get extinct. Let us notice from the beginning that the componentwise nonnegative solution (h, p) to (1) satisfies 0 ≤ h(x, t) ≤ h(x, t),

∀t ≥ 0, a.e. x ∈ Ω ,

where h is the solution to  2  ∂t h − d1 ∆h = r h − kh , ∂η h = 0,  h(x, 0) = h (x), 0

x ∈ Ω, t > 0 x ∈ ∂Ω , t > 0 x ∈Ω

(we have used a comparison principle for solutions to parabolic equations, cf. [12]). Since, limt→+∞ h(t) = rk in L ∞ (Ω ) if kh 0 k L ∞ (Ω ) > 0 and limt→+∞ h(t) = 0 in L ∞ (Ω ) if h 0 (x) = 0 a.e. in Ω (see [3]), one gets that ∀ε > 0, ∃ t (ε) > 0 such that r r  , 0 , ∀t ≥ t (ε), a.e. x ∈ Ω . f 2 (h(x, t), p(x, t))h(x, t) ≤ ε + f 2 k k p

Let λ1 be the principal eigenvalue to the problem ( r r −d2 ∆ψ + aψ − f 2 ( , 0)ψ = λψ in Ω k k ∂η ψ = 0 in ∂Ω . p

Assuming λ1 > 0 one may check that lim p(t) = 0

t→+∞

in L ∞ (Ω ). p

As a conclusion, when λ1 is positive the predator population dies out without any control and the question now is p “what happens when λ1 ≤ 0 ?” In the rest of this work we shall assume p

(H5) λ1 ≤ 0. Denote by λω1 , the principal eigenvalue to the problem  r r  −d2 ∆ψ + aψ − f 2 ( , 0)ψ = λψ in Ω \ ω k k ψ =0 in ∂ω   in ∂Ω . ∂η ψ = 0

(6)

The following result concerning the p-zero stabilizability will be proved in Section 2: Theorem 1. If the predator population is p-zero stabilizable, then λω1 ≥ 0. Conversely, if λω1 > 0, then the predator population is p-zero stabilizable and for γ ≥ 0 large enough, the feedback control u := −γ p realizes (3) and (4), where (h, p) is the solution to (2) corresponding to u := −γ p. Remark 1. It is an immediate consequence of Rayleigh’s principle that λω1 > λ1 . By Theorem 1 we deduce that if p λω1 > 0 (even if λ1 ≤ 0), then the predator population is p-zero stabilizable. p

The question of h-zero stabilization is more delicate. For any γ ≥ 0, we denote by µω1γ the principal eigenvalue to the problem  −d1 ∆ϕ − r ϕ + m(x)γ ϕ = µϕ in Ω ∂η ϕ = 0 in ∂Ω .

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Again a comparison principle for parabolic equations yields µω1γ > 0 ⇒ lim kh(t)k L ∞ (Ω ) = 0 ⇒ lim k p(t)k L ∞ (Ω ) = 0, t→+∞

t→+∞

where (h, p) is the nonnegative solution to (5) corresponding to the control v := −γ h. The mapping γ 7→ µω1γ is increasing and, see below, lim µω1γ = µω1 ,

γ →+∞

where µω1 is the principal eigenvalue to the problem  −d1 ∆ϕ − r ϕ = µϕ in Ω \ ω ϕ=0 in ∂ω  ∂η ϕ = 0 in ∂Ω . These are consequences of Rayleigh’s principle. Hence µω1 > 0 implies µω1γ > 0 for large γ and extinction of both populations. Now, condition µω1γ > 0 implying the extinction of the prey population to remove the predator population is much too severe. We now look for h-stabilization conditions preserving the prey population to the expense of a suitable depletion in spatial density. Noting that lim µω1γ = µω10 = −r < 0,

γ →0

there exists a γ2 ≥ 0 (γ2 could be eventually +∞) such that µω1γ > 0 µω1γ < 0

for γ > γ2 (if γ2 < +∞), for 0 ≤ γ < γ2 .

We shall now only consider µω1γ < 0 with 0 ≤ γ < γ2 . In that setting let K γ be the unique nontrivial and nonnegative solution to the semilinear elliptic boundary value problem  −d1 ∆K γ = r K γ − k K γ2 − m(x)γ K γ in Ω ∂η K γ = 0 in ∂Ω . Note that since µω1γ < 0, it follows that this problem has two nonnegative solutions, the trivial one and K γ , elsewhere the trivial solution is the unique nonnegative solution. γ ,ω Let ν1 be the principal eigenvalue of the problem  −d2 ∆ψ + aψ − K γ f 2 (K γ , 0)ψ = νψ in Ω ∂η ψ = 0 in ∂Ω . The following result concerning the h-zero stabilizability will be proved in Section 3: γ ,ω

Theorem 2. If ν1 > 0, then the predator population is h-zero stabilizable and the feedback control v := −γ h realizes (3) and (4), where (h, p) is the solution to (5) corresponding to v := −γ h. It has been proved in [3] in the case of a Holling II functional response to predation, that there exists a deep ρ2 r relationship between the p-zero stabilizability and the sign of d2 λ1 (ω) + a − k+qr , where λ1 (ω) is the principal eigenvalue of the problem  −∆ψ = λψ in Ω \ ω ψ =0 in ∂ω  in ∂Ω . ∂η ψ = 0 Some internal stabilizability results for parabolic equations and age-dependent population dynamics with state constraints have been obtained in [1,5]. A stabilization result for a reaction–diffusion system modelling a class of spatially structured epidemic model (introduced by V. Capasso—see [6,7]) has been obtained in a different manner

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in [4]. Basic definitions and results concerning the prey-predator systems can be found in [2,3]. For basic results concerning the elliptic equations and the parabolic equations in L k frame we refer to [10] and [12]. For stabilization of parabolic systems we refer to the classic monograph by Lions [13]. The main goal of this paper is to discuss two different strategies to stabilize to zero the predator population. This is an actual problem when it comes to protect native species from invading predator species, e.g. cf. [9]. The first stabilizing strategy consists in harvesting the predators in the subdomain ω, with a constant harvesting effort γ (with γ large enough). So, the stabilizing control will be u := −γ p. The second strategy consists in reducing the size of of the prey via a harvesting procedure in ω, with a constant harvesting effort γ (with γ large enough). If we manage to reduce the size of the prey under a certain level that cannot sustain the predator population, then the predator population will become extinct. The stabilizing control has the form v := −γ h. Our paper is organized as follows: Section 2 is devoted to the p-zero stabilizability of the predators and to the proof of Theorem 1, while Section 3 concerns the h-zero stabilizability of the predator population and the proof of Theorem 2. Section 4 concerns some evaluations of λω1 . Some final comments are given in the fifth Section. 2. The p-zero stabilization of the predator population Given γ ≥ 0, let λω1γ be the principal eigenvalue for the problem: ( r r −d2 ∆ψ + aψ − f 2 ( , 0)ψ + m(x)γ ψ = λψ, in Ω k k ∂η ψ = 0 in ∂Ω . In order to prove Theorem 1 we need to establish some auxiliary results. Lemma 1. lim λω γ →+∞ 1γ

= λω1 .

Proof of Lemma 1. By Rayleigh’s principle one has  Z Z r r  ω λ1γ = min d2 ,0 ; |∇ψ|2 dx + γ |ψ|2 dx + a − f 2 k k Ω ω



ψ ∈ H (Ω ), kψk L 2 (Ω ) = 1 . 1

Hence one may conclude that 0 ≤ γ1 ≤ γ2 ⇒ λω1γ1 ≤ λω1γ2 .

(7)

Let ψ1 be a solution to (6) corresponding to λ := λω1 and satisfying kψ1 k L 2 (Ω \ω) = 1. By Rayleigh’s principle ψ1 is a minimum point for  Z  r r  2 1 ω λ1 = min d2 |∇ψ| dx + a − f 2 , 0 ; ψ ∈ H (Ω \ ω), ψ = 0 on ∂ω, kψk L 2 (Ω \ω) = 1 . k k Ω \ω Consider the following trivial extension of ψ1 to Ω :  ψ1 (x), x ∈ Ω \ ω ˜ ψ(x) = 0, x ∈ ω. ˜ L 2 (Ω ) = 1. On the other hand It is obvious that ψ˜ ∈ H 1 (Ω ) and kψk Z Z r r  ˜ 2 dx + γ ˜ 2 dx + a − f 2 λω1 = d2 |∇ ψ| |ψ| , 0 ≥ λω1γ , k k Ω ω

(8)

for any γ ≥ 0. By (7) and (8) one gets lim λω γ →+∞ 1γ

≤ λω1 .

(9)

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Let us prove that equality actually holds in (9). Toward this end let ψγ ∈ H 1 (Ω ) be such that kψγ k L 2 (Ω ) = 1 and Z Z r r  2 ω |ψγ |2 dx + a − f 2 |∇ψγ | dx + γ λ1γ = d2 , 0 ≤ λω1 , ∀γ ≥ 0. k k ω Ω It follows that there exists a constant M ≥ 0 such that Z Z 2 |ψγ |2 dx ≤ M, |∇ψγ | dx ≤ M and γ ω

Ω

∀γ ≥ 0.

We may conclude to the existence of a subsequence, also denoted by {ψγ }, such that: ψγ → ψ ∗ weakly in H 1 (Ω ), ψγ → ψ in L (Ω ) ∗

2

and

as γ → +∞

ψγ → 0 in L 2 (ω),

as γ → +∞.

Hence ψ ∗ ∈ H 1 (Ω ), kψ ∗ k L 2 (Ω \ω) = 1, ψ ∗ = 0 in ω, and one may also infer that ψ ∗ = 0 in ∂ω. Now, Z r r  ω lim λ1γ ≥ d2 |∇ψ ∗ |2 dx + a − f 2 ,0 γ →+∞ k k Ω Z r r  , 0 ≥ λω1 . = d2 |∇ψ ∗ |2 dx + a − f 2 k k Ω \ω From (9) and (10) one gets the conclusion.

(10)



The proofs of the following two lemmas follow closely those of Lemmas 3.1 and 3.2 in [3]. Lemma 2. Let (h, p) be a nonnegative solution of (2) corresponding to the control u ∈ L ∞ loc (ω × [0, +∞)). Then, for each ε > 0, there exists T (ε) > 0, independent of ω, such that r 0 ≤ h(x, t) ≤ + ε, ∀t ≥ T (ε), a.e. x ∈ Ω . k Lemma 3. Assume that the assumptions in Lemma 2 hold. If in addition kh 0 k L ∞ (Ω ) > 0 and lim p(t) = 0 in L ∞ (Ω ),

t→+∞

then lim h(t) =

t→+∞

r k

in L ∞ (Ω ).

Proof of Theorem 1. Assume that kh 0 k L ∞ (Ω ) > 0 and p0 (x) > 0 a.e. in Ω and let (h, p) be the nonnegative solution to (2) corresponding to the p-stabilizing control u ∈ L ∞ loc (ω × [0, +∞)). Since limt→+∞ p(t) = 0 in L ∞ (Ω ), it follows from Lemma 3 that r in L ∞ (Ω ). lim h(t) = t→+∞ k This implies that for any ε > 0, there a exists Tε ≥ 0 such that r r  kh(t) f 2 (h(t), p(t)) − f 2 , 0 k L ∞ (Ω ) < ε, ∀t ≥ Tε . k k Given an arbitrarily fixed ε > 0 let p˜ be a solution to the following problem:  r r   ˜ x ∈ Ω \ ω, t > Tε f2 , 0 p˜ − ε p, ∂ p ˜ − d ∆ p ˜ = −a p ˜ +  t 2   k k p˜ = 0, x ∈ ∂ω, t > Tε  ∂ p ˜ = 0, x ∈ ∂Ω , t > Tε    η p(x, ˜ Tε ) = p(x, Tε ), x ∈ Ω \ ω. A comparison result for solutions to parabolic equations, cf. [12], implies that 0 ≤ p(x, ˜ t) ≤ p(x, t)

∀t ≥ Tε , a.e. x ∈ Ω \ ω.

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Since p(t) → 0 in L ∞ (Ω ), as t → +∞, one may conclude that lim p(t) ˜ =0

t→+∞

in L ∞ (Ω \ ω).

Now p(·, Tε ) is not identically 0 on Ω \ ω (by the backward uniqueness theorem) and one may infer that λω1 + ε > 0, for any ε > 0. This implies λω1 ≥ 0.  Conversely, when λω1 > 0, then for ε > 0 small enough and for γ > 0 large enough one has (by Lemma 1) λω1γ − ε > 0.

(11)

Set u := −γ p. System (2) becomes  2  ∂t h − d1 ∆h = r h − kh − f 1 (h, p)hp,  ∂t p − d2 ∆ p = −ap + f 2 (h, p)hp − m(x)γ p, ∂ h = ∂η p = 0,    η h(x, 0) = h 0 (x), p(x, 0) = p0 (x),

x x x x

∈ Ω, t > 0 ∈ Ω, t > 0 ∈ ∂Ω , t > 0 ∈ Ω.

(12)

Let (h, p) be the componentwise nonnegative solution to (12). One may deduce from Lemma 2 that for any ε > 0 there exists a T˜ε ≥ 0 such that r r  , 0 + ε, ∀t ≥ T˜ε , a.e. x ∈ Ω . h(x, t) f 2 (h(x, t), p(x, t)) < f 2 k k Let p be a solution to  r r   f2 , 0 p + ε p − m(x)γ p, x ∈ Ω , t > T˜ε p − d ∆ p = −a p + ∂ t 2  k k ∂ p = 0, x ∈ ∂Ω , t > T˜ε   η ˜ p(x, Tε ) = p(x, T˜ε ), x ∈ Ω. The comparison result for parabolic equations allows to conclude that 0 ≤ p(x, t) ≤ p(x, t)

∀t ≥ T˜ε , a.e. x ∈ Ω .

Since λω1γ − ε > 0 (by (11)), one gets lim p(t) = 0

t→+∞

in L 2 (Ω )

and in L ∞ (Ω ). This implies that p(t) → 0

in L ∞ (Ω ),

as t → +∞, at the same rate as exp(−(λω1γ − ε)t). Remark 2. By Theorem 1 we obtain as a particular case (corresponding to the Holling II functional response functional to predation) the main result in [3]. 3. The h-zero stabilization of the predator population First of all we assume that µω1 > 0. Let us consider the becomes  ∂ h − d1 ∆h = r h − kh 2 − f 1 (h, p)hp − m(x)γ h,    t ∂t p − d2 ∆ p = −ap + f 2 (h, p)hp, ∂ h = ∂η p = 0,    η h(x, 0) = h 0 (x), p(x, 0) = p0 (x),

feedback control v := −γ h, where γ ≥ 0. System (5) x x x x

∈ Ω, t > 0 ∈ Ω, t > 0 ∈ ∂Ω , t > 0 ∈ Ω.

(13)

Let (h, p) be the (nonnegative) solution of (13). By the comparison principle for parabolic equations we get 0 ≤ h(x, t) ≤ h 1 (x, t),

∀t ≥ 0, a.e. x ∈ Ω ,

(14)

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where h 1 is the solution to  ∂t h 1 − d1 ∆h 1 = r h 1 − m(x)γ h 1 , ∂ h = 0,  η 1 h 1 (x, 0) = h 0 (x),

x ∈ Ω, t > 0 x ∈ ∂Ω , t > 0 x ∈ Ω.

Since µω1 > 0, then by a similar argument as that in Lemma 1 we get that µω1γ > 0 (for γ ≥ 0 large enough). As a consequence we obtain that lim h 1 (t) = 0

in L 2 (Ω ) and in L ∞ (Ω ).

t→+∞

(15)

By (14) and (15) we conclude that lim h(t) = 0

t→+∞

in L ∞ (Ω ).

We may infer that for any ε > 0, there exists tε ≥ 0 such that: 0 ≤ h(x, t) f 2 (h(x, t), p(x, t)) < ε,

∀t ≥ tε , a.e. x ∈ Ω

and this implies via the comparison result for parabolic equations that 0 ≤ p(x, t) ≤ gε (x, t),

∀t ≥ tε , a.e. x ∈ Ω ,

where gε is the solution to  ∂t g − d2 ∆g = −ag + εg, ∂η g = 0,  g(x, tε ) = p(x, tε ),

x ∈ Ω , t > tε x ∈ ∂Ω , t > tε x ∈ Ω.

Function gε (t) is tending to 0 (in L 2 (Ω ) and in L ∞ (Ω )), as t → +∞, at the same rate as exp(−(a − ε)t) and so does p(t), for any ε > 0 small enough. Remark 3. We notice that in the case µω1 > 0, if we use the feedback control v := −γ h (for a certain γ ≥ 0 satisfying µω1γ > 0; this means for a γ ≥ 0 large enough), then we get extinction for both populations and p(t) → 0

in L ∞ (Ω ),

as t → +∞, at the same rate as exp((−a + ε)t). Remark 4. If a > λω1 , then this second strategy (when the control acts only on prey) is better than the first one (when the control acts only on predators), i.e. the convergence to zero is faster for this second strategy. Remark 5. Since µω1 = d1 λ1 (ω) − r , condition µω1 > 0 is equivalent to λ1 (ω) >

r d1 .

Proof of Theorem 2. If for a certain γ ≥ 0 we have µω1γ < 0

and

γ ,ω

ν1

> 0,

then we consider the feedback control v := −γ h in (5). Let (h, p) be the (nonnegative) solution of (13). If h 0 ≡ 0, then the conclusion of the theorem is obvious.  Assume now that kh 0 k L ∞ (Ω ) > 0. Using again the comparison principle for parabolic equations we get 0 ≤ h(x, t) ≤ h 2 (x, t),

∀t ≥ 0, a.e. x ∈ Ω ,

where h 2 is the solution to  2  ∂t h 2 − d1 ∆h 2 = r h 2 − kh 2 − m(x)γ h 2 , ∂η h 2 = 0,  h (x, 0) = h (x), 2 0

x ∈ Ω, t > 0 x ∈ ∂Ω , t > 0 x ∈ Ω.

(16)

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It follows in a standard manner that lim h 2 (t) = K γ

t→+∞

in L ∞ (Ω ).

We conclude that ∀ε > 0, ∃t1 (ε) ≥ 0 such that h 2 (x, t) ≤ K γ (x) + ε,

∀t ≥ t1 (ε), a.e. x ∈ Ω

and consequently by (16) we obtain that h(x, t) ≤ K γ (x) + ε,

∀t ≥ t1 (ε), a.e. x ∈ Ω .

We may infer that for any ε > 0, there exists t1ε ≥ 0 such that: 0 ≤ h(x, t) f 2 (h(x, t), p(x, t)) < K γ (x) f 2 (K γ (x), 0) + ε,

∀t ≥ t1ε , a.e. x ∈ Ω

and this implies via the comparison result for parabolic equations that 0 ≤ p(x, t) ≤ g1ε (x, t),

∀t ≥ t1ε , a.e. x ∈ Ω ,

where g1ε is the solution to  ∂t g − d2 ∆g = −ag + K γ f 2 (K γ , 0)g + εg, ∂η g = 0,  g(x, t1ε ) = p(x, t1ε ), γ ,ω

x ∈ Ω , t > t1ε x ∈ ∂Ω , t > t1ε x ∈ Ω. γ ,ω

Since ν1 > 0, then for ε > 0 small enough we have ν1 − ε > 0. So, function g1ε (t) is tending to 0 (in L 2 (Ω ) γ ,ω and in L ∞ (Ω )), as t → +∞, at the same rate as exp(−(ν1 − ε)t) and so does p(t). Remark 6. Conversely, if the predator population is h-zero stabilizable, then it is possible to prove using the comparison principle that ν1ω ≥ 0, where ν1ω is the principal eigenvalue to the problem  −d2 ∆ψ + aψ − K f 2 (K , 0)ψ = νψ in Ω \ ω ψ =0 in ∂ω  ∂η ψ = 0 in ∂Ω and K is the maximal solution to the problem  in Ω \ ω −d1 ∆K = r K − k K 2 K =0 in ∂ω  ∂η K = 0 in ∂Ω . 4. Evaluations of λω1 The results and remarks in Section 2 show how important is to find the position and geometry of ω and Ω in order to get a big value for λω1 . The goal of this section is to present three approaches to evaluate λω1 . The first approach. Let us discuss the problem of maximizing of λω1 subject to all translations of ω. Let ω∗ be a nonempty open subset of Rn (n ≥ 2) with a smooth boundary and consider O = {ω ⊂ Rn ; ω ⊂⊂ Ω and ∃V ∈ Rn such that ω = V + ω∗ } (O is the set of all translations ω ⊂⊂ Ω of ω∗ ). This yields λω1 = d2 λ1 (ω) + a − equivalent to maximizing λ1 (ω). For any ω ∈ O and V ∈ Rn we define the derivative dλ1 (ω)(V ): λ1 (εV + ω) − λ1 (ω) . ε&0 ε

dλ1 (ω)(V ) = lim

r r k f 2 ( k , 0).

So, maximizing λω1 is

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Since we wish to find the maximal value of λ1 (ω), subject to ω ∈ O, it is obvious that the evaluation of the derivative dλ1 (ω)(V ) would be of great importance. Our first goal is to evaluate the derivative dλ1 (ω)(V 0 ). For basic definitions and results concerning the optimal shape design we refer to the classic monograph [11]. Theorem 3. For any ω ∈ O such that Ω \ ω is a domain and for any V 0 = (V10 , V20 , . . . , Vn0 ) ∈ Rn we have Z |ψ1 (x)|2 (V 0 · η(x))dσ, dλ1 (ω)(V 0 ) = − ∂ω

where ψ1 is the eigenfunction corresponding to λ1 (ω) and satisfying kψ1 k L 2 (Ω \ω) = 1, ψ1 (x) > 0 a.e. x ∈ Ω \ ω and η(x) is the inward normal versor to ∂ω at x. For the proof we refer to [11]. The last Theorem allows us to obtain a gradient type algorithm to find the solution to the following problem: Maximize {λ1 (ω); ω ∈ O}. The second approach. Since λ1 (ω) depends on Ω , we may denote it by λ1 (ω, Ω ). So, λ1 (ω, Ω ) is the principal eigenvalue for the following problem:  −∆ψ(x) = λψ(x), x ∈ Ω \ ω ψ(x) = 0, x ∈ ∂ω (17)  ∂η ψ(x) = 0, x ∈ ∂Ω . Theorem 4. Assume that ψ ∗ is an eigenfunction of (17), corresponding to λ1 (ω, Ω ), that satisfies in addition:  0 < ψ ∗ (x) < M, ∀x ∈ Ω \ ω (18) ψ ∗ (x) = M, ∀x ∈ ∂Ω , where M > 0 is a constant. Then we have that λ1 (ω, Ω ) > λ1 (ω, Ω˜ ), for any domain Ω˜ ⊂ Rn with smooth boundary and such that ω ⊂⊂ Ω˜ , meas(Ω˜ ) = meas(Ω ) and Ω˜ 6≡ Ω . Remark 7. If ω and Ω are balls with the same centre, there exist such ψ ∗ . Proof. It is obvious that λ1 (ω, Ω ) > 0. Let Ω˜ ⊂ Rn be a domain with smooth boundary such that ω ⊂⊂ Ω˜ , Ω˜ 6≡ Ω and meas(Ω˜ ) = meas(Ω ). Due to Rayleigh’s principle we have Z  λ1 (ω, Ω ) = Min |∇ψ|2 dx; ψ ∈ H 1 (Ω \ ω), kψk L 2 (Ω \ω) = 1, ψ = 0 on ∂ω Ω \ω

and this minimum is attained for ψ ∗ . Consider the function ψ˜ given by  ∗ ψ (x), x ∈ Ω ∩ Ω˜ \ ω ˜ ψ(x) = M, x ∈ Ω˜ \ Ω . It is obvious that ψ˜ ∈ H 1 (Ω˜ \ ω) and ψ˜ = 0 on ∂ω, ∗ ∗ ˜ ˜ 2 ˜ kψk ˜ \ω) ≤ k∇ψ k L 2 (Ω \ω) . L (Ω \ω) > kψ k L 2 (Ω \ω) and k∇ ψk L 2 (Ω This yields R R ∗ 2 ˜ 2 Ω \ω |∇ψ | dx Ω˜ \ω |∇ ψ| d x < R = λ1 (ω, Ω ). R ∗ 2 ˜ 2 Ω \ω |ψ | dx Ω \ω |ψ| dx

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By Rayleigh’s principle we get that λ1 (ω, Ω˜ ) < λ1 (ω, Ω ).



Remark 8. If there exists ψ ∗ an eigenfunction of (17) corresponding to λ1 (ω, Ω ) and satisfying (18), then λ1 (ω, Ω ) = Max{λ1 (ω, Ω˜ ); Ω˜ ⊂ Rn is a domain with smooth boundary and satisfying ω ⊂⊂ Ω˜ , meas(Ω˜ ) = meas(Ω )} = Max {λ1 (ω, ˜ Ω ); ω˜ ⊂⊂ Ω is an isometric transform of ω}. Remark 9. If ω is a ball, ω ⊂⊂ Ω , then we may conclude by Theorem 4 that λ1 (ω, Ω ) ≤ λ1 (ω, B), where B is a ball such that meas(B) = meas(Ω ) and B and ω have the same center. Moreover, we have equality only for Ω = B and we conclude that the maximal value for λ1 (ω, Ω ), subject to all domains Ω ⊂ Rn with smooth boundary and satisfying ω ⊂⊂ Ω and meas(Ω ) = L, is attained for the ball B of measure L and with the same centre as ω. The third approach. It is possible to prove that the following Poincar´e inequality holds: Z Z 2 |ψ(x)| dx ≤ ρ |∇ψ(x)|2 dx, Ω \ω

(19)

Ω \ω

∀ψ ∈ H 1 (Ω \ ω) with ψ = 0 on ∂ω (here ρ > 0 is a constant independent of ψ). Relation (19) implies that R 2 1 Ω \ω |∇ψ| dx ≤ R , 2 ρ Ω \ω |ψ| dx ∀ψ ∈ H 1 (Ω \ ω) such that ψ 6≡ 0, ψ = 0 on ∂ω, and this implies that 1 ≤ λ1 (ω, Ω ). ρ It is important to notice that ρ depends on the “distance” between ω and ∂Ω . If this “distance” is small, then ρ is small and ρ1 and λ1 (ω, Ω ) are large. So, if we wish to find a position of ω for which λ1 (ω, Ω ) to be great, then it is important to find a position of ω ⊂⊂ Ω for which the “distance” between ω and ∂Ω to be small. Remark 10. In practice it is important to be precise how “close” to ∂Ω should we take ω in order to obtain a desired value δ > 0 for the principal eigenvalue for (17). For this reason we may consider the following Cauchy problem:  −∆ψ(x) = δψ(x) ψ(x) = 1, x ∈ ∂Ω  ∂η ψ(x) = 0, x ∈ ∂Ω (which has of course a local solution, which in addition satisfies ψ(x) < 1, ∀x 6∈ ∂Ω ). If there exists ω ⊂⊂ Ω such that ψ(x) = 0 on ∂ω, then it follows by the maximum principle for elliptic operators that ψ(x) > 0, ∀x ∈ Ω \ ω (if Ω is a ball, then for any δ > 0 such a set ω exists and is a ball with the same centre as Ω ).

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In conclusion, if such an ω exists, then λ1 (ω, Ω ) = δ and for any ω˜ domain in Rn with a smooth boundary and satisfying ω ⊂ ω˜ ⊂⊂ Ω , we have λ1 (ω, ˜ Ω ) ≥ δ. Remark 11. If there exists a ball β such that ω ⊂ β ⊂⊂ Ω , then by Rayleigh’s principle we get that λ1 (ω, Ω ) ≤ λ1 (β, Ω ). By Remark 8 we have that λ1 (β, Ω ) ≤ λ1 (β, B), where B is the ball with the same centre as η and meas(B) = meas(Ω ). So, we have that λ1 (ω, Ω ) ≤ λ1 (β, B). This gives an upper bound for λ1 (ω, Ω ). 5. Final comments We have presented two strategies for stabilizing to zero the predator population. The first one consists in acting directly on the predator population via a harvesting process (with a constant harvesting effort γ ). The second one consists in acting on the prey population (so, indirectly) via a harvesting process (with a constant harvesting effort γ ) in order to reduce the prey population density up to a level that cannot sustain anymore the predator population. We notice that the mapping γ 7→ µω1γ is increasing and continuous on [0, +∞) and µω,h 10 = −r < 0 and ω ω limγ →+∞ µ1γ = µ1 . It is possible to prove that the mapping γ 7→ K γ is nonincreasing and continuous from γ ,ω [0, +∞) to L ∞ (Ω ). This implies that the mapping γ 7→ ν1 is nondecreasing and continuous on [0, +∞) and p 0,ω ν1 = λ1 . The results concerning the second stabilizing strategy show the importance of finding the position and the γ ,ω geometry of ω for which there exists a γ ≥ 0 such that µω1γ < 0 and ν1 > 0. It is important to notice that we may combine both strategies: we may harvest the prey population in the subdomain ω1 (with a constant harvesting effort γ1 ) and harvest the predator population in the subdomain ω2 (with a harvesting effort γ2 ). In this way it is possible to stabilize the predator population to 0 even in situations when the previous two strategies fail. Similar stabilizability results can be obtained for this new strategy. So, in case of zero-stabilizability of the predator population (in all situations), the stabilizing controls are very simple feedback controls. Numerical methods for all the problems investigated here will be presented in a forthcoming paper. Acknowledgements The research of the first author was supported by the CEEX grant 05-D11-36/2005 and by the CNCSIS grant 1416/2005. The authors would like to thank the referees for their valuable comments and suggestions. References [1] B. Ainseba, S. Anit¸a, M. Langlais, Internal stabilizability of some diffusive models, J. Math. Anal. Appl. 265 (2002) 91–102. [2] B. Ainseba, F. Heiser, M. Langlais, A mathematical analysis of a prey-predator system in a highly heterogeneous environment, Differential Integral Equations 15 (2002) 385–404. [3] B. Ainseba, S. Anit¸a, Internal stabilizability for a reaction–diffusion problem modelling a predator–prey system, Nonlinear Anal. TMA 61 (2005) 491–501. [4] S. Anit¸a, V. Capasso, Internal stabilizability problem for a reaction–diffusion system modelling a class of spatially structured epidemic model, Nonlinear Anal. Real World Appl. 3 (2002) 453–464. [5] S. Anit¸a, J.-P. Raymond, Positive stabilization of a parabolic equation by controls localized on a curve, J. Math. Anal. Appl. 286 (2003) 107–115. [6] V. Capasso, Mathematical Structures of Epidemic Systems, in: Lecture Notes Biomath., vol. 97, Springer-Verlag, Heidelberg, 1993. [7] V. Capasso, R.E. Wilson, Analysis of a reaction–diffusion system modelling man-environment-man epidemics, SIAM J. Appl. Math. 57 (1997) 327–346. [8] W. Chen, M. Wang, Qualitative analysis of predator–prey models with Beddington-De Angelis functional response and diffusion, Math. Comput. Modelling 42 (2005) 31–44.

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