Protection Zones for Survival of Species in Random Environment

It is widely recognized that unregulated harvesting and hunting of biological resources can be harmful and endanger ecosystems. Therefore, various measures to prevent the biological resources from destruction and protect the ecological environment have been taken. An effective resolution is to designate protection zones where harvesting and hunting are prohibited. Assuming that migration can occur between protected areas and unprotected ones, a fundamental question is: How large a protection zone should be so that the species in both of the protection subregion and natural environment are able to survive. This paper aims to address this question for the case where the ecosystems are subject to random noise represented by a Brownian motion. Sufficient conditions for permanence and extinction are obtained, which are sharp and are close to necessary conditions. Moreover, ergodicity, convergence of probability measures to that of the invariant measure under total variation norm, and rates of convergence are obtained

Protection Zones for Survival of Species in Random Environment N. T. Dieu,∗ N. H. Du,† H. D. Nguyen,‡ G. Yin§ July 28, 2015 Abstract It is widely recognized that unregulated harvesting and hunting of biological resources can be harmful and endanger eco-systems. Therefore, various measures to prevent the biological resources from destruction and protect the ecological environment have been taken. An effective resolution is to designate protection zones where harvesting and hunting are prohibited. Assuming that migration can occur between protected areas and unprotected ones, a fundamental question is: How large a protection zone should be so that the species in both of the protection sub-region and natural environment are able to survive. This paper aims to address this question for the case where the eco-systems are subject to random noise represented by a Brownian motion. Sufficient conditions for permanence and extinction are obtained, which are sharp and are close to necessary conditions. Moreover, ergodicity, convergence of probability measures to that of the invariant measure under total variation norm, and rates of convergence are obtained. Keywords. Biodiversity; protection zone; extinction; permanence; ergodicity. Subject Classification. 34C12, 60H10, 92D25. ∗ Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam, dieunguyen2008@gmail.com. The author would like to thank Vietnam Institute for Advance Study in Mathematics (VIASM) for supporting and providing a fruitful research environment and hospitality. † Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi Vietnam, dunh@vnu.edu.vn. This research was supported in part by Vietnam National Foundation for Science and Technology Development (NAFOSTED) n0 101.03-2014.58. ‡ Department of Mathematics, Wayne State University, Detroit, MI 48202, USA, dangnh.maths@gmail.com. This research was supported in part by the National Science Foundation under grant DMS-1207667. This work was finished when the author was visiting VIASM. He is grateful for the support and hospitality of VIASM. § Corresponding author: Department of Mathematics, Wayne State University, Detroit, MI 48202, USA, gyin@math.wayne.edu. This research was supported in part by the National Science Foundation under grant DMS-1207667. 1 1 Introduction There is an alarming threat to wild life and biodiversity due to the pollution of environment as well as unregulated harvesting and hunting. Different measures have been taken to protect endangered species and their habitats. Among the effective measures, the approach of providing protected areas has become most popular over the past decades. Indeed, the Convention on Biological Diversity recognizes protected areas as a fundamental tool for safeguarding biodiversity, life itself. (“Convention on Biological Diversity” is a multilateral treaty, which has three main goals: conservation of biological diversity or biodiversity, sustainable use of its components, and fair and equitable sharing of benefits arising from genetic resources.) Recently, many scholars have used mathematical models to investigate the effect of protection zones in renewing biological resources and protect the population in both deterministic and stochastic models; see [3, 4, 5, 10, 31, 32] and references therein. The main idea of their work can be described as follows. The region Ω, where the species live in, is divided into two sub-regions Ω1 and Ω2 . The sub-region Ω1 is the nature environment and Ω2 is the nature reserve. The population densities between Ω1 and Ω2 are notably different. Migration can occur between Ω1 and Ω2 , which is assumed to be proportional to the difference of the densities with the proportional constant being D > 0. Denote the densities of population in Ω1 and Ω2 by X(t) and Y (t), respectively. Assume that the size of Ω1 is H and the size of Ω2 is h. Use D(X(t) − Y (t)) to represent the diffusing capacity that is the total biomass caused by diffusion effect. In the deterministic cases, this model can be formulated as  D  ˙ X(t) = X(t)(a − bX(t)) − (X(t) − Y (t)) − EX(t),  H (1.1)  D  ˙ Y (t) = Y (t)(a − bY (t)) + (X(t) − Y (t)), h a is the carry capacity of the environment and E is the comprehensive effects of the b unfavorable factors of biological growth relative to the biological growth in the protection where zone. This model has been studied in details in [5, 10]. It is well-recognized that the environment is always subject to random effect, so it is important to take into account the impact of stochastic perturbations on the evolution of the species. In the literature, Zou and Wang in [31] considered the following stochastic model for a single species with protection 2 zone.  D   dX(t) = X(t)(a − bX(t)) − (X(t) − Y (t)) − EX(t) dt + αX(t)dW (t) H  D  dY (t) = Y (t)(a − bY (t)) + (X(t) − Y (t)) dt + αY (t)dW (t), h (1.2) where a, b, D, H, h and α are appropriate constants, and W (·) is a standard real-valued D Brownian motion. To simplify the notation, we introduce = D∗ . Then H H D = D∗ β, β = . h h By substituting D∗ and β into system (1.2), we obtain dX(t) = [X(t)(a − bX(t)) − D∗ (X(t) − Y (t)) − EX(t)] dt + αX(t)dW (t) dY (t) = [Y (t)(a − bY (t)) + D∗ β(X(t) − Y (t))] dt + αY (t)dW (t). (1.3) Note that X(t) and Y (t) are fully correlated because the same Brownian motion is used in both equations. As a result, the system of diffusions is degenerate. Evidently, when we designate a protection zone, the larger the zone is, the higher the survival opportunity of the species is. However, setting up and maintaining a large protection zone is costly. It is therefore important to know what the threshold for the area of the protection zone is in order that the species survive permanently. Since β is the ratio of the area of Ω1 to that of Ω2 , the threshold should be a value β ∗ that can be calculated from a, b, D∗ , α, E such that if β < β ∗ the species will survive while it will reach extinction in case 2 β > β ∗ . In [31], it is proved that for any initial value (X(0), Y (0)) ∈ R2,◦ + (the interior of R+ ), 2,◦ there exists a unique global solution to (1.3) that remains in R+ almost surely. Although they provided sufficient conditions for the persistence in mean and extinction of the species, their conditions appear to be too restrictive to address the question of main interest. The goal of this paper is to provide a formula for calculating the threshold value β ∗ , which provides a sufficient and almost necessary condition for the size of the protection, for a more general class. We also go further than [31] by investigating important asymptotic properties of the solution such as the existence and uniqueness of an invariant probability measure, the convergence in total variation of the transition probability, the rate of convergence as well as the ergodicity of the solution process. Our contributions of the paper thus can be summarized as follows. 3 (a) We are dealing with a case of fully degenerate diffusions, which allows correlations of the species and is thus more suitable for the intended ecological applications. (b) In contrast to the usual approach of using Lyapunov function type argument, we derive a threshold value that characterize the size of the protection region. The conditions are sharp in that not only are the conditions obtained sufficient, but also they are almost necessary. (c) Then we go a step further than the existing results in the literature by investing the ergodicity of the systems under consideration. First, we give a sufficient condition for the ergodicity. Our result establishes the existence of the invariant probability measure. In addition, it describes precisely the support of the invariant probability measure. Second, we prove that the convergence in total variation to the invariant measure. Moreover, precise exponential upper bounds are obtained. Finally, a strong law of large numbers is obtained. Our result will be of important utility for the study of large time behavior of the dynamics of the spices. In practical terms, it indicates that when time is large enough, one can replace the instantaneous probability measure by that of the invariant measure that leads to much simplified treatment. The study on dynamics of species in ecological systems has received much attention. While many works were devoted to various aspects of deterministic systems with concentration on stability issues [1, 14, 15, 16, 21, 25, 29], there is an increasing effort to take steps treating systems that involve randomness [2, 6, 20, 28, 30]. Along this line, the current paper examines an important issue from the perspectives of protection zones and biodiversity. The rest of the paper is organized as follows. In Section 2, we provide a sufficient and almost necessary condition for the permanence of the species. The threshold β ∗ is determined. The existence and uniqueness of an invariant probability measure and the convergence in total variation of the transition probability are also proved. Moreover, an error bound of the convergence is provided. Section 3 is devoted to some discussion and comparison to existing results. Some numerical examples and figures are also provided to illustrate our results. Finally, further remarks are issued in Section 4, which point out possible future directions for investigations. 4 2 Sufficient and Almost Necessary Conditions for Permanence Let (Ω, F, {Ft }t≥0 , P) be a complete probability space with a filtration {Ft }t≥0 satisfying the usual condition, i.e., it is increasing and right continuous while F0 contain all P-null sets. Let W (t) be an Ft -adapted standard, real-valued Brownian motion. In lieu of considering X(t) (1.3), we set Z(t) = , and consider the following equation derived from Itˆo’s formula Y (t) dZ(t) = [bY (t)Z(t)(1 − Z(t)) + D∗ (1 − Z(t))(βZ(t) + 1) − EZ(t)] dt dY (t) = Y (t) [a − bY (t) + D∗ β(Z(t) − 1)] dt + αY (t)dW (t). (2.1) First, we note that dZ(t) < −EZ(t)dt ≤ −Edt if Z(t) ≥ 1. (2.2) Let z be the solution to dz(t) = [D∗ (1 − z(t))(βz(t) + 1) − Ez(t)] dt. By the comparison theorem for differential equation we can check that Z(t) ≥ z(t) ∀t ≥ 0 a.s. provided that Z(0) = z(0) ∈ (0, 1). Note that z(t) → z ∗ , where z∗ = (D∗ β − D∗ − E)2 + 4D∗ 2 β + D∗ β − D∗ − E 2D∗ β is the unique root of the equation D∗ (1 − z)(βz + 1) − Ez = 0 (2.3) lim inf Z(t) ≥ lim inf z(t) = z ∗ . (2.4) on (0, 1). As a result, t→∞ t→∞ For (z, y) ∈ R2+ , denote by (Z z,y (t), Y z,y (t)) the solution of (2.1) with the initial condition 2,◦ (Z(0), Y (0)) = (z, y). Let B(R2,◦ + ) be the σ-algebra of Borel subsets of R+ , and µ be the Lebesgue measure on R2,◦ + . To proceed, we first rewrite equation (2.1) in Stratonovich’s form:  dZ(t) = [bY (t)Z(t)(1 − Z(t)) + D∗ (1 − Z(t))(βZ(t) + 1) − EZ(t)] dt 2 (2.5) dY (t) = Y (t) a − α − bY (t) + D∗ β(Z(t) − 1) dt + αY (t) ◦ dW (t). 2 Let     ∗ byz(1 − z) + D (1 − z)(βz + 1) − Ez A1 (z, y) =  A(z, y) =  α2 A2 (z, y) y(a − − by + D∗ β(z − 1)) 2 5 and  B(z, y) =  B1 (z, y)    0  =  . B2 (z, y) αy Let L be the Lie algebra generated by vector fields A(z, y) and B(z, y) and L0 be the ideal in L generated by B(z, y). We will verify that the H¨ormander condition (see [22, 23]) holds for the diffusion (2.1), that is, dimL0 (z, y) = 2 at every (z, y) ∈ R2,◦ + . In other words, we will show that the set of vector fields B, [A, B], [A, [A, B]], [B, [A, B]], . . . spans R2 at every 2,◦ (z, y) ∈ R+ where [·, ·] is the Lie bracket that is defined as follows. If Φ(z, y) = (Φ1 (z, y), Φ2 (z, y))T and Ψ(z, y) = (Ψ1 (z, y), Ψ2 (z, y))T are vector fields on R2 (where z T denotes the transpose of z), then the Lie bracket [Φ; Ψ] is a vector field given by [Φ; Ψ]j (z, y) = ∂Ψj ∂Φj (z, y) − Ψ1 (z, y) (z, y) ∂z ∂z ∂Φj ∂Ψj (z, y) − Ψ2 (z, y) (z, y) , j = 1, 2. + Φ2 (z, y) ∂y ∂y Φ1 (z, y) By direct calculation,  C(z, y) := [A, B](z, y) =  and −αbyz(1 − z) αby 2  C(z, y) := [A, C1 (z, y)    1 , C](z, y) =  αb C2 (z, y) where ∂A1 (z, y) ∂z 2 + A2 (z, y)z(z − 1) − y [bz(1 − z)]. C1 (z, y) =A1 (z, y)(−y)(1 − 2z) + yz(1 − z) 2,◦ It can be clearly seen that, B(x, y), C(z, y) span R2 for all (z, y) ∈ R+ satisfying z = 1. When z = 1, we have C1 (1, y) = A1 (1, y)(−y)(1 − 2) = −Ey = 0 hence B(1, y) and C(1, y) span R2 for all y > 0. As a result, we obtain the following lemma. 2,◦ Lemma 2.1. The H¨ormander condition holds for the solution of (2.1) in R+ . As a result, the transition probability P (t, z0 , y0 , ·) of (Z(t), Y (t)) has density p(t, z0 , y0 , z, y) which is smooth in (z0 , y0 , z, y) ∈ R4,◦ + . 6 To proceed, we analyze the following control system corresponding to (2.5). z˙φ (t) = g(zφ (t), yφ (t)) y˙ φ (t) = h(zφ (t), yφ (t)) + αyφ (t)φ(t). (2.6) where g(z, y) = byz(1 − z) + D∗ (1 − z)(βz + 1) − Ez and α2 h(z, y) = y[a − − by + D∗ β(z − 1)] 2 with φ being from the set of piecewise continuous real valued functions defined on R+ . Let (zφ (t, z, y), yφ (t, z, y)) be the solution to Equation (2.6) with control φ and initial value (z, y). We have the following claims. Claim 1. For any y0 , y1 , z0 ∈ (0, ∞) and ε > 0, there exists a control φ and T > 0 such that yφ (T, z0 , y0 ) = y1 , |zφ (T, z0 , y0 ) − z0 | < ε. Suppose that y0 < y1 and let ρ1 = sup{|g(z, y)|, |h(z, y)| : y0 ≤ y ≤ y1 , |z − z0 | ≤ ε}. We αρ2 y0 choose φ(t) ≡ ρ2 with − 1 ε ≥ y1 − y0 . It is easy to check that with this control, ρ1 ε there is a 0 ≤ T ≤ such that yφ (T, z0 , y0 ) = y1 , |zφ (T, z0 , y0 ) − z0 | < ε. If y0 > y1 , we can ρ1 construct φ(t) similarly. Claim 2. For any 0 < z0 < z1 < 1, there is a y0 > 0, a control φ, and a T > 0 such that zφ (T, z0 , y0 ) = z1 and that yφ (T, z0 , y0 ) = y0 ∀ 0 ≤ t ≤ T . Indeed, if y0 is sufficiently large, there is a ρ3 > 0 such that g(z, y0 ) > ρ3 ∀ z0 ≤ z ≤ z1 < 1. This property, combining with (2.6), implies the existence of a feedback control φ and T > 0 satisfying the desired claim. Claim 3. Assume that z ∗ ≤ z1 < z0 . Since D∗ (1 − z)(βz + 1) − Ez < 0 ∀ z ∈ [z1 , z0 ], if y0 is sufficiently small, we have sup {g(z, y0 )} ≤ by0 sup {|z(1 − z)|} + sup {D∗ (1 − z)(βz + 1) − Ez} < 0. z∈[z1 ,z0 ] z∈[z1 ,z0 ] z∈[z1 ,z0 ] As a result, there is a feedback control φ and a T > 0 satisfying zφ (T, z0 , y0 ) = z1 and yφ (t, z0 , y0 ) = y0 ∀ 0 ≤ t ≤ T . Claim 4. For any 0 < z1 < z0 < z ∗ and we have D∗ (1 − z1 )(βz1 + 1) − Ez1 ≥ 0, which implies inf {h(z1 , y)} ≥ 0. Thus, we cannot find a control φ and a T > 0 satisfying y∈(0,∞) 7 zφ (T, z0 , y) = z1 . Similarly, if z1 > max{z0 , 1}, we cannot find a control φ and a T > 0 satisfying zφ (T, z0 , y) = z1 . Claim 5. It can be easily seen that there is z1∗ ∈ (z ∗ , 1) satisfying g(z1∗ , 1) = 0 and that the equilibrium (z1∗ , 1) of the system z˙ = g(z, y) y˙ = y(b − by), (2.7) is a sink. By the stable manifold theorem (see [24, p.107]), for any δ > 0, (z1∗ , 1) has a neighborhood Sδ ⊂ (z1∗ − δ, z1∗ + δ) × (1 − δ, 1 + δ) which is invariant under (2.7). Let (˜ z (t, z, y), y˜(t, z, y)) be the solution to (2.7) with initial value (z, y). With the feedback control φ satisfying a− α2 + D∗ β(˜ z (t, z, y) − 1)) + αφ(t) = b ∀ t ≥ 0, 2 we have (zφ (t, z, y), yφ (t, z, y)) = (˜ z (t, z, y), y˜(t, z, y)) ∀ t ≥ 0. As a result, (zφ (t, z, y), yφ (t, z, y) ∈ Sδ ∀(z, y) ∈ Sδ . for any t ≥ 0 with this control. Claim 6. For any z > 0 and δ > 0, there is T > 0 satisfying zφ (T, z, 0) ∈ (z ∗ − δ, z ∗ + δ) and clearly yφ (T, z, 0) = 0. We now provide a condition for the existence of a unique invariant probability measure for the process (Z(t), Y (t)) and investigate some properties of the invariant probability measure. Theorem 2.2. Let (Z(t), Y (t)) be the solution to equation (2.1). α2 +D∗ β(z ∗ −1) > 0, the process (Z(t), Y (t)) has an unique invariant probability 2 measure whose support is [z ∗ , 1] × (0, ∞). (i) If a− (ii) There exists γ > 0 and a function H(z, y) : R2,◦ + → R+ such that P (t, z, y, ·) − π ∗ (·) ≤ H(z, y)e−γt ∀ t ≥ 0. where · (2.8) is the total variation norm. (iii) Moreover, for any π ∗ -integrable function f , and (z, y) ∈ R2,◦ + we have 1 P lim t→∞ t t f (u, v)π ∗ (du, dv) = 1. f Z z,y (s), Y z,y (s) ds = R2 0 8 (2.9) To prove (2.8), we will apply [19, Theorem 6.1]. Hence, we need the following lemma. 2,◦ Lemma 2.3. For every compact set K ⊂ R+ is petite for the Markov chain (Z(n), Y (n)) 2,◦ (n ∈ N), that is, for all compact set K there exists a measure ψ with ψ(R+ ) > 0 and a proba- bility distribution ν(·) concentrated on N such that the kernel K(z, y, ·) := ∞ n=1 P (n, z, y, ·)ν(n) 2,◦ satisfying K(z, y, Q) ≥ ψ(Q) ∀(z, y) ∈ K, Q ∈ B(R+ ). We refer to [19] for more details on petite sets. Proof. Let the point (z1∗ , 1) be as in Claim 5. Since (z ∗ , 1) × (0, ∞) is invariant under (2.1), we have P (1, z1∗ , 1, (z ∗ , 1) × (0, ∞)) = 1 then p(1, z1∗ , 1, z2 , y2 ) > 0 for some (z2 , y2 ) ∈ (z ∗ , 1) × (0, ∞). In view of Claim 5 and the smoothness of p(1, ·, ·, ·, ·), there exists a neighborhood Sδ (z1∗ , 1), that is invariant under (2.7) and a open set G (z2 , y2 ) such that p(1, z, y, z , y ) ≥ m > 0 ∀ (z, y) ∈ Sδ , (z , y ) ∈ G. (2.10) For any (z, y) ∈ K, we derive from claims 1 − 3 that there is T > 0 and a control φ satisfying (zφ (T, z, y), yφ (T, z, y)) ∈ Sδ . Let nz,y be a positive integer greater than T . In view of Claim 5, we can extend control φ after T such that (zφ (nz,y , z, y), yφ (nz,y , z, y)) ∈ Sδ . By the support theorem (see [8, Theorem 8.1, p. 518]) P (nz,y , z, y, Sδ ) := 2ρz,y > 0. Since (Z(t), Y (t)) is a Markov-Feller process, there exists a open set Vz,y (z, y) such that P (nz,y , z , y , Sδ ) ≥ ρx,y ∀(z , y ) ∈ Vz,y . Since K is a compact set, there is a finite number l i=1 of Vzi ,yi , i = 1, . . . , l satisfying K ⊂ Vzi ,yi . Let ρK = min{ρzi ,yi , i = 1, . . . , l}. For each (z, y) ∈ K, there exists nzi ,yi such that P (nzi ,yi , z, y, Sδ ) ≥ ρK . (2.11) From (2.10) (2.11), for all (z, y) ∈ K there exists nzi ,yi such that p(nzi ,yi + 1, z, y, z , y ) ≥ ρK m ∀ (z , y ) ∈ G. Define the kernel K(z, y, Q) := 1 l l P (nzi ,yi + 1, z, y, Q) ∀Q ∈ B(R2,◦ + ). i=1 9 (2.12) We derive from (2.12) that 1 2,◦ K(z, y, Q) ≥ ρK m µ(G ∩ Q) ∀Q ∈ B(R+ ), l (2.13) 2,◦ where µ(·) is the Lebesgue measure on R+ . (2.13) means that every compact set K ⊂ R2,◦ + is petite for the Markov chain (Z(n), Y (n)). Proof of Theorem 2.2. Since a − α2 + D∗ β(z ∗ − 1) > 0, there exist q, δ ∗ ∈ (0, z ∗ ) such that 2 a − (q + 1) α2 + D∗ β(z ∗ − 1 − δ ∗ ) > 0. 2 First, we consider equation (2.5) in the invariant set M = {z ∗ − δ ∗ ≤ z ≤ 1, y > 0}. Denote by L the generator of the diffusion corresponding to (2.1). Letting U (z, y) = y −q + y + 1, we have lim U (z, y) = lim U (z, y) = ∞ y→0 y→∞ and α2 + D∗ β(z ∗ − 1 − δ ∗ ) − qD∗ β(z − z ∗ + δ ∗ )y −q + qby 1−q 2 + y(a − by + D∗ β(z − 1)) α2 ≤ − qy −q a − (q + 1) + D∗ β(z ∗ − 1 − δ ∗ ) + qby 1−q + y(a − by) 2 −q ≤ − θ1 (y + y) + θ2 ∀(z, y) ∈ M LU (z, y) = − qy −q a − (q + 1) where α2 + D∗ β(z ∗ − 1 − δ ∗ ) > 0, and 2 θ2 = sup{−θ1 y −q + qby 1−q + y(a − by) + θ1 y} < ∞. θ1 = q a − (q + 1) y>0 Similarly, we can estimate LU (z, y) ≤ θ3 U (z, y) ∀(z, y) ∈ R2,◦ + (2.14) for some θ3 > 0. By [19, Theorem 6.1], we derive from Lemma 2.3 and (2.14) that the Markov-Feller process (Z(t), Y (t)) has an unique invariant probability measure π ∗ in M. satisfying P (t, z, y, ·) − π ∗ (·) ≤ H0 (y −q + y + 1)e−γt ∀t ≥ 0, (z, y) ∈ M. 10 (2.15) Moreover, in light of the support theorem or [13, Lemma 4.1], we can easily imply from Claims 1-4 that the support of π ∗ is [z ∗ , 1] × (0, ∞). In view of (2.14) and standard arguments (see e.g. [12, Theorem 3.5, p. 75]), there is H1 , γ1 > 0 such that EU (Z z,y (t), Y z,y (t)) ≤ H1 U (z, y)eγ1 t ∀t > 0. (2.16) 2,◦ In view of (2.2) and (2.4), for any (z0 , y0 ) ∈ R+ , there is a non-random moment t0 = t0 (z0 , y0 ) > 0 such that (Z z,y (t), Y z,y (t)) ∈ M ∀t ≥ t0 with probability 1. Thus, we have from (2.15) and (2.16) the following estimate p(t0 , z0 , y0 , z, y) P (t, z, y, ·) − π ∗ (·) dzdy P (t + t0 , z0 , y0 , ·) − π ∗ (·) ≤ M p(t0 , z0 , y0 , z, y)H0 (y −q + y + 1)e−γt dzdy ≤ M =H0 e−γt EU (Z z0 ,y0 (t0 ), Y z0 ,y0 (t0 )) ≤ H0 H1 eγ1 t0 U (z0 , y0 )e−γt ∀t ≥ 0. Which proves (2.8). Finally, in view of Lemmas 2.1, 2.3 and estimates (2.2), (2.4), the strong law of large numbers (2.9) follows from the well-knows results in [18, Theorem 8.1] or [13]. Next, we give conditions for the extinction of the population density in both the protection zone and the natural environment. Theorem 2.4. Let (Z z,y (t), Y z,y (t)) be the solution to equation (2.1) with the initial conα2 dition (z, y) ∈ R2,◦ . If a − + D∗ β(z ∗ − 1) < 0, then (Z z0 ,y0 (t), Y z0 ,y0 (t)) → (z ∗ , 0) a.s. + 2 as t → ∞ for all (z0 , y0 ) ∈ R2,◦ + , that is, the species will be extinct. Moreover, for any (z0 , y0 ) ∈ R2,◦ + , we have with probability 1 that ln Y z0 ,y0 (t) α2 ln X z0 ,y0 (t) = lim =a− + D∗ β(z ∗ − 1) < 0. lim t→∞ t→∞ t t 2 (2.17) Proof. We proceed in the following steps. i) By using Lyapunov function method, we can show that the equilibrium (z ∗ , 0) is asymptotically stable in probability. z∗ , 1] × 2 [0, H]. For the control system, given δ > 0, there exists a T > 0 such that for any z∗ (z, y) ∈ [ , 1] × [0, H], there exists a control φ such that (zφ (t, z, y), yφ (t, z, y)) ∈ 2 (z ∗ − δ, z ∗ + δ) × [0, δ) for some t ∈ [0, T ]. z0 ,y0 ii) For any (z0 , y0 ) ∈ R2,◦ (t), Y z0 ,y0 (t)) is recurrent relative to [ + , the process (Z 11 iii) Using Markov property of the solution and the support theorem we obtain the desired conclusion. First, we prove that for any ε > 0, there exists a δ > 0 such that P{ lim (Z z,y (t), Y z,y (t)) = (z ∗ , 0)} ≥ 1 − ε ∀ (z, y) ∈ (z ∗ − δ, z ∗ + δ) × [0, δ). t→∞ (2.18) (2.18) is clearly true for the case y = 0. We only need to show (2.18) for (z, y) ∈ Nδ := (z ∗ − δ, z ∗ + δ) × (0, δ). Denote f1 (z, y) = byz(1 − z) + D∗ (1 − z)(βz + 1) − Ez, f2 (z, y) = y[a − by + D∗ β(z − 1)]. By computing partial derivatives of f1 (z, y) at the equilibrium (z ∗ , 0), we obtain ∂f1 ∗ (z , 0) = D∗ (β − 2βz ∗ − 1) − E := −c1 < 0; ∂z ∂f1 ∗ (z , 0) = bz ∗ (1 − z ∗ ) := c2 > 0. ∂y We have the Taylor expansion of f1 (z, y) in the vicinity of (z ∗ , 0), f1 (z, y) = −c1 (z − z ∗ ) + c2 y + o (z − z ∗ )2 + y 2 , where o lim z→z ∗ y→0 Otherwise, since a − (z − z ∗ )2 + y 2 (z − z ∗ )2 + y 2 = 0. α2 + D∗ β(z ∗ − 1) < 0, 2 −c3 := a − (1 − p) α2 + D∗ β(z ∗ − 1) < 0 2 for sufficiently small p > 0. Consider the Lyapunov function V (z, y) = (z − z ∗ )2 + y p which is twice differentiable in (z, y) ∈ R2,◦ + . By direct calculation, we have  2   ∂ V (z, y)  f (z, y) 2 ∂V (z, y) ∂V (z, y)  1  + 1 0, αy  2 ∂z LV (z, y) = ,  ∂ V (z, y) 2 ∂z ∂y f2 (z, y) ∂y∂z = 2(z − z ∗ ) −c1 (z − z ∗ ) + c2 y + o (z − z ∗ )2 + y 2 + py p [a − by − (1 − p) ≤ −c1 (z − z ∗ )2 +  ∂ 2 V (z, y)   0 ∂z∂y    2 ∂ V (z, y)  αy ∂y 2 α2 + D∗ β(z − 1)] 2 c22 2 y + o[(z − z ∗ )2 + y 2 ] − pc3 y p + py p D∗ β(z − z ∗ ). c1 12 Since y 2 = o(y p ) for small y, when y 2 + (z − z ∗ )2 is small, we have c22 2 2c3 py p y − pc1 y p + py p D∗ β(z − z ∗ ) ≤ − ; c1 3 c3 py p 2c1 (z − z ∗ )2 + . o[(z − z ∗ )2 + y 2 ] ≤ 3 3 Therefore, 1 LV (z, y) ≤ − [c1 (z − z ∗ )2 + c3 py p ] ≤ −θ4 V (z, y) ∀(z, y) ∈ Nδ 3 for some θ4 > 0 and sufficiently small δ. By [17, Theorem 2.3. p. 112], for any ε > 0, there is δ > 0 such that P{ lim (Z z,y (t), Y z,y (t)) = (z ∗ , 0)} ≥ 1 − ε ∀(z, y) ∈ Nδ . t→∞ (2.19) Next, we derive item ii). Let ϕ(t) be the solution to the following equation dϕ(t) = ϕ(t)[a + α2 − bϕ(t)]dt + αϕ(t)dW (t). 2 (2.20) If Z(0) = z0 ∈ (0, ∞), there is t0 > 0 such that Z(t) ≤ 1 ∀t > t0 , it is easy to check that Y (t) ≤ ϕ(t) ∀t ≥ t0 a.s. provided that Y (t0 ) = ϕ(t0 ) > 0 by the comparison theorem [8, Theorem 1.1, p. 352]. In view of [2], ϕ(t) has a unique stationary distribution µ∗ (·) which is 2a 2b a Gamma distribution with parameters α := 2 and β := 2 , that is, µ∗ (·) has the density α α φ∗ (x) = αβ α−1 x exp{−βx}, x > 0, Γ(α) where Γ(·) is the Gamma function. By the strong law of large number type result [26, Theorem 3.16, p. 46], we deduce that 1 lim t→∞ t ∞ t xα e−βx dx = ϕ(s)ds = 0 0 Consequently, 1 lim sup t→∞ t α β := K1 a.s. (2.21) t Y (s)ds ≤ K1 , (2.22) 0 which implies 1 lim sup t→∞ t t 0 1 1 1{Y (s)≥H} ds ≤ lim sup H t→∞ t t Y (s)ds ≤ 0 K1 . H (2.23) 2,◦ For any initial condition (z0 , y0 ) ∈ R+ . Let H > K1 , from (2.23) we have 1 lim inf t→∞ t t 1{Y z0 ,y0 (s)∈[0,H]} ds ≥ 1 − 0 13 K1 > 0 a.s. H (2.24) z∗ , 1]×[0, H]. It follows 2 ∗ from Claims 1-6, that for each (z, y) ∈ [ z2 , 1]×[0, H], we can choose a control φ(·) and Tz,y > 0 In view of (2.4) and (2.24), (Z z0 ,y0 (t), Y z0 ,y0 (t)) recurrent relative to [ such that (zφ (Tz,y , z, y), yφ (Tz,y , z, y)) ∈ Uδ . In view of the support theorem, for all (z, y) ∈ [ z∗ , 1] × [0, H] there is a Tz,y > 0 such that 2 P {(Z z,y (Tz,y ), Y z,y (Tz,y )) ∈ Uδ } > 2pz,y > 0. Since the process (Z(t), Y(t)) has the Feller property, there is a neighborhood Vz,y of (z, y) such that P (Z z ,y (Tz,y ), Y z ,y (Tz,y )) ∈ Uδ > pz,y , for all (z , y ) ∈ Vz,y . z∗ Because [ , 1] × [0, H] is a compact set, there is a finite number of Vzi ,yi , i = 1, . . . , n such 2 z∗ that [ , 1] × [0, H] ⊂ ni=1 Vzi ,yi . Put 2 T ∗ = max{Tzi ,yi , i = 1, . . . , n, }, p∗ = min{pzi ,yi , i = 1, . . . , n}. For (z, y) ∈ (0, ∞) × (0, ∞)), set τδz,y = inf{t > 0 : (Z z,y (t), Y z,y (t)) ∈ Uδ }. Then z∗ , 1] × [0, H]. (2.25) 2 z∗ Moreover, since (Z z0 ,y0 (t), Y z0 ,y0 (t)) is recurrent relative to [ , 1] × [0, H], we can define 2 finite stopping times P{τδz,y < T ∗ } ≥ p∗ > 0 ∀ (z; y) ∈ [ η0 = 0, ηk = inf t > ηk−1 + T ∗ : (Z z0 ,y0 (t), Y z0 ,y0 (t)) ∈ [ z∗ , 1] × [0, H] , k ∈ N. 2 Consider the events Ak = {(Z z0 ,y0 (t), Y z0 ,y0 (t)) ∈ / Uδ ∀ t ∈ [ηk , ηk + T ∗ ]}. We deduce from the strong Markov property of (Z(t), Y (t)) and (2.25) that n Ak ≤ (1 − p∗ )n → 0 as n → ∞. P k=1 14 As a result, P{τδz0 ,y0 < ∞} = 1. (2.26) In light of the strong Markov property of (Z(t), Y (t)), (2.19) and (2.26) yield P{ lim (Z z0 ,y0 (t), Y z0 ,y0 (t)) = (z ∗ , 0)} ≥ 1 − ε. t→∞ (2.27) Since ε can be taken arbitrarily, we obtain P{ lim (Z z0 ,y0 (t), Y z0 ,y0 (t)) = (z ∗ , 0)} = 1 ∀(z0 , y0 ) ∈ R2,◦ + . t→∞ (2.28) Finally, it follows from Itˆo’s formula that ln Y z0 ,y0 (t) ln y0 1 = + t t t t a− 0 W (t) α2 + D∗ β(Z z0 ,y0 (s) − 1) − bY z0 ,y0 (s) ds + . (2.29) 2 t Hence (2.17) follows straightforward from (2.28) and (2.29). 3 Discussion and Numerical Examples Now, we fix all the coefficients of (1.3) except for β. We want to answer the question “for what α2 values of β, does the species survive permanently?” First, consider the case a − − E > 0. 2 We obtain that even without a protection zone β = ∞, the species will survive permanently. α2 If a − − E = 0, for any β > 0, the inequality 2 a− α2 + D∗ β(z ∗ − 1) > 0 2 holds. Hence the species will survive if there is a protection zone even its area is small. On α2 the other hand, if a − ≤ 0, the species will die out even it is completely protected. We 2 α2 therefore focus on the case 0 < a − < E, in which the equation 2 a− where z∗ = α2 + D∗ β(z ∗ − 1) = 0, 2 (D∗ β − D∗ − E)2 + 4D∗ 2 β + (D∗ β − D∗ − E) 2D∗ β has a unique positive root β∗ = (2D∗ + 2E + α2 − 2a)(2a − α2 ) . 2D∗ (2E + α2 − 2a) 15 Moreover, α2 + D∗ β(z ∗ − 1) > 0 if and only if β < β ∗ , 2 which answers the aforementioned question. Intuitively, if β is less than but very close to β ∗ , 1 t in view of the ergodicity of (Z(t), Y (t)) (see (2.9)), we can show that lim Y (s)ds is small t→∞ t 0 1 t 1 t 1 t X(s)ds. In order to guarantee that lim Y (s)ds and lim X(s)ds and so is lim t→∞ t 0 t→∞ t 0 t→∞ t 0 are not too small, we need β to be considerably less than β ∗ . Let us compare our results to a− that used in [31]. α2 − D∗ β > 0, then the population 2 α2 in Ω2 is persistent in mean; in [31, Theorem 4] they show that if a − − E > 0 then 2 2 α population in Ω1 is persistent in mean, whereas we only need a − + D∗ β(z ∗ − 1) > 0 2 to obtain such results. Clearly • In [31, Theorem 3], the authors show that if a − a− α2 α2 + D∗ β(z ∗ − 1) > a − − D∗ β 2 2 and a− α2 D∗ β(z ∗ − 1)(βz ∗ + 1) α2 α2 + D∗ β(z ∗ − 1) > a − + = a − − E. 2 2 z∗ 2 The assumptions of the theorems in [31] are more restrictive than that of Theorem 2.2. • Moreover, α2 α2 >a− + D∗ β(z ∗ − 1), 2 2 which means that the condition for the extinction given in [31, Theorem 5] is also more a− respective than that of Theorem 2.4. Example 3.1. Consider (2.1) with parameters a = 3, β = 1, b = 4, α = 1, D∗ = 4, E = 3. Direct calculation shows that z ∗ = 0.693000468; β ∗ = 5.625 > β. We obtain a− α2 + D∗ β(z ∗ − 1) = 1.272001872 > 0. 2 By virtue of Theorem 2.2, (2.1) has a unique invariant probability measure µ∗ whose support is [0.693000468, 1] × R+ . Consequently, the strong law of large numbers and the convergence in total variation norm of the transition probability hold. A sample path of solution to (2.1) α2 is illustrated by Figures 1. It is easy to see that a − − D∗ β = −1.5 < 0, it means that the 2 condition of [31, Theorem 3] is not satisfied but the density population is still persistent. 16 Figure 1: Trajectories of X(t), Y (t), Z(t) in Example 3.1 respectively. Example 3.2. Consider (2.1) with parameters a = 3, β = 6, b = 4, α = 1, D∗ = 4, E = 3. Direct calculation shows that z ∗ = 0.8946300737, β ∗ = 5.625 < β. We obtain a− α2 + D∗ β(z ∗ − 1) = −0.02887823 < 0. 2 In view of Theorem 2.4, (Z(t), Y (t)) → (z ∗ , 0) a.s. as t → ∞. This claim is justified in α2 = 2.5 > 0, it means that the condition of [31, Figures 2. In this example, we see that a − 2 Theorem 5] does not hold but the density population will eventually reach extinction. Figure 2: Trajectories of Y (t), Z(t) in Example 3.2. 17 4 Further Remarks Aiming at conservation of biodiversity and stemming from the consideration of protection zones for species in ecological systems, this paper pinpoints the size of the protection region. We provide sufficient conditions that are very close to necessary. Dealing with degenerate diffusions, we obtain ergodicity of the systems using the ideas from geometric control theory. Convergence to the invariant measure under total variation norm is obtained together with an exponential error bound. The results obtained may facilitate future study of such ecological systems. For future study, a number of questions are of particular interests from both practical and theoretical point of view. • It is natural to consider controlled systems with protection zones. One may pose the question about what is the minimal size of the protection zone so as to maintain the permanence of the population. • One may study the protection zones that depend on controls. • Using the invariant measure obtained, we may also treat various long-run control objectives so that we can replace the instantaneous probability measures by that of the invariant measure. • To accommodate the random environment and to take into consideration of continuous dynamics and interactions with the discrete events, we may consider more complex models, in which the parameters a, b, D, H etc. are no longer fixed but are modulated by a continuous-time Markov chain. That is, in lieu of (1.2), we can consider  D(η(t))   dX(t) = X(t)(a(η(t)) − b(η(t))X(t)) − (η(t))(X(t) − Y (t))   H       −E(η(t))X(t) dt + α(η(t))X(t)dW (t) (4.1)   D(η(t))   dY (t) = Y (t)(a(η(t)) − b(η(t))Y (t)) + (η(t))(X(t) − Y (t)) dt   h     +α(η(t))Y (t)dW (t), where η(t) is a continuous-time Markov chain taking values in a finite set M = {1, . . . , m0 } for some m0 > 1. The Markov chain models the random environment 18 that cannot be modeled by the usual stochastic differential equations. All the questions studied in this paper are important issues to address for the Markov modulated model. • In the aforementioned model, if we allow the Markov chain to be hidden, further filtering techniques need to be bought in to analyze the protection zones. • One may also consider eco-systems of two or more interacting species with protection zones created to protect some of the species. For instance, it is interesting to study the behavior of a predator-prey model in which there is a region where only the prey can access and avoid predation. All of these questions deserve careful consideration and open up a new domain for further study. References [1] E. Beretta, F. Solimano, A generalization of Volterra models with continuous time delay in population dynamics: boundedness and global asymptotic stability, SIAM J. Appl. Math., 48 (1988), 607-626. [2] N.H. Du, D.H. Nguyen, G. Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models, to appear in J. Appl. Probab. [3] Y. Du, X. Liang, A diffusive competition model with a protection zone, J. Differential Eqs., 244 (2008) 61-86. [4] Y. Du, R. Peng, M. Wang, Effect of a protection zone in the diffusive Leslie predatorprey model, J. Differential Eqs., 246 (2009), 3932-3956. [5] M. Fan, and K. Wang, Study on harvested population with diffusional migration, J. Syst. Sci. Comput., 14 (2001) 139-148. [6] P. Hinow, F. Le Foll, P. Magal, G.F. Webb, Analysis of a model for transfer phenomena in biological populations, SIAM J. Appl. Math. 70 (2009), no. 1, 40-62. 19 [7] K. Ichihara, H. Kunita, A classification of the second order degenerate elliptic operators and its probabilistic characterization, Z. Wahrsch. Verw. Gebiete 30 (1974), 235-254. Corrections in 39 81-84 (1977). [8] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, second edition, North-Holland Publishing Co., Amsterdam, 1989. [9] V. Jurdjevic, Geometric Control Theory, (Cambridge Studies in Advanced Mathematics vol 52) Cambridge University Press 1997. [10] T.K. Kar, M. Swarnakamal, Influence of prey reserve in a prey-predator fishery, Nonlinear Analysis, 65 (2006) 1725-1735. [11] R.Z. Khas’minskii, Ergodic properties of recurrent diffusion processes and stabilization of the Cauchy problem for parabolic equations. Theory Probab. Appl., 5 (1960), 179196. [12] R.Z. Khas’minskii, Stochastic Stability of Differential Equations. Springer-Verlag Berlin Heidelberg. [13] W. Kliemann, Recurrence and invariant measures for degenerate diffusions, Ann. Probab., 15 (1987), no. 2, 690-707. [14] R. Kon, Age-structured Lotka-Volterra equations for multiple semelparous populations,SIAM J. Appl. Math. 71 (2011), no. 3, 694713. [15] S. Liu, E. Beretta, A stage-structured predator-prey model of Beddington-DeAngelis type. SIAM J. Appl. Math. 66 (2006), no. 4, 11011129 [16] J. Llibre, D. Xiao, Global dynamics of a Lotka-Volterra model with two predators competing for one prey. SIAM J. Appl. Math. 74 (2014), no. 2, 434453. [17] X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing Chichester, 1997. [18] S.P. Meyn, R.L. Tweedie, Stability of Markovian processes II: Continuous-time processes and Sampled chains, Adv. Appl. Prob. 25, (1993), 487-517. 20 [19] S.P. Meyn, R.L. Tweedie, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob. 25 (1993), 518-548. [20] D.H. Nguyen, N.H. Du, G. Yin, Existence of stationary distributions for Kolmogorov systems of competitive type under telegraph noise, J. Differential Eqs., 257 (2014), 2078–2101. [21] V.W. Noonburg, A neural network modeled by an adaptive Lotka-Volterra system, SIAM J. Appl. Math., 49 (1989), 1779-1792. [22] J. Norris, Simplified Malliavin calculus. In: S´eminaire de probabiliti´es XX, Lecture Notes in Mathematics, Vol. 1204. Springer, New York, (1986), 101-130. [23] D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, Berlin Heidelberg., 2006. [24] L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, New York. Inc., 2006. [25] S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), no. 4, 1445-1472. [26] A.V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Vol. 78. American Mathematical Soc., 1989. [27] L. Stettner, On the existence and uniqueness of invariant measure for continuous time Markov processes, LCDS Report No. 86-16, April 1986, Brown University, Providence. [28] F. Wu, Y. Xu, Stochastic Lotka-Volterra Population Dynamics with Infinite Delay, SIAM J. Appl. Math., 70(3), 641-657. [29] Y. Xia, M. Han, New conditions on the existence and stability of periodic solution in Lotka-Volterra’s population system, SIAM J. Appl. Math. 69 (2009), no. 6, 1580-1597. [30] C. Zhu, G. Yin, On competitive Lotka-Volterra model in random environments, J. Math. Anal. Appl., 357 (2009), 154-170. 21 [31] X. Zou, K. Wang, A robustness analysis of biological population models with protection zone, Appl. Math. Modelling, 35 (2011) 5553-5563. [32] X. Zou, K. Wang, and M. Liu, Can protection zone potentially strengthen protective effects in random environments? Appl. Math. Comput., 231 (2014), 26-38. 22 [...]... bought in to analyze the protection zones • One may also consider eco-systems of two or more interacting species with protection zones created to protect some of the species For instance, it is interesting to study the behavior of a predator-prey model in which there is a region where only the prey can access and avoid predation All of these questions deserve careful consideration and open up a new domain... condition of [31, Figures 2 In this example, we see that a − 2 Theorem 5] does not hold but the density population will eventually reach extinction Figure 2: Trajectories of Y (t), Z(t) in Example 3.2 17 4 Further Remarks Aiming at conservation of biodiversity and stemming from the consideration of protection zones for species in ecological systems, this paper pinpoints the size of the protection region We... theoretical point of view • It is natural to consider controlled systems with protection zones One may pose the question about what is the minimal size of the protection zone so as to maintain the permanence of the population • One may study the protection zones that depend on controls • Using the invariant measure obtained, we may also treat various long-run control objectives so that we can replace the instantaneous... continuous-time Markov chain taking values in a finite set M = {1, , m0 } for some m0 > 1 The Markov chain models the random environment 18 that cannot be modeled by the usual stochastic differential equations All the questions studied in this paper are important issues to address for the Markov modulated model • In the aforementioned model, if we allow the Markov chain to be hidden, further filtering... instantaneous probability measures by that of the invariant measure • To accommodate the random environment and to take into consideration of continuous dynamics and interactions with the discrete events, we may consider more complex models, in which the parameters a, b, D, H etc are no longer fixed but are modulated by a continuous-time Markov chain That is, in lieu of (1.2), we can consider  D(η(t)) ... close to necessary Dealing with degenerate diffusions, we obtain ergodicity of the systems using the ideas from geometric control theory Convergence to the invariant measure under total variation norm is obtained together with an exponential error bound The results obtained may facilitate future study of such ecological systems For future study, a number of questions are of particular interests from both... conditions for the extinction of the population density in both the protection zone and the natural environment Theorem 2.4 Let (Z z,y (t), Y z,y (t)) be the solution to equation (2.1) with the initial conα2 dition (z, y) ∈ R2,◦ If a − + D∗ β(z ∗ − 1) < 0, then (Z z0 ,y0 (t), Y z0 ,y0 (t)) → (z ∗ , 0) a.s + 2 as t → ∞ for all (z0 , y0 ) ∈ R2,◦ + , that is, the species will be extinct Moreover, for any... straightforward from (2.28) and (2.29) 3 Discussion and Numerical Examples Now, we fix all the coefficients of (1.3) except for β We want to answer the question for what α2 values of β, does the species survive permanently?” First, consider the case a − − E > 0 2 We obtain that even without a protection zone β = ∞, the species will survive permanently α2 If a − − E = 0, for any β > 0, the inequality... Norris, Simplified Malliavin calculus In: S´eminaire de probabiliti´es XX, Lecture Notes in Mathematics, Vol 1204 Springer, New York, (1986), 101-130 [23] D Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, Berlin Heidelberg., 2006 [24] L Perko, Differential Equations and Dynamical Systems, Springer-Verlag, New York Inc., 2006 [25] S Ruan and D Xiao, Global analysis in a predator-prey system... (Cambridge Studies in Advanced Mathematics vol 52) Cambridge University Press 1997 [10] T.K Kar, M Swarnakamal, Influence of prey reserve in a prey-predator fishery, Nonlinear Analysis, 65 (2006) 1725-1735 [11] R.Z Khas’minskii, Ergodic properties of recurrent diffusion processes and stabilization of the Cauchy problem for parabolic equations Theory Probab Appl., 5 (1960), 179196 [12] R.Z Khas’minskii, Stochastic ... bought in to analyze the protection zones • One may also consider eco-systems of two or more interacting species with protection zones created to protect some of the species For instance, it is interesting... extinction Figure 2: Trajectories of Y (t), Z(t) in Example 3.2 17 Further Remarks Aiming at conservation of biodiversity and stemming from the consideration of protection zones for species in. .. However, setting up and maintaining a large protection zone is costly It is therefore important to know what the threshold for the area of the protection zone is in order that the species survive