DSpace at VNU: PROTECTION ZONES FOR SURVIVAL OF SPECIES IN RANDOM ENVIRONMENT

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Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php SIAM J APPL MATH Vol 76, No 4, pp 1382–1402 c 2016 Society for Industrial and Applied Mathematics PROTECTION ZONES FOR SURVIVAL OF SPECIES IN RANDOM ENVIRONMENT∗ N T DIEU† , N H DU‡ , H D NGUYENĐ , AND G YINả Abstract 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 to 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 and unprotected areas, a fundamental question is, how large should a protection zone be so that the species in both the protection subregion and natural environment are able to survive Devoted to answering the question, this paper aims at studying ecosystems that are subject to random noise represented by Brownian motion Sufficient conditions for permanence and extinction are obtained, which are sharp and 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 Key words biodiversity, protection zone, extinction, permanence, ergodicity AMS subject classifications 34C12, 60H10, 92D25 DOI 10.1137/15M1032004 Introduction There is an alarming threat to wild life and biodiversity due to the pollution of the 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 researchers have used advanced mathematics to investigate the effect of protection zones in renewing biological resources and protecting the population in both deterministic and stochastic models; see [10, 11, 16, 34, 35, 36] and references therein The main idea of their work can be described as follows The region Ω, where the species live, is divided into two subregions Ω1 and Ω2 The subregion Ω1 is the ∗ Received by the editors July 22, 2015; accepted for publication (in revised form) May 16, 2016; published electronically July 21, 2016 http://www.siam.org/journals/siap/76-4/M103200.html † 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 author’s research was supported in part by Vietnam National Foundation for Science and Technology Development (NAFOSTED) 101.032014.58 § Department of Mathematics, Wayne State University, Detroit, MI 48202 (dangnh.maths@gmail com) This author’s 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 (gyin@math.wayne.edu) This author’s research was supported in part by the National Science Foundation under grant DMS-1207667 1382 Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php PROTECTION ZONES IN RANDOM ENVIRONMENT 1383 unprotected environment and Ω2 is the protected one Migration can occur between Ω1 and Ω2 , which is assumed to be proportional to the difference of the densities with the proportional constant D > Denote the densities of population in Ω1 and Ω2 by X(t) and Y (t), respectively Assume that the areas of Ω1 and Ω2 are H and h, respectively Use D(X(t) − Y (t)) to represent the diffusing capacity that is the total biomass caused by the diffusion effect In the deterministic cases, this model can be formulated as ⎧ D ⎪ ˙ ⎪ = X(t)(a − bX(t)) − (X(t) − Y (t)) − EX(t), ⎨X(t) H (1.1) ⎪ D ⎪ ⎩Y˙ (t) = Y (t)(a − bY (t)) + (X(t) − Y (t)), h where ab is the carrying capacity of the environment and E is the comprehensive effect of the unfavorable factors of biological growth relative to the biological growth in the protection zone This model has been studied in [36, 16] To capture the main ingredient, we recall the following theorem obtained in [36] Theorem 1.1 The following results hold a(H+h−ah) EH (a) If a < H , then the origin is the unique globally asymph and D > H−ah totically stable equilibrium of the system (1.1) a(H+h−ah) EH or a ≥ β, then there is a unique positive (b) If a < H h and D < H−ah equilibrium, which attracts all positive solutions of (1.1) The theorem above provides characterizations of the ecosystems The inequalities above can be viewed as “threshold”-type conditions, which give a precise description on the asymptotic behavior of different equilibria Statement (a) indicates that if the area of the protection region satisfies the given inequality, the population will reach extinction, whereas (b) states that if the condition is met, the population will reach a steady state eventually The results in Theorem 1.1 focuses on deterministic models It is, however, well recognized that the environment is always subject to random disturbances, so it is important to take the impact of stochastic perturbations on the evolution of the species into consideration An immediate question is, can we still characterize the protection zone so as to delineate the conditions for permanence and extinction similar to Theorem 1.1? In addition, how can we characterize the equilibrium or steady state behavior of the ecosystems? For stochastic systems, because randomness is involved, in addition to equilibria, stationary distributions also come into play We need to answer the question, under what conditions is there a stationary distribution The situation becomes more complex Our main objectives and contributions of this paper are to provide conditions similar to Theorem 1.1 so as to characterize the qualitative properties protection regions In fact, we obtain sufficient conditions that are close to necessary for permanence and extinction Furthermore, we also investigate the convergence and rates of convergence to the invariant or stationary or steady state distribution In the literature, Zou and Wang in [34] considered the following stochastic model for a single species with protection zone: (1.2) ⎧ 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 Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1384 N T DIEU, N H DU, H D NGUYEN, AND G YIN where a, b, D, H, h, and α are appropriate constants, and W (·) is a standard realD valued Brownian motion To simplify the notation, we introduce H = D∗ Then H D = D∗ β, β = h h ∗ By substituting D and β into system (1.2), we obtain (1.3) 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) 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 When we designate a protection zone, the larger the zone is, the higher the survival opportunity of the species gets 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 should be to make 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 the case β > β ∗ In [34], it is proved that for any initial value (X(0), Y (0)) ∈ R2,◦ + (the interior of R+ ), there exists a unique global solution to 2,◦ (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 For the deterministic case (1.1), the threshold β ∗ can be derived easily from Theorem 1.1 The goal of this paper is to provide a formula for calculating the threshold value β ∗ for the stochastic systems and to provide a sufficient and almost necessary condition for the permanence of the species In other words, a parameter λ, which is given as a function of the coefficients of system (1.2), will be introduced We show that if λ > then the species in both protected and unprotected areas will survive permanently while if λ < 0, the species will die out Thus, the threshold β ∗ will be obtained from the equation λ = We also reveal how the white noise influences the system and compare the deterministic and stochastic models in section Moreover, we go a step further than [34] 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 In recent years, the study of 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 [2, 19, 20, 21, 26, 32, 31], there is an increasing effort treating systems that involve randomness [9, 12, 13, 14, 25, 30, 33] Along this line, the current paper examines an important issue from the perspectives of protection zones and biodiversity Our contributions of the paper can be summarized as follows (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 a Lyapunov function-type argument, we derive a threshold value that characterizes the size of the protection region The conditions are sharp in that not only are the conditions obtained sufficient, but also they are close to necessary Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php PROTECTION ZONES IN RANDOM ENVIRONMENT 1385 (c) In contrast to the existing results in the literature, we invest the ergodicity of the systems under consideration First, we give a sufficient condition for the ergodicity Our result establishes the existence of an invariant probability measure In addition, it describes precisely the support of the invariant probability measure Second, we prove 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 important for the study of long-time behavior of the dynamics of the species 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 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 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 Sufficient conditions for permanence In this section, we obtain sufficient conditions for permanence The conditions are in fact close to necessary 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 To gain insight into the growth rates of species in the two areas, we first rewrite (1.3) in the form ⎧ ∗ ∗ Y (t) ⎪ ⎪ ⎨dX(t) = X(t) a − D − E − bX(t) + D X(t) dt + αX(t)dW (t), (2.1) ⎪ X(t) ⎪ ⎩dY (t) = Y (t) a − D∗ − bY (t) + D∗ β dt + αY (t)dW (t) Y (t) In this form, one can easily see that the growth rates of X(t) and Y (t) depend on the ratio X(t) Y (t) Thus, instead of working directly on (1.3), we use the transform Z(t) = (2.2) X(t) Y (t) and consider the following equation derived from Itˆo’s formula, 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) First, we note that (2.3) dZ(t) < −EZ(t)dt ≤ −Edt if Z(t) ≥ Let z be the solution to the first equation of (2.2) on the boundary {(z, y) : z > 0, y = 0} That is, (2.4) dz(t) = [D∗ (1 − z(t))(βz(t) + 1) − Ez(t)] dt By the comparison theorem for differential equations, we can check that Z(t) ≥ z(t) for all t ≥ a.s provided that Z(0) = z(0) ∈ (0, 1) Note that z(t) → z ∗ , Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1386 N T DIEU, N H DU, H D NGUYEN, AND G YIN Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php where (D∗ β − D∗ − E)2 + 4D∗ β + D∗ β − D∗ − E ∗ z = (2.5) 2D∗ β is the unique root of the equation D∗ (1 − z)(βz + 1) − Ez = on (0, 1) As a result, lim inf Z(t) ≥ lim inf z(t) = z ∗ (2.6) t→∞ t→∞ For (z, y) ∈ R2+ , denote by (Z z,y (t), Y z,y (t)) the solution of (2.2) with the initial 2,◦ condition (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 use the ideas in geometric control theory to study the dynamic systems To this end, it is more convenient to use the stochastic integral in the Stratonovich form Then we use the idea of reachable sets in control theory to overcome the difficulty of evaluating the systems Thus we rewrite (2.2) as ⎧ ⎨dZ(t) = [bY (t)Z(t)(1 − Z(t)) + D∗ (1 − Z(t))(βZ(t) + 1) − EZ(t)] dt, (2.7) ⎩dY (t) = Y (t) a − α − bY (t) + D∗ β(Z(t) − 1) dt + αY (t) ◦ dW (t) Let A(z, y) = A1 (z, y) A2 (z, y) ⎞ ⎛ byz(1 − z) + D∗ (1 − z)(βz + 1) − Ez ⎠ =⎝ α2 y(a − − by + D∗ β(z − 1)) and B(z, y) = B1 (z, y) = B2 (z, y) αy To use the ideas of reachable sets, we need the notion of Hăormanders condition The diusion (2.7) is said to satisfy Hă ormanders condition if the set of vector elds B, [A, B], [A, [A, B]], [B, [A, B]], spans R2 at every (z, y) ∈ R2,◦ + , where [·, ·] is the Lie bracket that is defined as follows (see [1, 27] for more details) 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 ∂Ψj ∂Φj + Φ2 (z, y) (z, y) − Ψ2 (z, y) (z, y) , j = 1, ∂y ∂y Φ1 (z, y) We next verify that Hă ormanders condition holds for the diusion given by (2.7) By direct calculation, C(z, y) := [A, B](z, y) = −αbyz(1 − z) αby Copyright © by SIAM Unauthorized reproduction of this article is prohibited PROTECTION ZONES IN RANDOM ENVIRONMENT 1387 Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php and C(z, y) := A, C (z, y) = αb C1 (z, y) , C2 (z, y) where ∂A1 (z, y) ∂z + A2 (z, y)z(z − 1) − y [bz(1 − z)] C1 (z, y) = A1 (z, y)(−y)(1 − 2z) + yz(1 − z) It can be seen that B(x, y), C(z, y) span R2 for all (z, y) ∈ R2,◦ + satisfying z = When z = 1, we have C1 (1, y) = A1 (1, y)(−y)(1 − 2) = −Ey = hence B(1, y) and C(1, y) span R2 for all y > As a result, we obtain the following lemma Lemma 2.1 Hă ormanders condition holds for the solution of (2.2) in R2,◦ + Remark 2.1 As a consequence of Lemma 2.1, [1, Corollary 7.2] yields that 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,◦ + To proceed, we analyze the following control system corresponding to (2.7): (2.8) where z˙φ (t) = g(zφ (t), yφ (t)), y˙ φ (t) = h(zφ (t), yφ (t)) + αyφ (t)φ(t), g(z, y) = byz(1 − z) + D∗ (1 − z)(βz + 1) − Ez and h(z, y) = y a − α2 − by + D∗ β(z − 1) 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 (2.8) with control φ and initial value (z, y) To establish our results, the main idea stems from the use of the notion of reachable sets Roughly, a reachable set can be illustrated as follows Starting with initial point (z0 , y0 ), the collection of all points (z1 , y1 ) = (zφ (t, z0 , y0 ), yφ (t, z0 , y0 )) under piecewise continuous controls φ forms the reachable set of (z0 , y0 ) In light of the support theorem (see [15, Theorem 8.1, p 518]), to obtain the desired properties of the transition probability and invariant probability measure of (2.2), we investigate the reachable sets of different initial values The results are given in the following claims Before getting to the detailed argument, let us first provide some illustrations on these claims Claim shows that we can control vertically while Claims and state that a point can be reached horizontally from the left and the right under suitable conditions To be more precise, Claim indicates that for any initial points y0 and z0 , there is a control so that yφ can reach any given point y1 while zφ will stay in a neighborhood of z0 in finite time Claim states that if the initial point z0 is less than the final point z1 , there are a y0 > and a control so that zφ will reach z1 while yφ remains unchanged in a finite time Claim considers the opposite case when the initial point z0 is greater than the final point z1 It illustrates that under an appropriate condition, we can find a feedback control so that zφ will reach z1 and yφ will stay at y0 in finite time Claim inserts that under the said conditions, we Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1388 N T DIEU, N H DU, H D NGUYEN, AND G YIN cannot find a control, so that zφ reaches z1 in finite time Claim indicates that there is a point that can be approached from any nearby initial point (z0 , y0 ) using a suitable feedback control Finally, Claim is concerned with properties of the control system restricted on the boundary {(z, y) : y = 0} Claim For any y0 , y1 , z0 ∈ (0, ∞) and ε > 0, there exist a control φ and a T > 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 choose φ(t) ≡ ρ2 with ( αρρ21y0 − 1)ε ≥ y1 − y0 It is easy to check that with this control, there is a ≤ T ≤ ρε1 such that yφ (T, z0 , y0 ) = y1 , |zφ (T, z0 , y0 ) − z0 | < ε If y0 > y1 , we can construct φ(t) similarly In the next two claims (Claims and 3), we consider the reachable sets from initial conditions starting from different regions Claim For an For any < z0 < z1 < 1, there are a y0 > 0, a control φ, and a T > such that zφ (T, z0 , y0 ) = z1 and that yφ (T, z0 , y0 ) = y0 for all ≤ t ≤ T Indeed, if y0 is sufficiently large, there is a ρ3 > such that g(z, y0 ) > ρ3 for all z0 ≤ z ≤ z1 < This property, combining with (2.8), implies the existence of a feedback control φ and T > satisfying the desired claim Claim For an Assume that z ∗ ≤ z1 < z0 Since D∗ (1 − z)(βz + 1) − Ez < for all z ∈ [z1 , z0 ], if y0 is sufficiently small, we have sup {g(z, y0 )} ≤ by0 z∈[z1 ,z0 ] sup {|z(1 − z)|} + z∈[z1 ,z0 ] sup {D∗ (1 − z)(βz + 1) − Ez} < z∈[z1 ,z0 ] As a result, there is a feedback control φ and a T > satisfying zφ (T, z0 , y0 ) = z1 and yφ (t, z0 , y0 ) = y0 for all ≤ t ≤ T Claim For anFor any < z1 < z0 < z ∗ , we have D∗ (1−z1 )(βz1 +1)−Ez1 ≥ 0, which implies inf y∈(0,∞) {h(z1 , y)} ≥ Thus, we cannot find a control φ and a T > satisfying zφ (T, z0 , y) = z1 Similarly, if z1 > max{z0 , 1}, we cannot find a control φ and a T > satisfying zφ (T, z0 , y) = z1 Claim For an It can be seen that there is z1∗ ∈ (z ∗ , 1) satisfying g(z1∗ , 1) = and that the equilibrium (z1∗ , 1) of the system z˙ = g(z, y), y˙ = y(b − by), (2.9) is a sink By the stable manifold theorem (see [28, p 107]), for any δ > 0, (z1∗ , 1) has a neighborhood Sδ ⊂ (z1∗ − δ, z1∗ + δ) × (1 − δ, + δ) which is invariant under (2.9) Let (˜ z (t, z, y), y˜(t, z, y)) be the solution to (2.9) with initial value (z, y) With the feedback control φ satisfying a− α2 + D∗ β(˜ z (t, z, y) − 1)) + αφ(t) = b for all t ≥ 0, we have (zφ (t, z, y), yφ (t, z, y)) = (˜ z (t, z, y), y˜(t, z, y)) for all t ≥ As a result, (zφ (t, z, y), yφ (t, z, y) ∈ Sδ for all (z, y) ∈ Sδ for any t ≥ with this control Claim For an For any z > and δ > 0, there is a T > satisfying zφ (T, z, 0) ∈ (z ∗ − δ, z ∗ + δ) and yφ (T, z, 0) = Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php PROTECTION ZONES IN RANDOM ENVIRONMENT 1389 Using the discussion above enables us to 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 2.1 Let (Z(t), Y (t)) be the solution to (2.2) and z ∗ be given by (2.5) Suppose that λ := a − α2 + D∗ β(z ∗ − 1) > Then we have the following (i) The process (Z(t), Y (t)) has a unique invariant probability measure π ∗ whose support is [z ∗ , 1] × (0, ∞) (ii) There exists γ > and a function H(z, y) : R2,◦ + → R+ such that P (t, z, y, ·) − π ∗ (·) ≤ H(z, y)e−γt for all t ≥ 0, (2.10) where · is the total variation norm (iii) Moreover, for any π ∗ -integrable function f , and (z, y) ∈ R2,◦ + we have (2.11) P t→∞ t t lim f Z z,y (s), Y z,y (s) ds = f (u, v)π ∗ (du, dv) = R2 To proceed, we first recall some technical concepts and results in [23, 24] Let X be a locally compact and separable metric space, and B(X) be the Borel σ-algebra on X Let Φ = {Φt : t ≥ 0} be a homogeneous Markov process with state space (X, B(X)) and transition semigroup P(t, x, ·) We can consider the process Φ on a probability space (Ω, F , {Px }x∈X ), where the measure Px satisfies Px (Φt ∈ A) = P(t, x, A) for all x ∈ X, t ≥ 0, A ∈ B(X) Suppose further that Φ is a Feller process For a probability measure a on R+ , we define a sampled Markov transition function Ka of Φ by ∞ Ka (x, B) = P(t, x, B)a(dt) Ka is said to possess a nowhere-trivial continuous component if there is a kernel T : (X, B(X)) → R+ satisfying • for each B ∈ B(X), the function T (·, B) is lower semicontinuous; • for any x ∈ X, T (x, ·) is a nontrivial measure satisfying Ka (x, B) ≥ T (x, B) for all B ∈ B(X) Φ is called a T-process if for some probability measure a, the corresponding transition function Ka admits a nowhere-trivial continuous component A subset A ∈ B(X) is said to be petite for the δ-skeleton chain {Φnδ , n ∈ N} of Φ if there is a probability measure a on N and a nontrivial measure ψ(·) on X such that ∞ P(nδ, x, B)a(n) ≥ ψ(B) for all x ∈ A, B ∈ B(X) Ka (x, B) := n=1 The following theorem is extracted from [23, Theorem 8.1] and [24, Theorem 6.1] Theorem 2.2 Suppose that Φ is a T -process with generator A The following assertions hold If Φ is bounded in probability on average, that is, for any x ∈ X and ε > 0, t there is a compact set Cε,x satisfying lim inf t→∞ 1t P(t, x, Cε,x ) > − ε If all compact sets are petite for some skeleton chain and if there exists a positive function V (·) : X → R+ , and positive constants c, d such that V (x) → ∞ as x → ∞ and that AV (x) ≤ −cV (x) + d for all x ∈ X, then there exists an invariant probability measure π, positive constants b1 , b2 such that P(t, x, ·) − π(·) ≤ b1 (V (x) + 1) exp(−b2 t) for all x ∈ X Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1390 N T DIEU, N H DU, H D NGUYEN, AND G YIN Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php To apply Theorem 2.2 to our process (Z(t), Y (t)), we need the following lemma Lemma 2.2 The solution (Z(t), Y (t)) to (2.2) is a T -process Moreover, every compact set K ⊂ R2,◦ + is petite for the Markov chain (Z(n), Y (n)) (n ∈ N) Proof Recall from Lemma 2.1 that the transition probability P (t, z, y, ·) of (Z(t), Y (t)) has a smooth density function Hence, it is readily proved that the resolvent kernel (a special case of sampled transition kernel) ∞ R1 (z, y, A) := e−t P (t, z, y, A)dt is a continuous function in (z, y) for each measurable subset A ⊂ R2,◦ + As a result, (Z(t), Y (t)) is a T -process To prove the latter statement, let the point (z1∗ , 1) be as in Claim Since ∗ (z , 1) × (0, ∞) is invariant under (2.2), we have P (1, z1∗ , 1, (z ∗ , 1) × (0, ∞)) = then p(1, z1∗ , 1, z2 , y2 ) > for some (z2 , y2 ) ∈ (z ∗ , 1) × (0, ∞) In view of Claim and the smoothness of p(1, ·, ·, ·, ·), there exist a neighborhood Sδ (z1∗ , 1) that is invariant under (2.9), and an open set G (z2 , y2 ) such that (2.12) p(1, z, y, z , y ) ≥ m > for all (z, y) ∈ Sδ , (z , y ) ∈ G For any (z, y) ∈ K, we derive from Claims 1–3 that there is a T > 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 [15, Theorem 8.1, p 518]) P (nz,y , z, y, Sδ ) := 2ρz,y > Since (Z(t), Y (t)) is a Markov–Feller process, there exists an open set Vz,y (z, y) such that P (nz,y , z , y , Sδ ) ≥ ρx,y for all (z , y ) ∈ Vz,y Since K is a compact set, l there is a finite number of Vzi ,yi , i = 1, , l, satisfying K ⊂ i=1 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 (2.13) From (2.12) and (2.13), for all (z, y) ∈ K there exists nzi ,yi such that (2.14) p(nzi ,yi + 1, z, y, z , y ) ≥ ρK m for all (z , y ) ∈ G Define the kernel K(z, y, Q) := l l P (nzi ,yi + 1, z, y, Q) for all Q ∈ B(R2,◦ + ) i=1 We derive from (2.14) that (2.15) K(z, y, Q) ≥ ρK m μ(G ∩ Q) for all Q ∈ B(R2,◦ + ), l where μ(·) is the Lebesgue measure on R2,◦ + Equation (2.15) means that every compact 2,◦ set K ⊂ R+ is petite for the Markov chain (Z(n), Y (n)) Copyright © by SIAM Unauthorized reproduction of this article is prohibited PROTECTION ZONES IN RANDOM ENVIRONMENT 1391 Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Proof of Theorem 2.1 Since a − α2 + D∗ β(z ∗ − 1) > 0, there exist q, δ ∗ ∈ (0, z ∗ ) such that α2 + D∗ β(z ∗ − − δ ∗ ) > a − (q + 1) First, we consider (2.7) in the invariant set M = {z ∗ − δ ∗ ≤ z ≤ 1, y > 0} Denote by L the generator of the diffusion corresponding to (2.2) Letting U (z, y) = y −q + y + 1, we have lim U (z, y) = lim U (z, y) = ∞ y→∞ y→0 and (2.16) LU (z, y) = −qy −q a − (q + 1) α2 + D∗ β(z ∗ − − δ ∗ ) − qD∗ β(z − z ∗ + δ ∗ )y −q + qby 1−q + y(a − by + D∗ β(z − 1)) ≤ −qy −q a − (q + 1) α2 + D∗ β(z ∗ − − δ ∗ ) + qby 1−q + y(a − by) ≤ −θ1 (y −q + y) + θ2 ≤ −θ1 U (z, y) + θ2 for all (z, y) ∈ M, where α2 + D∗ β(z ∗ − − δ ∗ ) > 0, and θ2 = sup{−θ1 y −q + qby 1−q + y(a − by) + θ1 y} < ∞ θ1 = q a − (q + 1) y>0 Similarly, we can estimate (2.17) LU (z, y) ≤ θ3 U (z, y) for all (z, y) ∈ R2,◦ + for some θ3 > By Theorem 2.2, we derive from Lemma 2.2 and (2.16) that the Markov–Feller process (Z(t), Y (t)) has a unique invariant probability measure π ∗ in M satisfying (2.18) P (t, z, y, ·) − π ∗ (·) ≤ H0 (y −q + y + 1)e−γt for all t ≥ 0, (z, y) ∈ M Moreover, in light of the support theorem or [18, Lemma 4.1], we obtain from Claims 1– that the support of π ∗ is [z ∗ , 1] × (0, ∞) In view of (2.17) and standard arguments (see, for example, [17, Theorem 3.5, p 75]), there are H1 , γ1 > such that (2.19) EU (Z z,y (t), Y z,y (t)) ≤ H1 U (z, y)eγ1 t for all t > and (z, y) ∈ R2,◦ + In view of (2.3) and (2.6), for any (z0 , y0 ) ∈ R2,◦ + , there is a nonrandom moment t0 = t0 (z0 , y0 ) > such that (Z z,y (t), Y z,y (t)) ∈ M for all t ≥ t0 with probability Thus, we have from (2.18) and (2.19) the following estimate, P (t + t0 , z0 , y0 , ·) − π ∗ (·) ≤ ≤ M M p(t0 , z0 , y0 , z, y) P (t, z, y, ·) − π ∗ (·) dzdy p(t0 , z0 , y0 , z, y)H0 (y −q + y + 1)e−γt dzdy = H0 e−γt EU (Z z0 ,y0 (t0 ), Y z0 ,y0 (t0 )) ≤ H0 H1 eγ1 t0 U (z0 , y0 )e−γt for all t ≥ 0, Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1392 N T DIEU, N H DU, H D NGUYEN, AND G YIN which proves (2.10) Finally, the strong law of large numbers (2.11) is derived from part of Theorem 2.2 since the bounded in probability on average follows from the convergence in total variation norm Next, we give conditions for the extinction of the population densities in both the protection zone and the natural environment Theorem 2.3 Let (Z z,y (t), Y z,y (t)) be the solution to (2.2) with the initial condition (z, y) ∈ R2,◦ + If λ := a − α2 + D∗ β(z ∗ − 1) < 0, then (Z z0 ,y0 (t), Y z0 ,y0 (t)) → (z ∗ , 0) a.s., as t → ∞ for all (z0 , y0 ) ∈ R2,◦ + , that is, the species will be extinct in the sense that z0 ,y0 (t) = limt→∞ Y z0 ,y0 (t) = a.s Moreover, for any (z0 , y0 ) ∈ R2,◦ limt→∞ X + , we have with probability that (2.20) ln X z0 ,y0 (t) ln Y z0 ,y0 (t) = lim = λ < t→∞ t→∞ t t lim Proof We prove the assertions in the following steps (i) By using the Lyapunov function method, we can show that the equilibrium (z ∗ , 0) is asymptotically stable in probability z0 ,y0 (ii) For any (z0 , y0 ) ∈ R2,◦ (t), Y z0 ,y0 (t)) is recurrent relative + , the process (Z z∗ to [ , 1] × [0, H] For the control system, given δ > 0, there exists a T > ∗ such that for any (z, y) ∈ [ z2 , 1] × [0, H], there exists a control φ satisfying (zφ (t, z, y), yφ (t, z, y)) ∈ (z ∗ − δ, z ∗ + δ) × [0, δ) for some t ∈ [0, T ] (iii) Using the 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 > such that (2.21) P lim (Z z,y (t), Y z,y (t)) = (z ∗ , 0) ≥ − ε for all (z, y) ∈ (z ∗ − δ, z ∗ + δ) × [0, δ) t→∞ When y = 0, (2.21) is clearly true We need only consider (2.21) 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 > ∂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 , where o lim z→z ∗ y→0 (z − z ∗ )2 + y (z − z ∗ )2 + y = Copyright © by SIAM Unauthorized reproduction of this article is prohibited PROTECTION ZONES IN RANDOM ENVIRONMENT Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Otherwise, since a − α2 1393 + D∗ β(z ∗ − 1) < 0, −c3 := a − (1 − p) α2 + D∗ β(z ∗ − 1) < for sufficiently small p > 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 LV (z, y) = ⎡ ∂V (z, y) , ∂z ∂ V (z, y) ⎢ ∂V (z, y) f1 (z, y) ∂z 0, αy ⎢ + ⎣ ∂ V (z, y) f2 (z, y) ∂y ∂y∂z = 2(z − z ∗ ) −c1 (z − z ∗ ) + c2 y + o + py p a − by − (1 − p) ≤ −c1 (z − z ∗ )2 + ⎤ ∂ V (z, y) ∂z∂y ⎥ ⎥ ∂ V (z, y) ⎦ αy ∂y (z − z ∗ )2 + y α2 + D∗ β(z − 1) c22 y + o[(z − z ∗ )2 + y ] − pc3 y p + py p D∗ β(z − z ∗ ) c1 Since y = o(y p ) for small y, when y + (z − z ∗ )2 is small, we have 2c3 py p c22 , y − pc1 y p + py p D∗ β(z − z ∗ ) ≤ − c1 2c1 c3 py p o[(z − z ∗ )2 + y ] ≤ (z − z ∗ )2 + 3 Therefore, LV (z, y) ≤ − [c1 (z − z ∗ )2 + c3 py p ] ≤ −θ4 V (z, y) for all (z, y) ∈ Nδ for some θ4 > and sufficiently small δ By [22, Theorem 2.3 p 112], for any ε > 0, there is a δ > such that (2.22) P lim (Z z,y (t), Y z,y (t)) = (z ∗ , 0) ≥ − ε for all (z, y) ∈ Nδ t→∞ Next, we derive item (ii) Let ϕ(t) be the solution to the following equation (2.23) dϕ(t) = ϕ(t) a + α2 − bϕ(t) dt + αϕ(t)dW (t) If Z(0) = z0 ∈ (0, ∞), there is a t0 > such that Z(t) ≤ for all t > t0 ; it is easy to check that Y (t) ≤ ϕ(t) for all t ≥ t0 a.s provided that Y (t0 ) = ϕ(t0 ) > by the comparison theorem [15, Theorem 1.1, p 352] In view of [9], ϕ(t) has a unique stationary distribution μ∗ (·) which is a gamma distribution with parameters α := α2a2 and β := α2b2 That is, μ∗ (·) has the density φ∗ (x) = αβ α−1 x exp{−βx}, x > 0, Γ(α) Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1394 N T DIEU, N H DU, H D NGUYEN, AND G YIN Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php where Γ(·) is the gamma function By the strong law of large-number-type result [29, Theorem 3.16, p 46], we deduce that t t→∞ t (2.24) ∞ ϕ(s)ds = lim xα e−βx dx = α := K1 a.s β Consequently, (2.25) lim sup t→∞ t t Y (s)ds ≤ K1 , which implies (2.26) lim sup t→∞ t t 1{Y (s)≥H} ds ≤ 1 lim sup H t→∞ t t Y (s)ds ≤ K1 H For any initial condition (z0 , y0 ) ∈ R2,◦ + Let H > K1 , from (2.26) we have (2.27) lim inf t→∞ t t 1{Y z0 ,y0 (s)∈[0,H]} ds ≥ − K1 > a.s H ∗ In view of (2.6) and (2.27), (Z z0 ,y0 (t), Y z0 ,y0 (t)) is recurrent relative to [ z2 , 1]× [0, H] ∗ It follows from Claims 1–6, that for each (z, y) ∈ [ z2 , 1]×[0, H], we can choose a control φ(·) and Tz,y > such that (zφ (Tz,y , z, y), yφ (Tz,y , z, y)) ∈ Uδ ∗ In view of the support theorem, for all (z, y) ∈ [ z2 , 1] × [0, H], there is a Tz,y > such that P {(Z z,y (Tz,y ), Y z,y (Tz,y )) ∈ Uδ } > 2pz,y > 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 ∗ Because [ z2 , 1]×[0, H] is a compact set, there are a finite number of Vzi ,yi , i = 1, , n, ∗ n such that [ z2 , 1] × [0, H] ⊂ i=1 Vzi ,yi Put T ∗ = max{Tzi ,yi , i = 1, , n, }, p∗ = min{pzi ,yi , i = 1, , n} For (z, y) ∈ (0, ∞) × (0, ∞)), set τδz,y = inf{t > : (Z z,y (t), Y z,y (t)) ∈ Uδ } Then (2.28) P{τδz,y < T ∗ } ≥ p∗ > for all (z; y) ∈ z∗ , × [0, H] ∗ Moreover, since (Z z0 ,y0 (t), Y z0 ,y0 (t)) is recurrent relative to [ z2 , 1] × [0, H], we can define a sequence of finite stopping times η0 = 0, ηk = inf t > ηk−1 + T ∗ : (Z z0 ,y0 (t), Y z0 ,y0 (t)) ∈ z∗ , × [0, H] , k ∈ N Copyright © by SIAM Unauthorized reproduction of this article is prohibited PROTECTION ZONES IN RANDOM ENVIRONMENT 1395 Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Consider the events / Uδ for all t ∈ [ηk , ηk + T ∗ ]} Ak = {(Z z0 ,y0 (t), Y z0 ,y0 (t)) ∈ We deduce from the strong Markov property of (Z(t), Y (t)) and (2.28) that n P ≤ (1 − p∗ )n → as n → ∞ Ak k=1 As a result, P{τδz0 ,y0 < ∞} = (2.29) In light of the strong Markov property of (Z(t), Y (t)), (2.22) and (2.29) yield P (2.30) lim (Z z0 ,y0 (t), Y z0 ,y0 (t)) = (z ∗ , 0) ≥ − ε t→∞ Since ε can be taken arbitrarily, we obtain (2.31) P lim (Z z0 ,y0 (t), Y z0 ,y0 (t)) = (z ∗ , 0) = for all (z0 , y0 ) ∈ R2,◦ + t→∞ Finally, it follows from Itˆo’s formula that (2.32) ln y0 t ln Y z0 ,y0 (t) α2 W (t) = + + D∗ β(Z z0 ,y0 (s) − 1) − bY z0 ,y0 (s) ds + a− t t t t Hence (2.20) follows from (2.31) and (2.32) Discussion and numerical examples Now, we fix all the coefficients of (1.3) except β We wish to answer the question, for what values of β does the species survive permanently? First, consider the case a − α2 − E > We obtain that even without a protection zone β = ∞, the species will survive permanently If a − α2 − E = for any β > 0, the inequality a− α2 + D∗ β(z ∗ − 1) > holds Hence the species will survive if there is a protection zone even if its area is small On the other hand, if a − α2 ≤ 0, the species will die out even it is completely protected We therefore focus on the case < a − α2 < E We aim to find the threshold β ∗ such that the species will survive permanently if β < β ∗ while it reaches extinction if β > β ∗ As a result of Theorems 2.1 and 2.3, β ∗ will be the root of λ = or, equivalently, ⎞ ⎛ ∗ β − D ∗ − E)2 + 4D ∗ β + (D ∗ β − D ∗ − E) (D α + D∗ β ⎝ − 1⎠ = a− 2D∗ β It can be shown that if < a − α2 < E the equation above has a unique positive root β∗ = (2D∗ + 2E + α2 − 2a)(2a − α2 ) 2D∗ (2E + α2 − 2a) Copyright © by SIAM Unauthorized reproduction of this article is prohibited 1396 N T DIEU, N H DU, H D NGUYEN, AND G YIN Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Moreover, α2 + D∗ β(z ∗ − 1) > if and only if β < β ∗ , which answers the aforementioned question Intuitively, if β is less than but very close to β ∗ , in view of the ergodicity of (Z(t), Y (t)) (see (2.11)), we can show t t that limt→∞ 1t Y (s)ds is small, so is limt→∞ 1t X(s)ds To guarantee that t t limt→∞ 1t Y (s)ds and limt→∞ 1t X(s)ds are not too small, we need β to be con∗ siderably less than β Let us compare our results with the deterministic case By solving an irrational inequality, it is seen that Theorem 1.1 is equivalent to the claim that if a+D∗ β(z ∗ −1) < 0, the species will reach extinction and if a + D∗ β(z ∗ − 1) > 0, the solution to (1.1) will converge to a positive stable equilibrium When the noise coefficient α = 0, our result reduced to that of Theorem 1.1 Moreover, it can be seen from the condition for permanence (a − α2 + D∗ β(z ∗ − 1) > 0) that the random noise is detrimental to the survival of the species Hence, in order to protect the species, we need a larger protection zone for the stochastic model than for the deterministic counterpart It should also mentioned that the unfavorable effect of random noise and periodically fluctuating habitat to discrete population dynamics has been shown in [3, 7, 6] We also refer to [8] for some other interesting influences of random noise to the dynamical behaviors of species in the discrete setting In this paper, we suppose that the protected and unprotected areas are subject to the same environmental noise because of the closeness in distance between them However, in many situations, they are far away from each other which is especially the case for migratory species such as birds, fish, and marine mammals To model this fact, it should be assumed that X(t) and Y (t) are driven by independent Brownian motions W1 (t), W2 (t) as follows: (3.1) ⎧ D ⎪ ⎪dX(t) = X(t)(a − bX(t)) − (X(t) − Y (t)) − EX(t) dt + α1 X(t)dW1 (t), ⎨ H ⎪ D ⎪ ⎩dY (t) = Y (t)(a − bY (t)) + (X(t) − Y (t)) dt + α2 Y (t)dW2 (t) h a− By setting Z(t) = X(t) Y (t) , we obtain (3.2) ⎧ ∗ ⎪ ⎨dZ(t) = bY (t)Z(t)(1 − Z(t)) + D (1 − Z(t))(βZ(t) + 1) + (α2 − E)Z(t) dt + α1 Z(t)dW1 (t) − α2 Z(t)dW2 (t), ⎪ ⎩ dY (t) = Y (t) [a − bY (t) + D∗ β(Z(t) − 1)] dt + α2 Y (t)dW2 (t) Recall that to treat (2.2), we consider (2.4), which is the restriction of (2.2) on the boundary {(z, y) : y = 0, z > 0} Similarly, to determine the threshold of permanence for (3.2), we consider the equation on the boundary {(z, y) : y = 0, z > 0}, namely, (3.3) ˜ ˜ ˜ ˜ + 1) + (α2 − E)Z(t) ˜ ˜ = D∗ (1 − Z(t))(β Z(t) dt + α1 Z(t)dW dZ(t) (t) − α2 Z(t)dW2 (t) Note that (2.4) is a deterministic equation having a globally asymptotic equilibrium z ∗ On the other hand, (3.3) is a stochastic one with an invariant probability π ˜ whose density can be calculated from the Fokker–Planck equation This invariant probability measure plays the same role for (3.2) as z ∗ for (2.2) Hence, the conditions for permanence and extinction of (3.2) can be stated as follows Copyright © by SIAM Unauthorized reproduction of this article is prohibited PROTECTION ZONES IN RANDOM ENVIRONMENT 1397 Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php ∞ Theorem 3.1 Let z˜∗ = z π ˜ (dz) We have the following α2 ∗ ∗ ˜ z − 1) < then all positive solutions to (3.2) tend to • If λ := a − + D β(˜ the origin ˜ := a− α2 +D∗ β(˜ • If λ z ∗ −1) > then (3.2) has a invariant probability measure 2,◦ in R+ which is the limit in total variation of the transition probability It is clear that treating (3.3) is much more difficult than (2.4) Hence different techniques, improvements, and modifications are needed to facilitate the proof of Theorem 3.1 However as far as establishing ergodicity is concerned, the ideas used in this paper can still be utilized We provide some numerical comparison between (2.2) and (3.2) in Examples 3.3 and 3.4 Realizing the fact that a species may not be protected if its population is too low, many models with the “Allee effect” have been proposed (see, e.g., [4, 5]) Recall that the Allee effect is a phenomenon in biology characterized by a correlation between population size or density and the mean individual fitness of a population or species Such models have a property that small positive initial conditions lead to extinction while larger initial conditions result in the convergence to a positive equilibrium To examine the impacts of the Allee effect on the stochastic systems is both interesting and important However, adding the Allee effect makes the systems more challenging to investigate The detailed study requires much more careful thought Example 3.1 Consider (2.2) with parameters a = 5.5, β = 1, b = 3.5, α = 1.95, D∗ = 0.75, and E = 0.6 Direct calculation shows that z ∗ = 0.677 We obtain λ=a− α2 + D∗ β(z ∗ − 1) = 3.3565 > By virtue of Theorem 2.1, (2.2) has a unique invariant probability measure μ∗ whose support is [0.677, 1] × R+ Consequently, the strong law of large numbers and the convergence in total variation norm of the transition probability hold In fact, we not necessarily need a protection zone in this case since the solution to the model without protection zone, dX(t) = X(t)(a − bX(t) − E)dt + αX(t)dW (t), will not tend to when a − α2 > E as in this example However, the protection zone clearly increases the density of the population which can be shown by the comparison theorem for stochastic differential equations To provide better visualization of the process and its long-term behavior, we simulate the sample paths of (2.2) by numerically solving the pair of differential equations for (Z(t), Y (t)) with a small step size Δ (Δ = 0.0025) and a large number of steps N (N = × 106 ) using the well-known Euler–Maruyama method Denoting the numerical solutions at step i by (Zi , Yi ), then we divide the space (in fact, an appropriate subset [0, H] × [0, K] ⊂ R2+ is chosen rather than the whole space) to cells Ahk each of which has the same area A Then N we form Fhk := N1A i=1 1{(Zi ,Yi )∈Ahk } Interpolating {Fhk }, we obtain a function: F : [0, H] × [0, K] → R+ , which approximates the density of the invariant measure A sample path of solution to (2.2) is depicted in Figure 1, while the density function of an empirical measure of (Z(t), Y (t)) in time interval [0, 104 ] is shown in Figure In light of Theorem 2.1, the empirical measure will converge to μ∗ Example 3.2 Consider (2.2) with parameters a = 3, β = 6, b = 4, α = 1, D∗ = 4, and E = Direct calculation shows that z ∗ = 0.89463 and β ∗ = 5.625 < β We obtain λ = −0.0288 < In view of Theorem 2.3, (Z(t), Y (t)) → (z ∗ , 0) a.s as t → ∞ To protect the species, we need to increase the area of the protection zone, Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1398 N T DIEU, N H DU, H D NGUYEN, AND G YIN Fig Trajectories of Y (t), Z(t) in Example 3.1, respectively Fig The density function of an empirical measure of (Z(t), Y (t)) with time interval [0, 104 ] in Example 3.1 in 2-dimensional (2D) and 3-dimensional (3D) settings, respectively Fig The left figure shows the trajectory of Y (t) in Example 3.2 with β = It can be seen that Y (t) converges to In contrast, as shown in the right figure, the trajectory of Y (t) with β = does not converge to that is, reducing β to a number below β ∗ For instance, we take β = Figure illustrates two cases β = and β = 4, respectively Example 3.3 Consider (3.2) with the same parameters as in Example 3.1 For ˜ = 4.0915 > 0, so the species is permanent Similarly to Exthis set of parameters, λ ample 3.1, we simulate the sample paths of the two-component solution process The density function of a long-term empirical measure, which approximates the stationary density of (3.2), is shown in Figure Unlike (2.2) that is driven by only one Brownian motion, two sources of noise in (3.2) can push the dynamics in any direction Because of this nondegeneracy, the invariant density of (3.2) is more spread out than that of (2.2) Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php PROTECTION ZONES IN RANDOM ENVIRONMENT 1399 Fig The density function of an empirical measure in time interval [0, 104 ] of (3.2) in Example 3.3 in 2D and 3D settings, respectively Fig Trajectories of Y (t) of (2.2) and (3.2), respectively, in Example 3.4 Example 3.4 Consider both (2.2) and (3.2) with the same parameters a = 1.6, β = 4, b = 1, α = α1 = α2 = 1, D∗ = 1, E = We obtain that λ = −0.1639 < ˜ = 0.5476 > The population of (3.2) is permanent, while extinction takes while λ place in (2.2) as can be seen in Figure Further remarks It is recognized that protection zones provide many economic, social, environmental, and cultural values Aiming at conservation of biodiversity and stemming from designing 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 the necessary one 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 As an example, let us use (1.3) to illustrate the implication of our results to population dynamics of fisheries with a prohibited fishing area In such a case, the two equations in (1.3) describe the populations in the fishing area and the prohibited area, respectively, in which the coefficient E indicates the harvesting rate The threshold value λ obtained in this paper provides insight into how the prohibited zone and the harvesting rate impact the fish populations This in turn will help us to design a sustainable strategy for constructing the prohibited area as well as controlling the harvesting rate From another angle, our model is a special case of a spatially heterogeneous environment consisting of n patches Introducing and analyzing a stochastic patchy model, the authors in [12] aimed to answer questions about the interactive influ- Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 1400 N T DIEU, N H DU, H D NGUYEN, AND G YIN ence of random fluctuations and dispersal rates to the growth rate of the population However, the linear system model in [12] does not take into account the intrinsic competition As discussed in [12], extending the analysis in [12] to a model involving intrinsic competition may provide important understanding to the evolution of species in a heterogeneous environment System (1.3) is a special case of multiplepatch models with intrinsic competition coefficient b The method developed in this paper may open new avenues for future study to characterize the permanence and extinction of stochastic heterogeneous population models, so as to contribute to deeper understanding of the interactive effects of stochasticity and spatial heterogeneity to population dynamics, which is a central issue in population dynamics Moreover, for future study, a number of questions are of particular interests from both practical and theoretical points of view • As discussed in section 3, the nondegenerate model (3.1) is worth studying carefully Taking into account the Allee effect should also be done in the future • It is natural to consider controlled systems with protection zones Treating it as an optimal control problem, 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 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 (4.1) ⎧ 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), ⎪ ⎪ 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 > The Markov chain models the random environment 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 brought in to analyze the protection zones • Another problem of considerable interest is to treat ecosystems 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 predatorprey model in which there is a region which only the prey can access to avoid predation Copyright © by SIAM Unauthorized reproduction of this article is prohibited PROTECTION ZONES IN RANDOM ENVIRONMENT 1401 Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php All of these questions deserve careful consideration and open up a new domain for further study Acknowledgment We are very grateful to the editors and reviewers for evaluating our manuscript and for the constructive comments and suggestions, which have led to much improvement in the presentation REFERENCES [1] L R Bellet, Ergodic properties of Markov processes, in Open Quantum Systems II, Springer Berlin, (2006), pp 1–39 [2] E Beretta and 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), pp 607–626 [3] P H Bezandry, T Diagana, and S Elaydi, On the stochastic Beverton-Holt equation with survival rates, J Difference Equ Appl., 14 (2008), pp 175–190 [4] F Courchamp, L Berec, and J Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, Oxford, UK, 2008 [5] J M Cushing, The evolutionary dynamics of a population model with a strong Allee effect, Math Biosci Eng., 12 (2015), pp 643–660 [6] J M Cushing and S M Henson, A periodically forced Beverton-Holt equation, J Difference Equ Appl., (2002), pp 1119–1120 [7] J M Cushing and S M Henson, Global dynamics of some periodically forced, monotone difference equations, J Difference Equ Appl., (2001), pp 859–872 [8] B Dennis, R A Desharnais, J M Cushing, S M Henson, and R F Costantino, Estimating chaos and complex dynamics in an insect population, Ecol Monogr., 71 (2001), pp 277–303 [9] N H Du, D H Nguyen, and G Yin, Conditions for permanence and ergodicity of certain stochastic predator-prey models, J Appl Probab., 53 (2016), pp 187–202 [10] Y Du and X Liang, A diffusive competition model with a protection zone, J Differential Equations, 244 (2008), pp 61–86 [11] Y Du, R Peng, and M Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J Differential Equations, 246 (2009), pp 3932–3956 [12] S Evans, P Ralph, S Schreiber, and A Sen, Stochastic population growth in spatially heterogeneous environments, J Math Biol., 66 (2013), pp 423–476 [13] S Evans, A Hening, and S Schreiber, Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments, J Math Biol., 71 (2015), pp 325– 359 [14] P Hinow, F Le Foll, P Magal, and G F Webb, Analysis of a model for transfer phenomena in biological populations, SIAM J Appl Math., 70 (2009), pp 40–62 [15] N Ikeda and S Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd ed., North-Holland, Amsterdam, 1989 [16] T K Kar and M Swarnakamal, Influence of prey reserve in a prey-predator fishery, Nonlinear Anal., 65 (2006), pp 1725–1735 [17] R Z Khas’minskii, Stochastic Stability of Differential Equations, Springer-Verlag, Berlin, 2012 [18] W Kliemann, Recurrence and invariant measures for degenerate diffusions, Ann Probab., 15 (1987), pp 690–707 [19] R Kon, Age-structured Lotka–Volterra equations for multiple semelparous populations, SIAM J Appl Math., 71 (2011), pp 694–713 [20] S Liu and E Beretta, A stage-structured predator-prey model of Beddington-DeAngelis type, SIAM J Appl Math., 66 (2006), pp 1101–1129 [21] J Llibre and D Xiao, Global dynamics of a Lotka–Volterra model with two predators competing for one prey, SIAM J Appl Math., 74 (2014), pp 434–453 [22] X Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing, Chichester, 1997 [23] S P Meyn and R L Tweedie, Stability of Markovian processes II: Continuous-time processes and sampled chains, Adv Appl Prob., 25 (1993), pp 487–517 [24] S P Meyn and R L Tweedie, Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv Appl Prob., 25 (1993), pp 518–548 Copyright © by SIAM Unauthorized reproduction of this article is prohibited Downloaded 07/26/16 to 128.252.67.66 Redistribution subject to SIAM license or copyright; 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