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

re-Keywords Biodiversity; protection zone; extinction; permanence; ergodicity

Subject Classification 34C12, 60H10, 92D25

∗ Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam, guyen2008@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.

dieun-† 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) n 0 101.03-2014.58.

‡ Department of Mathematics, Wayne State University, Detroit, MI 48202, USA, gnh.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.

dan-§ 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.

ap-of protection zones in renewing biological resources and protect the population in both ministic and stochastic models; see [3, 4, 5, 10, 31, 32] and references therein The main idea

deter-of their work can be described as follows The region eΩ, where the species live in, is dividedinto two sub-regions eΩ1 and eΩ2 The sub-region eΩ1 is the nature environment and eΩ2 is thenature reserve The population densities between eΩ1 and eΩ2are notably different Migrationcan occur between eΩ1 and eΩ2, which is assumed to be proportional to the difference of thedensities with the proportional constant being D > 0 Denote the densities of population

in eΩ1 and eΩ2 by X(t) and Y (t), respectively Assume that the size of eΩ1 is H and the size

of eΩ2 is h Use D(X(t) − Y (t)) to represent the diffusing capacity that is the total biomasscaused by diffusion effect In the deterministic cases, this model can be formulated as

unfavorable factors of biological growth relative to the biological growth in the protection

environment is always subject to random effect, so it is important to take into account theimpact of stochastic perturbations on the evolution of the species In the literature, Zou andWang in [31] considered the following stochastic model for a single species with protection

where a, b, D, H, h and α are appropriate constants, and W (·) is a standard real-valued

a, b, D∗, α, E such that if β < β∗ the species will survive while it will reach extinction in case

β > β∗ In [31], it is proved that for any initial value (X(0), Y (0)) ∈ R2,◦+ (the interior of R2+),there exists a unique global solution to (1.3) that remains in R2,◦+ almost surely Althoughthey 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

provides a sufficient and almost necessary condition for the size of the protection, for a moregeneral 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, theconvergence 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

(a) We are dealing with a case of fully degenerate diffusions, which allows correlations ofthe 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 aresharp in that not only are the conditions obtained sufficient, but also they are almostnecessary

(c) Then we go a step further than the existing results in the literature by investing theergodicity of the systems under consideration First, we give a sufficient conditionfor the ergodicity Our result establishes the existence of the invariant probabilitymeasure In addition, it describes precisely the support of the invariant probabilitymeasure Second, we prove that the convergence in total variation to the invariantmeasure Moreover, precise exponential upper bounds are obtained Finally, a stronglaw 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 indicatesthat 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 concentra-tion on stability issues [1, 14, 15, 16, 21, 25, 29], there is an increasing effort to take stepstreating systems that involve randomness [2, 6, 20, 28, 30] Along this line, the current paperexamines an important issue from the perspectives of protection zones and biodiversity

convergence in total variation of the transition probability are also proved Moreover, anerror bound of the convergence is provided Section 3 is devoted to some discussion andcomparison to existing results Some numerical examples and figures are also provided toillustrate our results Finally, further remarks are issued in Section 4, which point outpossible future directions for investigations

2 Sufficient and Almost Necessary Conditions for manence

Per-Let (Ω, F , {Ft}t≥0, P) be a complete probability space with a filtration {Ft}t≥0 satisfying theusual condition, i.e., it is increasing and right continuous while F0 contain all P-null sets

a.s provided that Z(0) =z(0) ∈ (0, 1) Note thatb bz(t) → z∗, where

z∗ =

p(D∗β − D∗ − E)2+ 4D∗2β + D∗β − D∗− E



for the diffusion (2.1), that is, dimL0(z, y) = 2 at every (z, y) ∈ R2,◦+ In other words, wewill show that the set of vector fields 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

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 zT denotes the transpose of z), then the Lie bracket [Φ; Ψ] is a vector field givenby

C1(z, y)e

It can be clearly seen that, B(x, y), C(z, y) span R2 for all (z, y) ∈ R2,◦+ satisfying z 6= 1.When z = 1, we have eC1(1, y) = A1(1, y)(−y)(1 − 2) = −Ey 6= 0 hence B(1, y) and eC(1, y)span R2 for all y > 0 As a result, we obtain the following lemma

Lemma 2.1 The H¨ormander condition holds for the solution of (2.1) in R2,◦+ As a result,the transition probability P (t, z0, y0, ·) of (Z(t), Y (t)) has density p(t, z0, y0, z, y) which issmooth in (z0, y0, z, y) ∈ R4,◦+

To proceed, we analyze the following control system corresponding to (2.5).

(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

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 issufficiently small, we have

zφ(T, z0, y) = z1 Similarly, if z1 > max{z0, 1}, we cannot find a control φ and a T > 0satisfying zφ(T, z0, y) = z1.

Claim 5 It can be easily seen that there is z1∗ ∈ (z∗, 1) satisfying g(z∗1, 1) = 0 and that theequilibrium (z∗1, 1) of the system

(˜z(t, z, y), ˜y(t, z, y)) be the solution to (2.7) with initial value (z, y) With the feedbackcontrol φ satisfying

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) andclearly yφ(T, z, 0) = 0

We now provide a condition for the existence of a unique invariant probability measure forthe 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)

where k · k is the total variation norm

(iii) Moreover, for any π∗-integrable function f , and (z, y) ∈ R2,◦+ we have

P

nlim

t→∞

1t

To prove (2.8), we will apply [19, Theorem 6.1] Hence, we need the following lemma.

(n ∈ N), that is, for all compact set K there exists a measure ψ with ψ(R2,◦+ ) > 0 and a bility distribution ν(·) concentrated on N such that the kernel K(z, y, ·) :=P∞

proba-n=1P (n, z, y, ·)ν(n)satisfying K(z, y, Q) ≥ ψ(Q) ∀(z, y) ∈ K, Q ∈ B(R2,◦+ ) We refer to [19] for more details onpetite sets

Proof Let the point (z∗1, 1) be as in Claim 5 Since (z∗, 1) × (0, ∞) is invariant under (2.1),

we have P (1, z∗1, 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 3 (z∗

1, 1), that is invariant under (2.7) and a open set G 3 (z2, y2) such that

p(1, z, y, z0, y0) ≥ m0 > 0 ∀ (z, y) ∈ Sδ, (z0, y0) ∈ 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 3 (z, y) such that

P (nz,y, z0, y0, Sδ) ≥ ρx,y ∀(z0, y0) ∈ Vz,y Since K is a compact set, there is a finite number

We derive from (2.12) that

lρKm

0

where µ(·) is the Lebesgue measure on R2,◦+ (2.13) means that every compact set K ⊂ R2,◦+

is petite for the Markov chain (Z(n), Y (n))

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, wehave

{−θ1y−q+ qby1−q+ y(a − by) + θ1y} < ∞

Similarly, we can estimate

satisfying

Moreover, in light of the support theorem or [13, Lemma 4.1], we can easily imply from Claims1-4 that the support of π∗ is [z∗, 1] × (0, ∞) In view of (2.14) and standard arguments (seee.g [12, Theorem 3.5, p 75]), there is H1, γ1 > 0 such that

In view of (2.2) and (2.4), for any (z0, y0) ∈ R2,◦+ , there is a non-random moment t0 =

t0(z0, y0) > 0 such that (Zz,y(t), Yz,y(t)) ∈ M ∀t ≥ t0 with probability 1 Thus, we havefrom (2.15) and (2.16) the following estimate

Next, we give conditions for the extinction of the population density in both the protectionzone and the natural environment

Theorem 2.4 Let (Zz,y(t), Yz,y(t)) be the solution to equation (2.1) with the initial dition (z, y) ∈ R2,◦+ If a − α

Proof We proceed in the following steps

asymp-totically stable in probability

ii) For any (z0, y0) ∈ R2,◦+ , the process (Zz0 ,y 0(t), Yz0 ,y 0(t)) is recurrent relative to [z

∗

2, 1] ×[0, H] For the control system, given δ > 0, there exists a T > 0 such that for any(z, y) ∈ [z

∗

2, 1] × [0, H], there exists a control φ such that (zφ(t, z, y), yφ(t, z, y)) ∈(z∗− δ, z∗+ δ) × [0, δ) for some t ∈ [0, T ]

iii) Using Markov property of the solution and the support theorem we obtain the desiredconclusion.

First, we prove that for any ε > 0, there exists a δ > 0 such that

limz→z∗

is twice differentiable in (z, y) ∈ R2,◦+ By direct calculation, we have

c1y

2+ o[(z − z∗)2+ y2] − pc3yp+ pypD∗β(z − z∗)

Since y2 = o(yp) for small y, when y2+ (z − z∗)2 is small, we have

c2 2

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

where Γ(·) is the Gamma function By the strong law of large number type result [26,Theorem 3.16, p 46], we deduce that

lim

t→∞

1t

Z t

0

1{Y z0,y0(s)∈[0,H]}ds ≥ 1 − K1

In view of (2.4) and (2.24), (Zz 0 ,y 0(t), Yz 0 ,y 0(t)) recurrent relative to [z

∗

2 , 1] × [0, H] It followsfrom Claims 1-6, that for each (z, y) ∈ [z2∗, 1]×[0, H], we can choose a control φ(·) and Tz,y > 0such that

(zφ(Tz,y, z, y), yφ(Tz,y, z, y)) ∈ Uδ

In view of the support theorem, for all (z, y) ∈ [z

∗

2, 1] × [0, H] there is a Tz,y > 0 such that

P {(Zz,y(Tz,y), Yz,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

n(Zz0,y0(Tz,y), Yz0,y0(Tz,y)) ∈ Uδo> pz,y, for all (z0, y0) ∈ Vz,y.Because [z

η0 = 0, ηk = infnt > ηk−1+ T∗ : (Zz0 ,y 0(t), Yz0 ,y 0(t)) ∈ [z

∗

2, 1] × [0, H]

o, k ∈ N

Consider the events

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

2

therefore focus on the case 0 < a − α

z∗ =

p(D∗β − D∗− E)2+ 4D∗2β + (D∗β − D∗− E)

2D∗βhas a unique positive root

∗+ 2E + α2− 2a)(2a − α2)

in view of the ergodicity of (Z(t), Y (t)) (see (2.9)), we can show that lim

t→∞

1t

Rt

0 Y (s)ds is smalland so is lim

t→∞

1t

Rt

t→∞

1t

Rt

t→∞

1t

2 − D∗β > 0, then the population

in eΩ2 is persistent in mean; in [31, Theorem 4] they show that if a − α

2

2 − E > 0 thenpopulation in eΩ1 is persistent in mean, whereas we only need a −α

Direct calculation shows that z∗ = 0.693000468; β∗ = 5.625 > β We obtain

in total variation norm of the transition probability hold A sample path of solution to (2.1)

is illustrated by Figures 1 It is easy to see that a −α

2

2 − D∗β = −1.5 < 0, it means that thecondition of [31, Theorem 3] is not satisfied but the density population is still persistent

Figure 1: Trajectories of X(t), Y (t), Z(t) in Example 3.1 respectively.

2

∗β(z∗ − 1) = −0.02887823 < 0

For future study, a number of questions are of particular interests from both practicaland theoretical point of view.

• It is natural to consider controlled systems with protection zones One may pose thequestion about what is the minimal size of the protection zone so as to maintain thepermanence 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 jectives so that we can replace the instantaneous probability measures by that of theinvariant measure

ob-• To accommodate the random environment and to take into consideration of continuousdynamics and interactions with the discrete events, we may consider more complexmodels, 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)

where η(t) is a continuous-time Markov chain taking values in a finite set M =