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PROTECTION ZONES FOR SURVIVAL OF SPECIES IN RANDOM

N T DIEU†, N H DU‡, H D NGUYEN§, AND G YIN¶

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

1 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 becomemost popular over the past decades Indeed, the Convention on Biological Diversityrecognizes protected areas as a fundamental tool for safeguarding biodiversity, lifeitself (“Convention on Biological Diversity” is a multilateral treaty, which has threemain goals: conservation of biological diversity or biodiversity, sustainable use of itscomponents, 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 inboth deterministic and stochastic models; see [10, 11, 16, 34, 35, 36] and referencestherein The main idea of their work can be described as follows The region Ω, wherethe 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 tality.

hospi-‡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) 2014.58.

101.03-§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

unprotected environment and Ω2is the protected one Migration can occur between

Ω1and Ω2, which is assumed to be proportional to the difference of the densities with

the proportional constant D > 0 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 beformulated as

b 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 mainingredient, we recall the following theorem obtained in [36]

Theorem 1.1 The following results hold.

(a) If a < H

asymp-totically stable equilibrium of the system (1.1).

(b) If a < H

equilibrium, which attracts all positive solutions of (1.1).

The theorem above provides characterizations of the ecosystems The inequalitiesabove can be viewed as “threshold”-type conditions, which give a precise description

on the asymptotic behavior of different equilibria Statement (a) indicates that if thearea of the protection region satisfies the given inequality, the population will reachextinction, 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 els It is, however, well recognized that the environment is always subject to randomdisturbances, so it is important to take the impact of stochastic perturbations on theevolution of the species into consideration An immediate question is, can we stillcharacterize the protection zone so as to delineate the conditions for permanence andextinction similar to Theorem 1.1? In addition, how can we characterize the equi-librium or steady state behavior of the ecosystems? For stochastic systems, becauserandomness is involved, in addition to equilibria, stationary distributions also comeinto play We need to answer the question, under what conditions is there a sta-tionary distribution The situation becomes more complex Our main objectives andcontributions of this paper are to provide conditions similar to Theorem 1.1 so as tocharacterize the qualitative properties protection regions In fact, we obtain sufficientconditions that are close to necessary for permanence and extinction Furthermore,

mod-we also investigate the convergence and rates of convergence to the invariant or tionary or steady state distribution

sta-In the literature, Zou and Wang in [34] considered the following stochastic modelfor a single species with protection zone:

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

real-valued Brownian motion To simplify the notation, we introduce D

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 vival opportunity of the species gets However, setting up and maintaining a largeprotection zone is costly It is therefore important to know what the threshold for thearea of the protection zone should be to make the species survive permanently Since

sur-β is the ratio of the area of Ω1to 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)) ∈ R 2,◦+ (the interior ofR2

+), there exists a unique global solution to

(1.3) that remains inR2,◦+ almost surely Although they provided sufficient conditionsfor 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 > 0 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 = 0 We also reveal how the white noise influences the

system and compare the deterministic and stochastic models in section 3 Moreover,

we go a step further than [34] by investigating important asymptotic properties of thesolution 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

In recent years, the study of dynamics of species in ecological systems has receivedmuch attention While many works were devoted to various aspects of deterministicsystems with concentration on stability issues [2, 19, 20, 21, 26, 32, 31], there is anincreasing 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 relations of the species and is thus more suitable for the intended ecologicalapplications

cor-(b) In contrast to the usual approach of using a Lyapunov function-type ment, we derive a threshold value that characterizes the size of the protectionregion The conditions are sharp in that not only are the conditions obtainedsufficient, but also they are close to necessary

(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 forthe ergodicity Our result establishes the existence of an invariant probabil-ity measure In addition, it describes precisely the support of the invariantprobability measure Second, we prove the convergence in total variation

to the invariant measure Moreover, precise exponential upper bounds areobtained Finally, a strong law of large numbers is obtained Our resultwill be important for the study of long-time behavior of the dynamics of thespecies It indicates that when time is large enough, one can replace the in-stantaneous 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 sufficientand 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 tosome discussion and comparison to existing results Some numerical examples andfigures are also provided to illustrate our results Finally, further remarks are issued

in section 4, which point out possible future directions for investigations

2 Sufficient conditions for permanence In this section, we obtain

suffi-cient conditions for permanence The conditions are in fact close to necessary Let

(Ω, F, {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0satisfyingthe usual condition, i.e., it is increasing and right continuous while F0 contain all

P-null sets Let W (t) be an F t-adapted standard, real-valued Brownian motion Togain insight into the growth rates of species in the two areas, we first rewrite (1.3) inthe form

By the comparison theorem for differential equations, we can check that Z(t) ≥

z(t) for all t ≥ 0 a.s provided that Z(0) = z(0) ∈ (0, 1) Note that z(t) → z ∗ ,

+, denote by (Z z,y (t), Y z,y (t)) the solution of (2.2) with the initial

condition (Z(0), Y (0)) = (z, y) Let B(R 2,◦+ ) be the σ-algebra of Borel subsets ofR2,◦+ ,

and μ be the Lebesgue measure onR2,◦+ .

To proceed, we use the ideas in geometric control theory to study the dynamicsystems To this end, it is more convenient to use the stochastic integral in theStratonovich form Then we use the idea of reachable sets in control theory to over-come the difficulty of evaluating the systems Thus we rewrite (2.2) as

2 − by + D ∗ β(z − 1))

⎞

⎠and

B, [A, B], [A, [A, B]], [B, [A, B]], spans R2 at every (z, y) ∈ R 2,◦+ , 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))Tand Ψ(z, y) = (Ψ1(z, y), Ψ2(z, y))Tare vector fields onR2(where

zTdenotes the transpose of z), then the Lie bracket [Φ; Ψ] is a vector field given by



It can be seen that B(x, y), C(z, y) span R2 for all (z, y) ∈ R 2,◦+ 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) spanR2 for all y > 0 As a result, we obtain the following lemma.

Lemma 2.1 H¨ ormander’s condition holds for the solution of (2.2) inR2,◦+ 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

with φ being from the set of piecewise continuous real-valued functions defined on

R+ Let (z φ (t, z, y), y φ (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 able sets Roughly, a reachable set can be illustrated as follows Starting with initial

reach-point (z0, y0), the collection of all points (z1, y1) = (z φ (t, z0, y0), y φ (t, z0, y0)) under

piecewise continuous controls φ forms the reachable set of (z0, y0) In light of thesupport theorem (see [15, Theorem 8.1, p 518]), to obtain the desired properties ofthe transition probability and invariant probability measure of (2.2), we investigatethe reachable sets of different initial values The results are given in the followingclaims Before getting to the detailed argument, let us first provide some illustrations

on these claims Claim 1 shows that we can control vertically while Claims 2 and

3 state that a point can be reached horizontally from the left and the right undersuitable conditions To be more precise, Claim 1 indicates that for any initial points

y0 and z0, there is a control so that y φ can reach any given point y1 while z φ will

stay in a neighborhood of z0 in finite time Claim 2 states that if the initial point z0

is less than the final point z1, there are a y0> 0 and a control so that z φ will reach

z1 while y φ remains unchanged in a finite time Claim 3 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 z φ will reach z1and

y φ will stay at y0 in finite time Claim 4 inserts that under the said conditions, we

cannot find a control, so that z φ reaches z1 in finite time Claim 5 indicates that

there is a point that can be approached from any nearby initial point (z0, y0) using a

suitable feedback control Finally, Claim 6 is concerned with properties of the controlsystem restricted on the boundary{(z, y) : y = 0}.

Claim 1 For any y0, y1, z0 ∈ (0, ∞) and ε > 0, there exist a control φ and a

y0> y1, we can construct φ(t) similarly In the next two claims (Claims 2 and 3), we

consider the reachable sets from initial conditions starting from different regions.Claim 2 For an For any 0 < z0< z1< 1, there are a y0> 0, a control φ, and a

T > 0 such that z φ (T , z0, y0) = z1and that y φ (T , z0, y0) = y0 for all 0≤ t ≤ T

Indeed, if y0 is sufficiently large, there is a ρ3 > 0 such that g(z, y0) > ρ3 for

all z0 ≤ z ≤ z1< 1 This property, combining with (2.8), implies the existence of a

feedback control φ and T > 0 satisfying the desired claim.

Claim 3 For an Assume that z ∗ ≤ z1 < z0 Since D ∗(1− z)(βz + 1) − Ez <

0 for all z ∈ [z1, z0], if y0 is sufficiently small, we have

Claim 4 For anFor any 0 < z1< z0< z ∗ , we have D ∗(1−z1)(βz1+1)−Ez1≥ 0,

which implies infy∈(0,∞) {h(z1, y) } ≥ 0 Thus, we cannot find a control φ and a T > 0

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

and a T > 0 satisfying z φ (T , z0, y) = z1.

Claim 5 For an It can be seen that there is z ∗ ∈ (z ∗ , 1) satisfying g(z ∗ , 1) = 0

and that the equilibrium (z ∗ , 1) of the system

a neighborhood S δ ⊂ (z ∗ − δ, z ∗ + δ) × (1 − δ, 1 + 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

2 + D

∗ β(˜ z(t, z, y) − 1)) + αφ(t) = b for all t ≥ 0,

we have (z φ (t, z, y), y φ (t, z, y)) = (˜ z(t, z, y), ˜ y(t, z, y)) for all t ≥ 0 As a result,

(z φ (t, z, y), y φ (t, z, y) ∈ S δ for all (z, y) ∈ S δ

for any t ≥ 0 with this control.

Claim 6 For an For any z > 0 and δ > 0, there is a T > 0 satisfying z φ (T , z, 0) ∈

(z ∗ − δ, z ∗ + δ) and y φ (T , z, 0) = 0.

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

2 + D ∗ β(z ∗ − 1) > 0 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 > 0 and a function H(z, y) :R2,◦+ → R+ such that

(2.10) P (t, z, y, ·) − π ∗(·) ≤ H(z, y)e −γt for all t ≥ 0, where · is the total variation norm.

(iii) Moreover, for any π ∗ -integrable function f , and (z, y) ∈ R 2,◦+ we have

(2.11) P

lim

To proceed, we first recall some technical concepts and results in [23, 24] Let X be

a locally compact and separable metric space, andB(X) be the Borel σ-algebra on X.

Let Φ ={Φ t : 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, {P x } x∈X ), where the measure P x satisfies P x(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

measurea on R+, we define a sampled Markov transition function Ka of Φ by

• 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 measurea, 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 {Φ nδ , n ∈ N} of Φ if there is a probability

measurea on N and a nontrivial measure ψ(·) on X such that

Ka(x, B) :=

∞



n=1

P(nδ, x, B) a(n) ≥ ψ(B) for all x ∈ A, B ∈ B(X).

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.

1 If Φ is bounded in probability on average, that is, for any x ∈ X and ε > 0,

there is a compact set C ε,x satisfying lim inf t→∞1tt

0P(t, x, C ε,x ) > 1 − ε.

2 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( −b2t) for all x ∈ X.

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 ⊂ R 2,◦+ 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 theresolvent kernel (a special case of sampled transition kernel)

To prove the latter statement, let the point (z ∗ , 1) be as in Claim 5. Since

(z ∗ , 1) × (0, ∞) is invariant under (2.2), we have P (1, z ∗ , 1, (z ∗ , 1) × (0, ∞)) = 1 then p(1, z ∗ , 1, z2, y2) > 0 for some (z2, y2)∈ (z ∗ , 1) × (0, ∞) In view of Claim 5 and the

smoothness of p(1, ·, ·, ·, ·), there exist a neighborhood S δ ∗ , 1) that is invariant

under (2.9), and an open set G 2, y2) such that

(2.12) p(1, z, y, z , y )≥ m > 0 for all (z, y) ∈ S δ , (z , y )∈ G.

For any (z, y) ∈ K, we derive from Claims 1–3 that there is a T > 0 and a control φ

satisfying (z φ (T , z, y), y φ (T , z, y)) ∈ S δ Let n z,y be a positive integer greater than T

In view of Claim 5, we can extend control φ after T such that

(z φ (n z,y , z, y), y φ (n z,y , z, y)) ∈ S δ

By the support theorem (see [15, Theorem 8.1, p 518])

P (n z,y , z, y, S δ ) := 2ρ z,y > 0.

Since (Z(t), Y (t)) is a Markov–Feller process, there exists an open set V z,y

such that P (n z,y , z , y , S δ) ≥ ρ x,y for all (z , y )∈ V z,y Since K is a compact set,

there is a finite number of V z i ,y i , i = 1, , l, satisfying K ⊂l i=1 V z i ,y i Let ρ K =

min{ρ z i ,y i , i = 1, , l } For each (z, y) ∈ K, there exists n z i ,y i such that

set K ⊂ R 2,◦+ is petite for the Markov chain (Z(n), Y (n)).

Proof of Theorem 2.1 Since a − α2

First, we consider (2.7) in the invariant setM = {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,

Similarly, we can estimate

(2.17) LU (z, y) ≤ θ3U (z, y) for all (z, y) ∈ R 2,◦+

for some θ3 > 0 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–

4 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> 0 such that

(2.19) EU(Z z,y (t), Y z,y (t)) ≤ H1U (z, y)e γ1t for all t > 0 and (z, y) ∈ R 2,◦+ .

In view of (2.3) and (2.6), for any (z0, y0) ∈ R 2,◦+ , there is a nonrandom moment

t0= t0(z0, y0) > 0 such that (Z z,y (t), Y z,y (t)) ∈ M for all t ≥ t0 with probability 1.Thus, we have from (2.18) and (2.19) the following estimate,

there is a point that can be approached from any nearby initial point... class="page_container" data-page="8">

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