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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. 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 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.
/>† Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam
(). 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 (). 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
(). This author’s research was supported in part by the National Science
Foundation under grant DMS-1207667.

1382

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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 > 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 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

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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
2
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 > 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 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.

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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 3 is devoted to
some discussion and comparison to existing results. Some numerical examples and
figures are also provided to illustrate our results. Finally, further remarks are issued
in section 4, which point out possible future directions for investigations.
2. 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) ≥ 1.

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

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1386

N. T. DIEU, N. H. DU, H. D. NGUYEN, AND G. YIN

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

∗

z =

(2.5)

2D∗ β

is the unique root of the equation
D∗ (1 − z)(βz + 1) − Ez = 0 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
(2.7)
⎩dY (t) = Y (t) a − α − bY (t) + D∗ β(Z(t) − 1) dt + αY (t) ◦ dW (t).
2
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))
2

and
B(z, y) =

B1 (z, y)
=
B2 (z, y)

0
.
αy

To use the ideas of reachable sets, we need the notion of H¨ormander’s condition.
The diffusion (2.7) is said to satisfy H¨
ormander’s condition if 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 (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, 2.

∂y
∂y
Φ1 (z, y)

We next verify that H¨
ormander’s condition holds for the diffusion given by (2.7). By
direct calculation,
C(z, y) := [A, B](z, y) =

−αbyz(1 − z)
αby 2

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PROTECTION ZONES IN RANDOM ENVIRONMENT

1387

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and
C(z, y) := A,

1
C (z, y) =
αb

C1 (z, y)
,
C2 (z, y)

where
∂A1 (z, y)
∂z
2
+ A2 (z, y)z(z − 1) − y [bz(1 − z)].

C1 (z, y) = A1 (z, y)(−y)(1 − 2z) + yz(1 − z)

It can be seen that B(x, y), C(z, y) span R2 for all (z, y) ∈ R2,◦
+ satisfying z = 1.
When z = 1, we have C1 (1, y) = A1 (1, y)(−y)(1 − 2) = −Ey = 0 hence B(1, y) and
C(1, y) span R2 for all y > 0. As a result, we obtain the following lemma.
Lemma 2.1. H¨
ormander’s 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)
2

with φ being from the set of piecewise continuous real-valued functions defined on
R+ . Let (zφ (t, z, y), yφ (t, z, y)) be the solution to (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 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 under
suitable conditions. To be more precise, Claim 1 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 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 zφ will reach
z1 while yφ 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 zφ will reach z1 and
yφ will stay at y0 in finite time. Claim 4 inserts that under the said conditions, we

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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 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 control
system 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
T > 0 such that yφ (T, z0 , y0 ) = y1 , |zφ (T, z0 , y0 ) − z0 | < ε.
Suppose that y0 < y1 and let ρ1 = sup{|g(z, y)|, |h(z, y)| : y0 ≤ y ≤ y1 , |z − z0 | ≤ ε}.
We choose φ(t) ≡ ρ2 with ( αρρ21y0 − 1)ε ≥ y1 − y0 . It is easy to check that with this
control, there is a 0 ≤ 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 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 zφ (T, z0 , y0 ) = z1 and that yφ (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
sup {g(z, y0 )} ≤ by0

z∈[z1 ,z0 ]

sup {|z(1 − z)|} +

z∈[z1 ,z0 ]

sup {D∗ (1 − z)(βz + 1) − Ez} < 0.

z∈[z1 ,z0 ]

As a result, there is a feedback control φ and a T > 0 satisfying zφ (T, z0 , y0 ) = z1 and
yφ (t, z0 , y0 ) = y0 for all 0 ≤ t ≤ T .
Claim 4. For anFor any 0 < z1 < z0 < z ∗ , we have D∗ (1−z1 )(βz1 +1)−Ez1 ≥ 0,
which implies inf y∈(0,∞) {h(z1 , y)} ≥ 0. Thus, we cannot find a control φ and a T > 0
satisfying zφ (T, z0 , y) = z1 . Similarly, if z1 > max{z0 , 1}, we cannot find a control φ
and a T > 0 satisfying zφ (T, z0 , y) = z1 .
Claim 5. For an It can be seen that there is z1∗ ∈ (z ∗ , 1) satisfying g(z1∗ , 1) = 0
and that the equilibrium (z1∗ , 1) of the system
z˙ = g(z, y),
y˙ = y(b − by),

(2.9)

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, 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
+ D∗ β(˜
z (t, z, y) − 1)) + αφ(t) = b for all t ≥ 0,
2

we have (zφ (t, z, y), yφ (t, z, y)) = (˜

z (t, z, y), y˜(t, z, y)) for all t ≥ 0. As a result,
(zφ (t, z, y), yφ (t, z, y) ∈ Sδ for all (z, y) ∈ Sδ
for any t ≥ 0 with this control.
Claim 6. For an For any z > 0 and δ > 0, there is a T > 0 satisfying zφ (T, z, 0) ∈
(z ∗ − δ, z ∗ + δ) and yφ (T, z, 0) = 0.

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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).
2
Suppose that λ := a − α2 + 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
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

1
t→∞ t

t

lim

f Z z,y (s), Y z,y (s) ds =

f (u, v)π ∗ (du, dv)

= 1.

R2

0

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).
0

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.
1. 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 0 P(t, x, Cε,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(−b2 t) for all x ∈ X.

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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

0

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 5. Since
∗
(z , 1) × (0, ∞) is invariant under (2.2), we have P (1, z1∗ , 1, (z ∗ , 1) × (0, ∞)) = 1 then
p(1, z1∗ , 1, z2 , y2 ) > 0 for some (z2 , y2 ) ∈ (z ∗ , 1) × (0, ∞). In view of Claim 5 and the
smoothness of p(1, ·, ·, ·, ·), there 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 > 0 for all (z, y) ∈ Sδ , (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 (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 > 0.
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) :=

1
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) ≥

1
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)).

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2

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 ∗ − 1 − δ ∗ ) > 0.
a − (q + 1)
2
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 ∗ − 1 − δ ∗ ) − qD∗ β(z − z ∗ + δ ∗ )y −q

2

+ qby 1−q + y(a − by + D∗ β(z − 1))
≤ −qy −q a − (q + 1)

α2
+ D∗ β(z ∗ − 1 − δ ∗ ) + qby 1−q + y(a − by)
2

≤ −θ1 (y −q + y) + θ2 ≤ −θ1 U (z, y) + θ2 for all (z, y) ∈ M,
where

α2
+ D∗ β(z ∗ − 1 − δ ∗ ) > 0, and
2
θ2 = sup{−θ1 y −q + qby 1−q + y(a − by) + θ1 y} < ∞.
θ1 = q a − (q + 1)
y>0

Similarly, we can estimate
(2.17)

LU (z, y) ≤ θ3 U (z, y) for all (z, y) ∈ R2,◦
+

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)) ≤ H1 U (z, y)eγ1 t for all t > 0 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 ) > 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,
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,

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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 1 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.,
2

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) = 0 a.s. Moreover, for any (z0 , y0 ) ∈ R2,◦
limt→∞ X
+ , we
have with probability 1 that
(2.20)

ln X z0 ,y0 (t)
ln Y z0 ,y0 (t)
= lim

= λ < 0.
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 [ 2 , 1] × [0, H]. For the control system, given δ > 0, there exists a T > 0
∗
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 > 0 such that
(2.21)
P lim (Z z,y (t), Y z,y (t)) = (z ∗ , 0) ≥ 1 − 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 > 0.
∂y
We have the Taylor expansion of f1 (z, y) in the vicinity of (z ∗ , 0),
f1 (z, y) = −c1 (z − z ∗ ) + c2 y + o

(z − z ∗ )2 + y 2 ,

where
o
lim

z→z ∗
y→0

(z − z ∗ )2 + y 2
(z − z ∗ )2 + y 2

= 0.

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Otherwise, since a −

α2
2

1393

+ D∗ β(z ∗ − 1) < 0,

−c3 := a − (1 − p)

α2
+ D∗ β(z ∗ − 1) < 0
2

for sufficiently small p > 0. Consider the Lyapunov function V (z, y) = (z − z ∗ )2 + y p
which is twice differentiable in (z, y) ∈ R2,◦
+ . By direct calculation, we have
LV (z, y)

=

⎡

∂V (z, y)
,
∂z

∂ 2 V (z, y)
⎢

1
∂V (z, y) f1 (z, y)
∂z 2
0, αy ⎢
+
⎣ ∂ 2 V (z, y)
f2 (z, y)
2
∂y
∂y∂z

= 2(z − z ∗ ) −c1 (z − z ∗ ) + c2 y + o
+ py p a − by − (1 − p)
≤ −c1 (z − z ∗ )2 +

⎤
∂ 2 V (z, y)
∂z∂y ⎥
⎥ 0
2
∂ V (z, y) ⎦ αy
∂y 2

(z − z ∗ )2 + y 2

α2
+ D∗ β(z − 1)
2

c22 2

y + o[(z − z ∗ )2 + y 2 ] − pc3 y p + py p D∗ β(z − z ∗ ).
c1

Since y 2 = o(y p ) for small y, when y 2 + (z − z ∗ )2 is small, we have
2c3 py p
c22 2
,
y − pc1 y p + py p D∗ β(z − z ∗ ) ≤ −
c1
3
2c1
c3 py p
o[(z − z ∗ )2 + y 2 ] ≤
(z − z ∗ )2 +
.
3
3
Therefore,
1
LV (z, y) ≤ − [c1 (z − z ∗ )2 + c3 py p ] ≤ −θ4 V (z, y) for all (z, y) ∈ Nδ
3
for some θ4 > 0 and sufficiently small δ. By [22, Theorem 2.3. p. 112], for any ε > 0,
there is a δ > 0 such that
(2.22)

P

lim (Z z,y (t), Y z,y (t)) = (z ∗ , 0) ≥ 1 − 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).
2

If Z(0) = z0 ∈ (0, ∞), there is a t0 > 0 such that Z(t) ≤ 1 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 ) > 0 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)

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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

1
t→∞ t

(2.24)

∞

ϕ(s)ds =

lim

0

xα e−βx dx =

0

α

:= K1 a.s.

β

Consequently,
(2.25)

lim sup
t→∞

1
t

t
0

Y (s)ds ≤ K1 ,

which implies
(2.26)

lim sup
t→∞

1
t

t
0

1{Y (s)≥H} ds ≤

1
1
lim sup
H t→∞ t

t
0

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→∞

1
t

t
0

1{Y z0 ,y0 (s)∈[0,H]} ds ≥ 1 −

K1
> 0 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 > 0 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 > 0 such
that
P {(Z z,y (Tz,y ), Y z,y (Tz,y )) ∈ Uδ } > 2pz,y > 0.
Since the process (Z(t), Y(t)) has the Feller property, there is a neighborhood Vz,y of
(z, y) such that
P (Z z ,y (Tz,y ), Y z ,y (Tz,y )) ∈ Uδ

> pz,y for all (z , y ) ∈ Vz,y .

∗

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 > 0 : (Z z,y (t), Y z,y (t)) ∈ Uδ }.
Then
(2.28)

P{τδz,y < T ∗ } ≥ p∗ > 0 for all (z; y) ∈

z∗
, 1 × [0, H].
2
∗

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∗
, 1 × [0, H] , k ∈ N.
2

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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 → 0 as n → ∞.

Ak
k=1

As a result,
P{τδz0 ,y0 < ∞} = 1.

(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) ≥ 1 − ε.

t→∞

Since ε can be taken arbitrarily, we obtain
(2.31)

P

lim (Z z0 ,y0 (t), Y z0 ,y0 (t)) = (z ∗ , 0) = 1 for all (z0 , y0 ) ∈ R2,◦
+ .

t→∞

Finally, it follows from Itˆo’s formula that
(2.32)

ln y0
1 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 0
2
t
Hence (2.20) follows from (2.31) and (2.32).
3. 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
2
survive permanently? First, consider the case a − α2 − E > 0. We obtain that
even without a protection zone β = ∞, the species will survive permanently. If
2
a − α2 − E = 0 for any β > 0, the inequality
a−

α2
+ D∗ β(z ∗ − 1) > 0
2

holds. Hence the species will survive if there is a protection zone even if its area is

2
small. On the other hand, if a − α2 ≤ 0, the species will die out even it is completely
2
protected. We therefore focus on the case 0 < 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 = 0
or, equivalently,
⎞
⎛
∗ β − D ∗ − E)2 + 4D ∗ 2 β + (D ∗ β − D ∗ − E)
2
(D
α
+ D∗ β ⎝
− 1⎠ = 0.
a−
2
2D∗ β
2

It can be shown that if 0 < a − α2 < E the equation above has a unique positive root
β∗ =

(2D∗ + 2E + α2 − 2a)(2a − α2 )
.
2D∗ (2E + α2 − 2a)

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Moreover,

α2
+ D∗ β(z ∗ − 1) > 0 if and only if β < β ∗ ,
2
which answers the aforementioned question. Intuitively, if β is less than but
very close to β ∗ , in view of the ergodicity of (Z(t), Y (t)) (see (2.11)), we can show
t
t
that limt→∞ 1t 0 Y (s)ds is small, so is limt→∞ 1t 0 X(s)ds. To guarantee that
t
t
limt→∞ 1t 0 Y (s)ds and limt→∞ 1t 0 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
2
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)
⎧
∗
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)
1 (t) − α2 Z(t)dW2 (t).
2

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.

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PROTECTION ZONES IN RANDOM ENVIRONMENT

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∞

Theorem 3.1. Let z˜∗ = 0 z π
˜ (dz). We have the following.
α2
∗
∗
˜
z − 1) < 0 then all positive solutions to (3.2) tend to
• If λ := a − 2 + D β(˜
the origin.
˜ := a− α2 +D∗ β(˜
• If λ
z ∗ −1) > 0 then (3.2) has a invariant probability measure
2
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 > 0.
2

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 do
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
2
tend to 0 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 = 4 × 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 2.
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 = 3. Direct calculation shows that z ∗ = 0.89463 and β ∗ = 5.625 < β.
We obtain λ = −0.0288 < 0. 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,

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1398

N. T. DIEU, N. H. DU, H. D. NGUYEN, AND G. YIN

Fig. 1. Trajectories of Y (t), Z(t) in Example 3.1, respectively.

Fig. 2. 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. 3. The left figure shows the trajectory of Y (t) in Example 3.2 with β = 6. It can be seen
that Y (t) converges to 0. In contrast, as shown in the right figure, the trajectory of Y (t) with β = 4
does not converge to 0.

that is, reducing β to a number below β ∗ . For instance, we take β = 4. Figure 3
illustrates two cases β = 6 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 4. 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).

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PROTECTION ZONES IN RANDOM ENVIRONMENT

1399

Fig. 4. 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. 5. 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 = 1. We obtain that λ = −0.1639 < 0
˜ = 0.5476 > 0. The population of (3.2) is permanent, while extinction takes
while λ
place in (2.2) as can be seen in Figure 5.
4. 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-

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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 > 1. 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.

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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.
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