DSpace at VNU: On the dynamics of a stochastic ratio-dependent predator–prey model with a specific functional response

J. Appl. Math. Comput.
DOI 10.1007/s12190-014-0812-3
ORIGINAL RESEARCH

On the dynamics of a stochastic ratio-dependent
predator–prey model with a specific functional response
Yan Zhang · Shujing Gao · Kuangang Fan ·
Yanfei Dai

Received: 14 May 2014
© Korean Society for Computational and Applied Mathematics 2014

Abstract In this paper, a new stochastic two-species predator–prey model which
is ratio-dependent and a specific functional response is considered in, is proposed.
The existence of a global positive solution to the model for any positive initial value
is shown. Stochastically ultimate boundedness and uniform continuity are derived.
Moreover, under some sufficient conditions, the stochastic permanence and extinction
are established for the model. At last, numerical simulations are carried out to support
our results.
Keywords

Itô’s formula · Stochastically permanent · Extinction · White noise

1 Introduction
Predator–prey relationship can be important in regulating the number of prey and
predators. And the dynamic relationship between predators and their preys has long
been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance [1]. In the past decades,
more and more mathematical models for predator–prey behavior are carried out, see
[2–4] and the references cited therein.
Y. Zhang (B) · S. Gao · Y. Dai
Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques,

Gannan Normal University, Ganzhou 341000, People’s Republic of China
e-mail:
S. Gao
e-mail:
K. Fan
School of Mechanical and Electrical Engineering, Jiangxi University of Science
and Technology, Ganzhou 341000, People’s Republic of China
e-mail:

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Y. Zhang et al.

When investigating biological phenomena, there are many factors which affect
dynamical properties of biological and mathematical models. One of the familiar
nonlinear factors is functional response [5]. There are many significant functional
responses in order to model various different situations and evidences show that when
predators have to search or compete for food, a more suitable functional response
depending on the densities of both prey and predator, which is called a ratio-dependent
functional response, should be introduced in realistic models [6,7]. In the past decades,
many authors proposed different forms of ratio-dependent functional responses to
model this process and the three classical predator-dependent functions are Crowley–
Martin type [8], Beddington–DeAngelis type by Beddington [9] and DeAngelis et al.
[10], as well as Hassell–Varley type [11]. In this paper, we consider the following
model with a specific functional response:
⎧
⎪
⎪
˙ = x(t) r − ax(t) −

⎨ x(t)

ωy(t)
,
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)
f ωx(t)
⎪
⎪
⎩ y˙ (t) = y(t) h − by(t) +
.
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)

(1.1)

where x(t), y(t) stand for the population densities of prey and predator at time t,
respectively. These parameter are defined as follows: r is the growth rate of the prey, a
measures the strength of competition among individuals of species x, ω is the capturing
rate of predator, f is the rate of conversion of nutrients into the production of predator,
h, b have the similar meaning to r, a, respectively. 1+α1 x(t)+αωx(t)
is the
2 y(t)+α3 x(t)y(t)
radio-dependent functional response, where α1 , α2 , α3 ≥ 0 are constants. It is very
important to note that this functional response becomes a linear mass-action function
response (or Holling type I functional response) if α1 = α2 = α3 = 0, the Holling
type-II functional response if α2 = α3 = 0, the modified Holling type-II functional
response proposed in [12,13] when α3 = 0, and Crowley–Martin functional response
presented in [6,8,14] if α3 = α1 α2 .
As a matter of fact, population systems in the real world are often inevitably subject
to environmental noises. Many researchers pointed out the fact that due to environmental noise, the birth rate, carrying capacity and other parameters involved in the
model exhibit random fluctuation to a greater or lesser extent. Therefore, more and

more interest is focused on stochastic systems and many research has been done in
this field , see e.g., [15–20]. In this paper, taking into the effect of randomly fluctuating
environment, we incorporate white noise in each equations of the system (1.1) and we
assume that fluctuations in environment will manifest themselves mainly as fluctuations in the growth rate and capturing rate of the prey population and the growth rate
and conversion rate of the predator population, therefore, the corresponding stochastic
system to Eq. (1.1) can be described as follows:
ωy(t)
dt
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)
σ2 y(t)
+ x(t) σ1 +
d B1 (t),
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)

d x = x(t) r − ax(t) −

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On the dynamics of a stochastic ratio-dependent predator–prey model

f ωx(t)
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)
δ2 x(t)
+ y(t) δ1 +
d B2 (t).
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)

dy = y(t) h − by(t) +

(1.2)

where r, a, h, b, ω, f, αi are positive parameters,i = 1, 2, 3. σi2 and δi2 represent
the intensities of the white noises, i = 1, 2. B1 (t), B2 (t) are independent Brownian
motions defined on a complete probability space ( , F, P) with a filtration {Ft }t∈R+
satisfying the usual conditions(i.e. it is right continuous and increasing while F0 con2 the positive cone in R 2 , and also denote by
tains all P−null sets). We denote by R+
1

X (t) = (x(t), y(t)) and |X (t)| = (x 2 (t) + y 2 (t)) 2 .
As far as we know, there is no work has been done on the stochastic model (1.2),
and the aim of this paper is to analyze the dynamical properties of the model and show
the impact of environmental noise on the population system (1.2).
The rest of the paper is arranged as follows. In this paper, we firstly show that the
system (1.2) has a unique global solution for any positive initial value in Sect. 2.1. The
stochastic boundedness of the positive solution is discussed in Sect. 2.2. Moreover, we
show that the solution is uniformly continuous in Sect. 2.3. In Sect. 3, by constructing
a suitable Lyapunov function, we establish the sufficient conditions on the stochastic
permanence and extinction of the model. At last, in Sect. 4, a numerical simulation
which verifies our qualitative results is given and the paper is ended with a discussion.
2 Properties of the solution
2.1 Existence, uniqueness and global positive solution
In this section, as x(t), y(t) in system (1.2) stand for the population densities of prey
and predator at time t, respectively, we will only interested in the positive solutions
of system (1.2). In order for a stochastic differential equation to have a unique global
solution for any given initial value, the coefficients of the equation are generally
required to satisfy the linear growth condition and local Lipschitz condition [21].
However, the coefficients of system (1.2) neither satisfy the linear growth condition,
nor local Lipschitz continuous. In the following, by making the change of variables,
existence, uniqueness of the positive solution will be shown.

Theorem 1 For any initial value x0 > 0, y0 > 0, there is a unique positive local
solution(x(t), y(t)) for t ∈ [0, τe ) of system (1.2) a.s.
Proof Let u(t) = lnx(t), v(t) = lny(t), then we obtain the following equations
du(t) = r − aeu(t) −
− 0.5 σ1 +

1 + α1

eu(t)

ωev(t)
+ α2 ev(t) + α3 eu(t) ev(t)

σ2 ev(t)
u(t)
1 + α1 e
+ α2 ev(t) + α3 eu(t) ev(t)

2

dt

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Y. Zhang et al.

+ σ1 +

σ2 ev(t)

d B1 (t),
1 + α1 eu(t) + α2 ev(t) + α3 eu(t) ev(t)

dv(t) = h − bev(t) +
− 0.5 δ1 +
+ δ1 +

f ωeu(t)
1 + α1 eu(t) + α2 ev(t) + α3 eu(t) ev(t)

δ2 eu(t)
1 + α1 eu(t) + α2 ev(t) + α3 eu(t) ev(t)

2

dt

δ2 eu(t)
d B2 (t).
1 + α1 eu(t) + α2 ev(t) + α3 eu(t) ev(t)

(2.1)

for t ≥ 0 with initial value u 0 = lnx0 , v0 = lny0 . Obviously, the coefficients of
model (2.1) satisfy the local Lipschitz condition. Therefore, there exists a unique local
solution u(t), v(t) on t ∈ [0, τe ), where τe is the explosion time. So, by Itô’s formula,
it is easy to obtain that x(t) = eu(t) , y(t) = ev(t) is the unique positive local solution
to system (1.2) with initial value x0 > 0, y0 > 0.
Next, we will show the unique positive solution of model (2.1) is global, i.e.,
τe = ∞.

Theorem 2 For any given value (x(0), y(0)) = X 0 ∈ R2+ , there is a unique solution
2 with probability
(x(t), y(t)) to Eq. (1.2) on t ≥ 0 and the solution will remain in R+
2
1, namely (x(t), y(t)) in R+ for all t ≥ 0 almost surely..
Proof Our Proof is motivated by the works of Mao et al. [22] and Liu and Wang
[23]. Let k0 > 0 be sufficiently large for X 0 lying within the interval [1/k0 , k0 ].
For each integer k > k0 , define the stopping times τk = inf{t ∈ [0, τe ] :
x(t)∈(1/k, k), or y(t)∈(1/k, k)}, where throughout this paper we set in f ∅ = ∞.
Obviously, τk is increasing as k → ∞. Let τ∞ = lim τk , then τ∞ ≤ τe . If we can
k→+∞

2 a.s. for
show τ∞ = ∞ a.s., then we can obtain that τe = ∞a.s. and (x(t), y(t)) ∈ R+
all t ≥ 0. In other words, to complete the proof all of we need to show is that τ∞ = ∞
a.s.

If this statement is false, then there exist constants T > 0 and ε ∈ (0, 1) such that
P{τ∞ ≤ T } > ε. Hence, there exists an integer k1 ≥ k0 such that
P{τk ≤ T } ≥ ε for all k ≥ k1

(2.2)

2 → R by V (x, y) = (x − 1 − lnx) + (y − 1 − lny).
Define a C 2 -function V : R+
+
Then the nonnegativity of the function V (x, y) can easily be seen from

u − 1 − lnu ≥ 0 on u > 0
ˆ formula, we obtain that

If (x(t), y(t)) ∈ R2+ , in view of I t o’s
d V (x, y) = Vx d x + 0.5Vx x (d x)2 + Vy dy + 0.5Vyy (dy)2
= (1 − 1/x)x r − ax(t) −

123

ωy(t)
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)


On the dynamics of a stochastic ratio-dependent predator–prey model

+ (1 − 1/y)y h − by(t) +
+ 0.5 σ1 +

f ωx(t)
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)

σ2 y(t)
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)

dt

2

dt

2
δ2 x(t)
dt

1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)
σ2 y(t)
d B1 (t)
+ (1 − 1/x)x σ1 +
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)
δ2 x(t)
+ (1 − 1/y)y δ1 +
d B2 (t)
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)

+ 0.5 δ1 +

≤ (r +a)x(t)−ax 2 (t)+
+ 0.5 δ1 +

δ2
α1

ω
fω
σ2
+
+(h +b)y −by 2 −h +0.5(σ1 + )2 dt
α2 α3
α2

2

dt +(x(t)−1) σ1 +

σ2 y(t)
d B1 (t)
1+α1 x(t)+α2 y(t)+α3 x(t)y(t)

δ2 x(t)
d B2 (t)
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)

+ (y(t) − 1) δ1 +

Thus, there exists a positive number G such that
σ2 y(t)
d B1 (t)
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)
δ2 x(t)
d B2 (t)
+ (y(t) − 1) δ1 +
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)

d V (x, y) ≤ Gdt + (x(t) − 1) σ1 +

Taking integral for each side of the above inequality from 0 to τk T yields that
τk T
0

τk T

d V (x, y) ≤

Gdt

0
τk T

+

(x(t)−1) σ1 +

0

+ (y(t)−1) δ1 +

σ2 y(t)
d B1 (t)
1+α1 x(t)+α2 y(t)+α3 x(t)y(t)

δ2 x(t)
d B2 (t)
1+α1 x(t)+α2 y(t)+α3 x(t)y(t)

where τk T = min{τk , T }. Whence taking expectation, we have that
E V (x(τk T ), y(τk T )) ≤ V (x(0), y(0)) + G 1 E(τk T ) ≤ V (x(0), y(0))
+ G1T

(2.3)

Set k = {τk ≤ T } for k ≥ k1 and by (2.2), P( k ) ≥ ε. Note that for every
ω ∈ k , there is x(τk , ω) or y(τk , ω) equals either k or 1/k, and hence
V (x(τk , ω), y(τk , ω)) ≥ min{k − 1 − lnk, 1/k − 1 + lnk}

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Y. Zhang et al.

It then follows from (2.3) that
V (x(0), y(0)) + G 1 T ≥ E[1 k (ω)V (x(ω), y(ω))]
≥ ε min{k − 1 − lnk, 1/k − 1 + lnk}
where 1

k

is the indicator function of

k.

Letting k → ∞ leads to the contradiction

∞ > V (x(0), y(0)) + G 1 T = ∞
So we must have τ∞ = ∞ a.s. This completes the proof of Theorem 2.
2.2 Stochastic boundedness
Definition 1 (See [24]) The solution X (t) = (x(t), y(t)) of Eq. (1.2) is said to be
stochastically ultimately bounded, if for any ε ∈ (0, 1), there is a positive constant
δ = δ(ε), such that for any given initial value X 0 ∈ R2+ , the solution X (t) to (1.2)
has the property that
lim sup P{|X (t)| > δ} < ε.
t→∞

Theorem 3 The solutions of model (1.2) are stochastically ultimately bounded for
any initial value X 0 = (x0 , y0 ) ∈ R2+ .
Proof Define Lyapunov functions V1 = et x p and V2 = et y p respectively, for (x, y) ∈

R2+ and p > 0. Then by the Itô formula, we compute
d(et x p ) = et x p dt + et px p−1 d x + 0.5 p( p − 1)x p−2 (d x)2
= et x p (t)dt + pet x p (t) r −ax(t)−

ωy(t)
dt
1+α1 x(t)+α2 y(t)+α3 x(t)y(t)

σ2 y(t)
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)
σ2 y(t)
d B1 (t)
+ pet x p (t) σ1 +
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)

2

+ 0.5 p( p − 1)et x p (t) σ1 +

dt

and
d(et y p ) = et y p (t)dt + pet y p (t) h −by(t)+

f ωx(t)
dt
1+α1 x(t)+α2 y(t)+α3 x(t)y(t)

δ2 x(t)
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)

δ2 x(t)
d B2 (t)
+ pet y p (t) δ1 +
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)
+ 0.5 p( p − 1)et y p (t) δ1 +

123

2

dt


On the dynamics of a stochastic ratio-dependent predator–prey model

Therefore
L V1 = et x p (t)dt + pet x p (t) r − ax(t) −
+ 0.5 p( p − 1)et x p (t) σ1 +

ωy(t)
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)

σ2 y(t)
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)

≤ et x p (t) 1 + pr + 0.5 p( p − 1) σ1 +

σ2
α2

2

2

− apx

≤ G 2 ( p)et

(2.4)

and
L V2 = et y p (t)dt + pet y p (t) h − by(t) +
+ 0.5 p( p − 1)et y p (t) δ1 +
≤ et y p (t) 1 + ph +

f ωx(t)
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)

δ2 x(t)
1 + α1 x(t) + α2 y(t) + α3 x(t)y(t)

2

2

pf ω
δ2
+ 0.5 p( p − 1) δ1 +
α1
α1

− pby

≤ G 3 ( p)et

(2.5)

where
1 + pr + 0.5 p( p − 1) σ1 +
G 2 ( p) =

σ2
α2

2

p+1

,

a p ( p + 1) p+1
1 + ph +

G 3 ( p) =

pf ω
α1

+ 0.5 p( p − 1) δ1 +

δ2
α1

2

p+1

b p ( p + 1) p+1

Thus, taking integral and expectations on both sides of (2.4) and (2.5), respectively,
we have the following equalities
lim sup E x p ≤ G 2 ( p) < +∞, lim sup E y p ≤ G 3 ( p) < +∞
t→∞

t→∞

On the other hand, for X (t) = (x(t), y(t)) ∈ R2+ , note that
|X (t)| p ≤ 2 max{x 2 (t), y 2 (t)}

p
2

p

≤ 2 2 (x p (t) + y p (t)).

Consequently,
lim sup E|X (t)| p ≤ G 4 ( p) < +∞
t→∞

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Y. Zhang et al.
p

where G 4 ( p) = 2 2 (G 2 ( p) + G 3 ( p)). By the Chebyshev inequality, the proof is
completed.

2.3 Uniform continuity
In this section, we continue to show the positive solution X (t) = (x(t), y(t)) is
uniformly Holder
¨
continuous. Main tools are to use appropriate Lyapunov functions
and fundamental inequalities.
Lemma 1 (See [25,26]) Suppose that a n− dimensional stochastic process X (t) on
t ≥ 0 satisfies the condition
E|X (t) − X (s)|α1 ≤ c|t − s|1+α2 , 0 ≤ s, t < ∞,
for some positive constants α1 , α2 and c. Then there exists a continuous modification
X˜ (t) of X (t) which has the property that for every υ ∈ 0, αα21 , there is a positive
random variable ψ(ω) such that
P ω:

| X˜ (t, ω) − X (t, ω)|
2
≤
υ
|t − s|
1 − 2−υ
0<|t−s|<ψ(ω),0≤s,t<∞

sup

= 1.

In other words, almost every sample path of X˜ (t) is locally but uniformly H older
¨
continuous with exponent υ.
Theorem 4 Let (x(t), y(t)) be a solution of system (1.2) on t ≥ 0 with initial value
2 , then almost every sample path of (x(t), y(t)) is uniformly continuous.
(x0 , y0 ) ∈ R+
Proof The first equation of system (1.2) is equivalent to the following stochastic
integral equation

t

+
0

ωy(s)
ds
1 + α1 x(s) + α2 y(s) + α3 x(s)y(s)
0
σ2 y(s)
d B1 (s),
x(s) σ1 +
1 + α1 x(s) + α2 y(s) + α3 x(s)y(s)
t

x(t) = x0 +

x(s) r − ax(s) −

Let
ωy(s)
,
1 + α1 x(s) + α2 y(s) + α3 x(s)y(s)
σ2 y(s)
f 2 (s) = x(s) σ1 +
1 + α1 x(s) + α2 y(s) + α3 x(s)y(s)
f 1 (s) = x(s) r − ax(s) −

123


On the dynamics of a stochastic ratio-dependent predator–prey model

then we have
p
ωy
1 + α1 x(s) + α2 y(s) + α3 x(s)y(s)
p
ωy
E |x| p r − ax −
1 + α1 x(s) + α2 y(s) + α3 x(s)y(s)
1
1
E|x|2 p + E|r + ax + ωy|2 p
2
2
1

1
E|x|2 p + 32 p−1 [r 2 p + a 2 p E|x|2 p + ω2 p E|y|2 p ]
2
2
1
32 p−1 2 p
G 2 (2 p) +
[r + a 2 p G 2 (2 p) + ω2 p G 3 (2 p)]
2
2
F1 ( p)
p
σ2 y
E x σ1 +
1 + α1 x(s) + α2 y(s) + α3 x(s)y(s)
σ2 p
σ2 p
σ1 +
E|x| p ≤ σ1 +
G 2 ( p)
α2
α2

E| f 1 (t)| p = E x r − ax −
=
≤
≤
≤

E| f 2 (t)| p =

≤

F2 ( p)
In view of the moment inequality for stochastic integrals, we can have that for
0 ≤ t1 ≤ t2 and p > 2,
t2

E

p

f 2 (s)d B1 (s)

≤

t1

≤

p( p − 1)
2

p
2

p( p − 1)
2

p
2

Then for 0 < t1 < t2 < ∞, t2 − t1 ≤ 1,
t2

E|x(t2 )−x(t1 )| p = E

≤ 2 p−1 E

t2

f 1 (s)ds +

t1

1
p

+

(t2 − t1 )

t2

p−2
2

E| f 2 (s)| p ds

t1

1
q

p
2

(t2 − t1 ) F2 ( p)
= 1, we have
p

f 2 (s)d B1 (s)

t1
t2

p

f 1 (s)ds

+ 2 p−1 E

t1

t2

p

f 2 (s)d B1 (s)

t1

p
q

≤ 2 p−1 (t2 − t1 ) E

t2

| f 1 (s)| p ds

t1

+2
≤2

p−1

p−1

p( p − 1)
2

p
2

p

(t2 − t1 ) 2 F2 ( p)

p
q

(t2 −t1 ) F1 ( p)(t2 −t1 )+2

= 2 p−1 (t2 − t1 ) p F1 ( p) + 2 p−1

p−1

p( p−1)
2

p( p − 1)
2

p
2

p
2

p

(t2 −t1 ) 2 F2 ( p)
p

(t2 − t1 ) 2 F2 ( p)

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Y. Zhang et al.

=2

p−1

(t2 − t1 )

p
2

(t2 − t1 ) F1 ( p) +

p

1+

≤ 2 p−1 (t2 − t1 ) 2

p
2

p( p − 1)
2

p( p − 1)
2

p
2

F2 ( p)

p
2

F( p)

where F( p) = max{F1 ( p), F2 ( p)}. From Lemma 1, we can obtain that almost every
sample path of x(t) is locally but uniformly H older
¨
-continuous with exponent υ for
every υ ∈ 0, p−2
2 p and thus almost every sample path of x(t) is uniformly continuous
on t ∈ R+ . In the same way, we can prove that almost every sample path of y(t) is also
uniformly continuous on t ∈ R+ . That is, almost every sample path of (x(t), y(t)) to
Eq. (1.2) is uniformly continuous on t ≥ 0.
3 The long behavior of system (1.2)
In this section, we will investigate the long time behavior of system (1.2) such as
permanence and extinction to discuss how the solution varies in R2+ in more detail.
Firstly, let us impose the following hypothesis which will be useful in the following.
2

σ2
δ2
ω
ω
+ 2 max σ1 , , δ1 ,
< min r − , h ;
α3
α1

α2
α2
fω
2
2
(H2) r − 0.5σ1 < 0, h +
− 0.5δ1 < 0.
α1

(H1)

3.1 Stochastic permanence
Now we are in the position to show the stochastic permanence which is defined below.
Definition 2 (See [24]) The solution X (t) = (x(t), y(t)) of Eq. (1.2) is said to be
stochastically permanent, if for any ε ∈ (0, 1), there exists a pair of positive constants
δ = δ(ε) and χ = χ (ε) such that for any initial value X (0) = (x(0), y(0)) ∈ R2+ , the
solution X (t) to (1.2) has the properties that
lim inf P{|X (t)| ≥ δ} ≥ 1 − ε, lim inf P{|X (t)| ≤ χ } ≥ 1 − ε.
t→∞

t→∞

Then we have the following theorem:
Theorem 5 Suppose that assumption (H1) holds, then system (1.2) is stochastically
permanent.
Proof Firstly, we will prove that for any initial value X (0) = (x(0), y(0)) ∈ R2+ , the
solution X (t) = (x(t), y(t)) satisfies that
lim sup E
t→∞

123

1
|X (t)|θ

≤ M,


On the dynamics of a stochastic ratio-dependent predator–prey model

where θ is an arbitrary positive constant satisfying
ω
σ2
δ2
+ 2(θ + 1) max σ1 , , δ1 ,
α3
α1
α2

2

< min r −

ω
,h .
α2
(3.1)

By (3.1), there exists an arbitrary positive constant p > 0 satisfying
θ min r −

ω
ω
σ2
δ2
,h − p − θ
− 2θ (θ + 1) max σ1 , , δ1 ,
α2
α3
α1
α2

2 and U (x, y) =
Define V (x, y) = x + y for (x, y) ∈ R+

we have

2

> 0. (3.2)

1
, by the Itô formula,
V (x, y)

ωy
1 + α1 x + α2 y + α3 x y
f ωx
dt
+ y h − by +

1 + α1 x + α2 y + α3 x y
σ2 x y
d B1 (t)
+ σ1 x +
1 + α1 x + α2 y + α3 x y
δ2 x y
d B2 (t).
+ δ1 y +
1 + α1 x + α2 y + α3 x y

d V (x, y) = x r − ax −

and
ωy
1 + α1 x + α2 y + α3 x y
f ωx
dt
+ y h − by +
1 + α1 x + α2 y + α3 x y

dU (X ) = −U 2 (X ) x r − ax −

+ U 3 (X ) x 2 σ1 +
+ y 2 δ1 +

σ2 y
1 + α1 x + α2 y + α3 x y

δ2 x
1 + α1 x + α2 y + α3 x y

2

2

dt

σ2 x y
d B1 (t)
1 + α1 x + α2 y + α3 x y
δ2 x y
d B2 (t)
+ δ1 y +
1 + α1 x + α2 y + α3 x y
σ2 x y
)d B1 (t)
= LU (X )dt − U 2 (X )(σ1 x +
1 + α1 x + α2 y + α3 x y
δ2 x y
)d B2 (t)
− U 2 (X )(δ1 y +
1 + α1 x + α2 y + α3 x y
− U 2 (X )

σ1 x +

123


Y. Zhang et al.

Under Assumption (H1), we can certainly choose a positive constant θ such that it
obeys (3.1). Applying the Itô formula again, we get
1
L(1 + U (X ))θ = θ (1 + U (X ))θ−1 LU (X ) + θ (θ − 1)(1 + U (X ))θ−2 U 4 (X )
2
2
σ2 y
× x 2 σ1 +
1 + α1 x + α2 y + α3 x y
+ y 2 δ1 +

δ2 x
1 + α1 x + α2 y + α3 x y

2

Now, we can choose p > 0 sufficiently small for (3.2) to hold. Then, let W (X ) =
e pt (1 + U (X ))θ and thus,
L W (X ) = pe pt (1 + U (X ))θ + e pt L(1 + U (X ))θ
= e pt L(1 + U (X ))θ−2
×

P(1 + U (X ))2 − θU 2 (X )x r − ax −

ωy
1 + α1 x + α2 y + α3 x y

f ωx
1 + α1 x + α2 y + α3 x y

ωy
− θU 3 (X ) x r − ax −
1 + α1 x + α2 y + α3 x y
f ωx
+ y h − by +
1 + α1 x + α2 y + α3 x y
− θU 2 (X )y h − by +

+ θU 3 (X ) x 2 σ1 +
+ y 2 δ1 +
+

σ2 y
1 + α1 x + α2 y + α3 x y

δ2 x
1 + α1 x + α2 y + α3 x y

2

2

σ2 y
θ (θ + 1) 4
U (X ) x 2 σ1 +
2
1 + α1 x + α2 y + α3 x y

+ y 2 δ1 +

δ2 x
1 + α1 x + α2 y + α3 x y

2

2

It is easy to see that
θU 3 (X ) x 2 σ1 +

≤ θU (X ) 2 max σ1 ,

123

2

σ2 y
1+α1 x +α2 y +α3 x y
σ2
δ2
, δ1 ,
α1
α2

+ y 2 δ1 +

2

;

δ2 x
1+α1 x +α2 y +α3 x y

2


On the dynamics of a stochastic ratio-dependent predator–prey model

θ (θ +1) 4
σ2 y
U (X ) x 2 σ1 +
2
1+α1 x +α2 y +α3 x y

2

+ y 2 δ1 +

δ2 x
1+α1 x +α2 y +α3 x y

2

2
σ2
δ2
θ (θ + 1) 2
;
U (X ) 2 max σ1 , , δ1 ,
2

α1
α2
ωy
f ωx
+ y h −by +
θU 2 (X ) x r −ax −
1+α1 x +α2 y +α3 x y
1+α1 x +α2 y +α3 x y
ω
≥ −θ max{a, b} + θ min{r, h}U (X ) − θ U 2 (X )
α3

≤

and
ωy
f ωx
+ y h −by +
1+α1 x +α2 y +α3 x y
1+α1 x +α2 y +α3 x y
ω
2
≥ −θ max{a, b}U (X ) + θ min{r − , h}U (X )
α2

θU 3 (X ) x r −ax −

Thus,
L W (X ) ≤ e pt (1 + U (X ))θ−2 {( p + θ max{a, b})
+ 2 p − θ min{r, h} + θ 2 max σ1 ,

+ θ max{a, b}U (X ) + p +
+

2

σ2
δ2
, δ1 ,
α1
α2

U (X )

ω
θω
− θ min r − , h
α3
α2

θ (θ + 1)
σ2
δ2
2 max σ1 , , δ1 ,
2
α1
α2

U 2 (X )

2

U 2 (X )

From (3.2), we know that there exists a positive constant S such that L W (X ) ≤
Se pt , which implies that
E[e pt (1 + U (X ))θ ] ≤ (1 + U (0))θ +

S(e pt − 1)
p

Therefore,
lim sup E[U θ (X (t))] ≤ lim sup E[(1 + U (X (t))θ ] ≤
t→∞

t→∞

S
p

Note that
θ

(x + y)θ ≤ 2θ (x 2 + y 2 ) 2 = 2θ |X |θ ,
where X = (x, y) ∈ R2+ . Consequently,
lim sup E
t→∞

1
|X (t)|θ

≤ 2θ lim sup EU θ (X ) ≤ 2θ
t→∞

S
:= M
p

123


Y. Zhang et al.

Thus, for any ε > 0, let δ = ( Mε ) θ , by Chebyshev inequality, we can obtain that
1

P{|X (t)| < δ} = P{|X (t)|−θ > δ −θ } ≤ E[|X (t)|−θ ]/δ −θ = δ θ E[|X (t)|−θ ]
that is,
lim inf P{|X (t)| ≥ δ} ≥ 1 − ε.
t→∞

Also, we can obtain that for any ε > 0, there exists χ > 0, such that lim inf P{|X (t)| ≤
χ } ≥ 1 − ε. Here we omit the proof. Theorem 5 is proved.

t→∞

3.2 Extinction
In the previous section, we have shown that the positive solutions of system (1.2) will
not explode to infinity in a finite time and will be ultimately bounded and permanent.
However, in this subsection, we will show that if the noise is sufficiently large, the
species will become extinct with probability one.

Theorem 6 Assume (H2) holds, then for any given initial value (x0 , y0 ) ∈ R2+ , the
solution (x(t), y(t)) of system (1.2) will be extinct with probability one.
Proof Let V3 = lnx, V4 = lny, then by the Itˆo’s formula, we can derive from (1.2)
that
r −ax −

dlnx =
+

σ1 +

σ2 y
1 + α1 x + α2 y + α3 x y

h −by +

dlny =
+

δ1 +

ωy
σ2 y
−0.5 σ1 +
1+α1 x +α2 y +α3 x y
1+α1 x +α2 y +α3 x y

dt

d B1 (t)

f ωx
δ2 x
−0.5 δ1 +
1+α1 x +α2 y +α3 x y
1+α1 x +α2 y +α3 x y

δ2 x
1 + α1 x + α2 y + α3 x y

2

2

dt

d B2 (t)

Hence, integrating them from 0 to t, yields
lnx(t) = lnx0 + (r − 0.5σ12 )t −
+

0

ax +

ωy
1 + α1 x + α2 y + α3 x y

σ1 σ2 y

σ2 y
+ 0.5
1 + α1 x + α2 y + α3 x y
1 + α1 x + α2 y + α3 x y
t

+
0

123

t

σ1 d B1 (s) + M1 (t)

2

ds
(3.3)


On the dynamics of a stochastic ratio-dependent predator–prey model
t

lny(t) = lny0 + (h − 0.5δ12 )t −
+

by −

0

f ωx
1 + α1 x + α2 y + α3 x y

δ2 x
δ1 δ2 x
+ 0.5
1 + α1 x + α2 y + α3 x y
1 + α1 x + α2 y + α3 x y
t

+

δ1 d B2 (s) + M2 (t)

2

ds
(3.4)

0
σ2 y
d B1 (s), M2 (t) =
where M1 (t) = 0 1+α1 x+α
2 y+α3 x y
martingales and the quadratic variations are
t

t

< M1 (t), M1 (t) > =
0

t

< M2 (t), M2 (t) > =
0

t
δ2 x
0 1+α1 x+α2 y+α3 x y d B2 (s)

σ2 y
1 + α1 x + α2 y + α3 x y

2

δ2 x
1 + α1 x + α2 y + α3 x y

2

are

ds,
ds.

By Virtue of the exponential martingale inequality(see [20]), for any positive constants
T, α and β, we have
P

sup
0≤t≤T

1
M(t) − α < M(t), M(t) >
2

> β ≤ e−αβ

Choose T = n, α = 1, β = 2lnn, we get
P

sup
0≤t≤n

P

sup
0≤t≤n

1
< M1 (t), M1 (t) >
2
1
M2 (t) − < M2 (t), M2 (t) >
2
M1 (t) −

1

n2
1
> 2lnn ≤ 2
n

> 2lnn ≤

An application of Borel-Cantelli lemma (see [20]) yields that for almost all ω ∈
there is a random integer n 0 = n 0 (ω) such that for n ≥ n 0
sup
0≤t≤n

sup
0≤t≤n

1
< M1 (t), M1 (t) >
2
1
M2 (t) − < M2 (t), M2 (t) >
2
M1 (t) −

,

≤ 2lnn
≤ 2lnn

That is to say,
t

M1 (t) ≤ 2lnn + 0.5
0

t

M2 (t) ≤ 2lnn + 0.5
0

σ2 y
1 + α1 x + α2 y + α3 x y

2

δ2 x
1 + α1 x + α2 y + α3 x y

2

ds,
ds

123


Y. Zhang et al.

for all 0 ≤ t ≤ n, n ≥ n 0 a.s. Substituting the above inequality into Eq. (3.3)
and (3.4),respectively, and we get that
lnx(t) ≤ lnx0 + (r − 0.5σ12 )t +

lny(t) ≤ lny0 + h +

t

σ1 d B1 (s) + 2lnn,

0

fω
− 0.5δ12 t +
α1

t

δ1 d B2 (s) + 2lnn.

0

It finally follows from (3.3) and (3.4) by dividing t (0 < n − 1 ≤ t ≤ n) on the both
side and letting t → ∞ that
lim sup
t→∞

fω
lnx(t)
lny(t)
≤ r − 0.5σ12 , lim sup
≤h+
− 0.5δ12 a.s.
t

t
α1
t→∞

By assumption (H2 ), the desired assertion is derived.
4 Numerical simulation and discussion
Numerical verification of the results is necessary for completeness of the analytical
study. In this section, we present some numerical simulations to substantiate and
augment our analytical findings of system (1.2) by means of the Milstein method
mentioned in Higham [27]. From model (1.2), we consider the discretization equations:
xk+1 = xk + xk r − axk −
+ xk σ1

yk+1

√

tξk +

ωyk
1 + α1 xk + α2 yk + α3 xk yk

t

√
σ12 2
σ2 yk
(ξk − 1) t +
tξk
2

1 + α1 xk + α2 yk + α3 xk yk

2
yk
(ξk2 − 1) t ,
1 + α1 xk + α2 yk + α3 xk yk
f ωxk
t
= yk + yk h − byk +
1 + α1 xk + α2 yk + α3 xk yk
√
√
δ2
δ2 xk
+ yk δ1
tηk + 1 (ηk2 − 1) t +
tηk
2
1 + α1 xk + α2 yk + α3 xk yk

+

σ22
2

+

δ22
2

xk
1 + α1 xk + α2 yk + α3 xk yk

2

(ηk2 − 1) t .

(4.1)

where time increment t > 0, and ξk , ηk , k = 1, 2, ..., n are independent Gaussian
random variables N (0, 1) which can be generated numerically by pseudorandom number generators. In order to understand their role on the dynamics, we use different
values of σ1 , σ2 , δ1 and δ2 .
In all the following figures, the imaginary lines and the real lines in the figures
represent solutions of the deterministic system (1.1) and the stochastic system (1.2),
respectively.

123


6

6

5

5

4

4

y(t)

x(t)

On the dynamics of a stochastic ratio-dependent predator–prey model

3

3

2

2

1

1

0
0

20

40

60

80

0
0

100

20

40

t

60

80

100

t

6

6

5

5

4

4

y(t)

x(t)

Fig. 1 The trajectories of deterministic system (1.1) and stochastic system (1.2) with σ1 = 0.3, σ2 =
0.2, δ1 = 0.2, δ2 = 0.3

3

3

2

2

1

1

0

0
0

20

40

60

80

100

0

20

40

t

60

80

100

t

6

6

5

5

4

4

y(t)

x(t)

Fig. 2 The trajectories of deterministic system (1.1) and stochastic system (1.2) with σ1 = 0.1, σ2 =
0.1, δ1 = 0.1, δ2 = 0.1

3

3

2

2

1

1

0
0

20

40

60

t

80

100

0
0

20

40

60

80

100

t

Fig. 3 The trajectories of deterministic system (1.1) and stochastic system (1.2) with σ1 = 0.3, σ2 =
0.02, δ1 = 0.2, δ2 = 0.03

In Figs. 1, 2, 3, 4, 5, 6, we always choose initial values x(0) = 0.3, y(0) = 0.2
and parameters r = 2, a = 1, ω = 1, f = 0.5, h = 1, b = 2, α1 = 4, α2 = 5, α3 =
6. The only difference between Figs. 1 and 2 is the different intensities of white

123

6

6

5

5

4

4

y(t)

x(t)

Y. Zhang et al.

3

3

2

2

1

1

0

0

20

40

60

80

0

100

0

20

40

t

60

80

100

t

Fig. 4 The trajectories of deterministic system (1.1) and stochastic system (1.2) with σ1 = 0, σ2 =
0.2, δ1 = 0, δ2 = 0.3
6

5

y(t)

4

3

2

1

0

0

0.5

1

1.5

2

2.5

3

3.5

x(t)
Fig. 5 The stationary distribution of stochastic population system (1.2) with σ1 = 0.3, σ2 = 0.2, δ1 =
0.2, δ2 = 0.3

noise which satisfy condition (H1). By Matlab software, we simulate the solution of
model (1.2) with different values of σ1 , σ2 , δ1 and δ2 and the solution of model (1.1).
In Fig. 1, we choose σ1 = 0.3, σ2 = 0.2, δ1 = 0.2, δ2 = 0.3 and in Fig. 2, we
choose σ1 = 0.1, σ2 = 0.1, δ1 = 0.1, δ2 = 0.1. From the Figs. 1 and 2, we can see
that the curves of model (1.2) always fluctuate around the curves of the deterministic
system (1.1) and moreover, by comparisons of the two figures, we can also obtain
that the larger the intensities of the white noises are, the larger the fluctuations of the
solutions will be.
Furthermore, in Fig. 3, we keep σ1 , δ1 the same value as in Fig. 1, but take σ2 , δ2
smaller values σ2 = 0.02, δ2 = 0.03. Then we can observe that the fluctuations will
be very smaller, which suggest that the intensities of σ2 , δ2 have little effect on the
fluctuation. Next, in Fig. 4, we choose σ1 = δ1 = 0, and from Figs. 1 and 4, we can see

123


6

6

5

5

4

4

y(t)

x (t)

On the dynamics of a stochastic ratio-dependent predator–prey model

3

3

2

2

1

1

0
0

20

40

60

t

80

100

0
0

20

40

60

80

100

t

Fig. 6 The solution of the stochastic system (1.2) with σ1 = 2.3, σ2 = 0.2, δ1 = 2, δ2 = 0.3. The system
is stochastic extinction

that the fluctuation for x(t) and y(t) almost can not be seen, then we can conjecture
that the model (1.2) is stochastically stable in the large. Moreover, the stationary
distribution and solutions of stochastic population system (1.2) are also given in Fig.
5, from which we can also obtain the stochastic permanence of system (1.2).
At last, in Fig. 6, we choose the intensities of white noise as σ1 = 2.3, σ2 =
0.2, δ1 = 2, δ2 = 0.3, which satisfy the condition (H2). Both species x and y will
go to extinction and the conclusion of Theorem 6 is verified. Also, by comparing
Figs. 1 and 6, we can obtain that if the environment noise is small, the stochastic
system can maintain permanent while the system can be extinct under sufficiently
large environmental noise.
Acknowledgments
The research have been supported by The Natural Science Foundation of China
(11261004) and The Natural Science Foundation of Jiangxi Province(20122BAB211010).

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