Periodic Solution for a Stochastic Predator Prey Model with Impulses and Holling II Functional Response

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https://www.scirp.org/journal/jamp ISSN Online: 2327-4379

ISSN Print: 2327-4352

DOI: 10.4236/jamp.2019.710152 Oct. 8, 2019 2212 Journal of Applied Mathematics and Physics

Periodic Solution for a Stochastic

Predator-Prey Model with Impulses and

Holling-II Functional Response

Yafei Yang, Yuanfu Shao, Mengwei Li

College of Physics, Guilin University of Technology, Guangxi, China

Abstract

Considering the mutual interference between species, a stochastic preda-tor-prey model with impulses and Holling-II functional response is proposed in this paper. Firstly, by constructing an equivalent system without impulses, the existence of a globally unique positive solution is proved. Secondly, in cases of the mutual coefficient m = 1 and 0 < m < 1, by constructing suitable Lyapunov functional, the existence of T-periodic solution is investigated un-der some certain conditions. Finally, numerical simulation is introduced to verify our main results.

Keywords

Stochastic, Impulses, Mutual Interference, Periodic Solution

1. Introduction

The interaction between predator and prey has long been one of the themes of mathematical biology because of the ubiquity and importance of predation. Due to the imbalance of species in the ecological environment, some species are en-dangered and many of them have become extinct. Therefore, protecting the di-versity of ecological species has become one of the main topics in today’s society. This will inspire more scholars to devote themselves to this research.

It is well known that, in the ecosystem, many factors affect the dynamics of ecological models. One of the key elements is called “functional response”, which represents the consumption per unit of time. Holling-II functional

re-sponse is one of the most important functional rere-sponses [1]. In the past few

decades, the deterministic predator model has attracted much attention. For

example, Li and Gao [2] introduced the following predator-prey system with

How to cite this paper: Yang, Y.F., Shao, Y.F. and Li, M.W. (2019) Periodic Solution for a Stochastic Predator-Prey Model with Impulses and Holling-II Functional Re-sponse. Journal of Applied Mathematics and Physics, 7, 2212-2230.

https://doi.org/10.4236/jamp.2019.710152

Received: August 31, 2019 Accepted: October 5, 2019 Published: October 8, 2019

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/jamp.2019.710152 2213 Journal of Applied Mathematics and Physics

Holling II functional response:

1

1 1

2

2 2

d d ,

1

d d ,

1

c y x x r b x t

x c x y y r b y t

x

 

= _ − − _ +

 

 

= − 

− + 



 + 



 

 

(1.1)

where x t

( ) ( )

,y t stand for prey and predator densities at time t, respectively. Parameters r r b b c1, , ,2 1 2, 1 and c2 are positive constants; r1 and r2 stand for

intrinsic growth rates of prey x t

( )

and predator y t

( )

respectively.

Parame-ters b1 and b2 describethe strength of competition among individuals of

spe-cies x t

( )

or y t

( )

. Parameters c1 and c2 represent the capture rate of

pre-dators and the reproduction rate of converting nutrients into prepre-dators, respec-tively.

However, in order to better describe the phenomenon in population dynamics, Hassell initially proposed a nonlinear function of the interaction size of species. He found that as the population grew, the interference became stronger.

There-fore, he introduced the concept of mutual interference constant m∈

(

0,1

]

(see

e.g. [3][4][5]). The deterministic predator-prey model with mutual interference

and Holling-II function response can be expressed as

( )

( ) ( )

1

1 1

2 2

d d ,

1

d d ,

1 m

m

c y

x x t r b x t

x

c x

y r y b y

t

t t

t

t t t

x y t t

 

= _ − − _ +

 

 

= − 

− +

 

 +

 

 

  

(1.2)

In recent years, system (1.2) and its various extension forms have been exten-sively studied by scholars (e.g. [6][7]).

The growth of species in nature is often restricted by the environment. Be-cause of environmental fluctuations, the parameters involved in the population model are not constant, and they may fluctuate around some average values. Based on this factor, more and more people begin to pay attention to the

ran-dom population system [8][9][10]. We assume that environmental fluctuations

mainly affect the internal growth rate r t1

( )

and the mortality r t2

( )

of

preda-tors, that is,

( )

( ) ( )

1 1 1 1

r t →r t +σ t B′ t , −r2

( )

t → −r2

( )

t +σ2

( ) ( )

t B2′ t .

where B1 and B2 are independent Brownian motions,

( )(

)

2

1, 2 i t i

σ =

de-notes the intensity of white noise. That is, we consider the following stochastic non-autonomous predator-prey system with Holling-II functional response and mutual interference:

( )

( ) ( )

( )

( ) ( )

( )

1

1 1 1 1

2 2

2 2 2 2

d d d

1

d d d

1

,

, m

m

c y

x x r b x t x B

x

c x

y r y b

t

t t t t t

t

t t y t y t t t y B

x t t

σ σ

 

= _ − − _ + +

 

 

= − − 

+ +

 ₊ 

 

     

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On the other hand, the life of species often has some big abrupt changes, such as drought, earthquake, typhoon and other big natural disasters, as well as inter-ference from human activities, such as large-scale hunting, policy protection, etc., which will bring great changes to the number and density of species in a short period of time. Therefore, the interference of impulses to the model needs to be considered, and the following model can be established:

( )

( ) ( )

( )

_{( ) ( )}

( ) ( ) ( )

( )

( ) ( )

( )

_{( ) ( ) ( )}

( ) ( )

( )

1

1 1 1 1

2 2

2 2 2 2

d d d

1

d d d

1

, 1, 2, .

,

3, m

k m

k k k k

k

k k k k

c

x x r b x y t x B

x

t t c

y r y b y x y t y B

x

x t x t x t

t

t t t

t t k

y t y t y t

t t t t t t

t

t t t t t t t t t t

t

σ σ α

β

+

 

= _ − − _ +

+

  _≠

 

= −_ − + _ +

+

 

− =

 

 

 



 _



= = − =

 

 _



 



 





(1.4) where r ti

( )

, b ti

( )

, c ti

( )

and

( )(

)

2

1, 2 i t i

σ = _{are positive and continuous}

T-periodic functions; and the time sequences satisfies 0< < < <t1 t2 t3 , and lim _k

k→∞t = +∞. In addition, Parameters α βk, k represent the impulsive effects,

and αk >0, βk >0 denote the planting of the species, and if αk <0, βk <0,

then they represent the harvest of the species. From a biological point of view, we are only looking at the positive solution of this equation. Therefore, it’s a natural constraint that

1+αk >0, 1+βk >0, k=1, 2, 3,.

For the periodicity, we assume that there exists a positive integer p such that

k p k

t + = +t T , αk+p =αk, βk+p =βk, k∈Z. Without loss of generality, we

as-sume

[

0,T

) {

 t kk, ∈Z

}

=

{

t t t1, ,2 3,tp

}

.

From the biological point of view, the population density will change with the changes of some factors, such as rainfall, drought, plague, which is random. All

the possible outcomes from a set Ω_{with typical element} w∈ Ω. A filtration

{ }

t t≥₀ is the smallest σ-algebra σ Ω

( )

, which contains Ω. Throughout this

paper, let

(

Ω,

{ }

t t≥0,

)

is a complete probability space with a filtration

{ }

t

satisfying the usual normal conditions. (i.e., it is increasing and right continuous

while 0 contains all  -null sets), and we define sup

( )

u t

f = _→∞ f t ,

( )

inf l

t

f = →∞ f t .

The main purpose of this paper is to study the existence and uniqueness of global positive periodic solutions as well as the permanence and extinction of species of system (1.4).

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2. Existence and Uniqueness of the Global Positive Solution

Definition 2.1 ([11]). Consider the following impulsive stochastic differential

equation (ISDE)

( )

(

( )

)

(

( )

)

( )

d , d , d , , 0,

, , 1, 2, 3, . k

k k k k k

x t f t x t t g t x t B t t t t

x t+ x t α x t t t k

 = + ≠ >

 

− = = =

  (2.1)

with the initial value

( )

0 0

n

x =x ∈ .

A stochastic process

( )

(

( ) ( )

( )

)

T

[

)

1 , 2 , , n , 0,

x t = x t x t  x t t∈ +∞ is said to be

a solution of ISDE (2.1), if x t

( )

satisfies

1) x t

( )

is t adapted and is continuous on

( )

0,t1 and each interval

(

t tk, k+1

)

,k∈ and

(

( )

)

(

)

1

, , n

f t x t ∈L  + , g t x t

(

,

( )

)

∈L2

(

 +, n

)

;

2) x t

( )

obeys the equivalent integral equation of (2.1) for almost every

\ k

t∈+ t and satisfies the impulsive conditions at each t∈+,k∈ a.s.;

3) For each t k_k, ∈,

( )

lim

( )

k

k t t

x t+ ₊x t

→

= and

( )

lim

( )

k

k t t

x t− ₋x t

→

= exist and

( )

k

( )

k

x t− =x t with probability one.

As to the existence and uniqueness of global positive solution of system (1.4), we have the following result.

Theorem 2.1 For any initial value

(

)

2

0, 0

x y ∈R+, system (1.4) has a unique

global positive solution

(

x t

( ) ( )

,y t

)

for t≥0 and the solution remains in +

 with probability one.

Proof. First, we construct the following stochastic differential equation(SDE) without impulses:

( )

( ) ( )

(

)

( ) ( ) ( )

( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

(

)

( ) ( ) ( )

( )

( ) ( ) ( )

( ) ( )

( )

( ) ( )

( )

1 1 1 1 1 1

1

2 2 1 1 1

1 1

2 2 2 2 2 2

1

1 1

1 2 1 2

2

2 2 2

1 1

1

d ln 1

d d ,

1

d ln 1

d d ,

1

p

j j

m m

p

j j

m m

y t y t r t b t A t y t

T

c t

A t y t t t y t B t

A t y t

y t y t r t b t A t y t

T

c t

A t A t y t y t t t y t B t

A t y t

α

σ β

σ

=

− −



=  + + −





− +

            



+ _



= _− + + −





+ +

 + 

∑

(2.2) with the initialvalue

(

y1

( ) ( ) (

0 ,y2 0 = x y0, 0

)

. According to the classic theory

of SDE without impulse, SDE (2.2) has a unique global positive solution

( )

t

(

1

( ) ( )

t , 2

)

y = y y t (more details see [11]). Let x t

( )

=A t y t₁

( ) ( )

₁ ,

( )

2

( ) ( )

2

y t =A t y t , then we claim that

(

x t

( ) ( )

,y t

)

is the solution of the

sys-tem (1.4).

In fact, it is easy to check that x t

( )

and y t

( )

are continuous on

( )

0,t1

and

(

t tk,k+1

)

⊂

[

0,+∞

)

,k∈N,

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( )

( ) ( )

( ) ( ) ( )

_{( ) ( )}

( )

( ) ( ) ( )

( )

1 1 1 1

1

1 1 1 1 1 1 2 2

1 1

1 1 1 1

d d d

d 1

d .

m m

x t A t y t t A t y t

c t

A t y t r t b t A t y t A y t A t y t

t A t y t w t

σ ′

= +

 

= _ − − _

+

 

 

+

Similarly, we have

( )

( ) ( )

2

( )

_{( ) ( )}

1

( )

( ) ( )

( )

2 2 2 2

d d d

1

m

c t

y t y t r t b t y t x t y t t t y t w t

x t σ

−

 

= _− − + _ −

+

 

  .

And for every k∈N,

( )

1

( ) ( )

1

(

)

(

)

1

( )

(

)

( )

1 0

lim 1 1 1

k

k _j _k

t

p T

k j j k j k

t t _j _t _t

x t A t y t α α y t α x t

+

−

+ +

→ ₌ _{≤ ≤}

 

= = +  + = +



∏



∏

,

( )

lim 1

( ) ( )

1 1

( )

1

( )

k

k k k k

t t

x t− ₋A t y t A t y t− x t

→

= = = _.

In the same way, we have

( )

_k

(

1 _j

)

( )

_k

y t+ = +β y t , y t

( )

_k− = y

( )

t_k .

This completes the proof.

3. Existence of Positive

T

-Periodic Solution

In this section, we give the existence of the positive periodic solution of the

sto-chastic system (1.4) with impulses. For convenience of readers, we first give the definition of the periodic solution of the impulsive stochastic differential equa-tion in the sense of distribuequa-tion and the results of the existence of periodic solu-tions (see [12][13]).

Definition 3.1 ([13]). A stochastic process ξ

( )

t =ξ ω

( )

t, is said to be

peri-odic with period T, if for every finite sequence of numbers t t1, ,2 ,tn, the joint

distribution of random variables ξ

(

t1+h

) (

,ξ t2+h

)

,,ξ

(

tn+h

)

is

indepen-dent of h, where h=kT (k= ± ±1, 2,).

Consider the following periodic stochastic differential equation without im-pulse:

( )

(

( )

)

(

( )

)

( )

dX t = f t X t, dt+g t X t, dB t t, ≥0, (3.1)

where

(

,

( )

)

n l

g t X t _× is a n l× matrix function, f t X t

(

,

( )

)

and the matrix

( )

(

,

)

n l

g t X t _× are T-periodic in t. Then, Itô’s formula can be applied to

(

,

)

F t X where X satisfies (3.1). This yields the stochastic differential for F of the form

( )

(

)

(

)

(

) (

)

(

)

(

)

(

) (

)

( )

2 2

2

, , 1 ,

d , , , d

2

,

, d .

F t X F t X F t X

F t X t f t X g t X t

t x x

F t X

g t X W t

x

∂ ∂ ∂ 

=_ + + _

∂ ∂ ∂

 

∂ +

∂

(3.2)

Lemma 3.1 ([14]). Assume that system (3.1) has a global solution, and there

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1) LV t x

( )

, ≤ −1 on the outside of some compact set, where

( )

2 2

2

1 ,

2

V V V

LV t x f g

t x x

∂ ∂ ∂

= + +

∂ ∂ ∂ .

2) inf_x_>_R→ ∞, as R→ ∞.

Then (3.1) has a T-periodic solution.

According to Lemma 3.1, we can obtain the main result in this section. Firstly, we can translate system (1.4) into the following two cases:

Case I. When m=1, we have

( )

( ) ( )

( )

_{( ) ( )}

( ) ( ) ( )

( )

( ) ( )

( )

_{( ) ( )}

( ) ( )

( )

1

1 1 1 1

2

2 2 2 2

d d d

1

d d d

1

, 1, 2, 3,

k

k k k k

k

k k k k

c t

x t x t r t b t x t y t t t x t B t

x t

t t c t

y t y t r t b t y t x t t t y t B t

x t

x t x t x t

t t k

y t y t y t

σ σ α

β

+

 

= _ − − _ + +

  _≠

 

= _− − + _ + +

 

 

 

 



 _



 

 _



 

 − =

= =

− = _

 

(3.3) Theorem 3.1 Assume that the following assumption hold

(H1): 1 2 1

( ) ( )

2

u u

c A >c t A t , (H2):

( )

(

)

(

)

2 2

1 2

2

1 2

0

1 1 1

1 1

1

d

2 2

1 1

ln 1 ln 1 0.

l T

u u u

p p

j j

t t

c

r r t

T r b A t t

T T

σ σ

λ

α β

= =

   

= _ −  _{ }− + _

+ _ _{ } _

+ + + + >

∫

∑

Then system (3.2) has a positive T-periodic solution.

Proof. We only need to prove the existence of a periodic solution of the equivalent system (3.3) without impulses as follows:

( )

( ) ( )

(

)

( ) ( ) ( )

( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

(

)

( ) ( ) ( )

( )

( ) ( ) ( ) ( )

( ) ( )

( )

1 1 1 1 1 1

1

2 2 1 1 1

1 1

2 2 2 2 2 2

1

1 1 2 2 2

1 1

2

, 1

d ln 1

d d

1

d ln 1

d ,

d 1

p

j j

p

j j

y y r b A y

T

c

A y t y B

A y

y y r

t t t t t t

t

t t t t t

t t

t t t b t A y

T

c

A y t y B

A y

t t

t

t t t t t

t t

α

σ β

σ

=



=  + + −





− _ +

+ _



= −

      

+ + −

 

+ +

+ 

 

 _

 _

 _



∑

(3.4)

The global existence of the solution has been ensured by Theorem 2.1. Then, we only have to verify the conditions of Lemma 3.1.

Define a C2_-function

(

)

2

, , :

V t x y +→+ as follows:

(

)

(

)

_{( )}

( )

1

2 1 2

1 1 1 2

1 2 3

, ,

ln ln

1 1

, ,

l u u

u u u l

x y x

V x y t

x py

c x c A

M y y MW t

x

r b A r

V V y V t

θ

+

  _ _  +

= − + ₊ −  ₊ + + ₊ +  

  

 

= + +

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where 1

2 l u c p c   =  

 , θ is a constant such that

( )

2 2 1 2

0 min 1,

l u r θ σ    

< < _ _

 

 

. Parameter

0

M > will be given later and W t

( )

satisfies

( )

2

( )

12

( )

22

( )

1 2

1 1 1 2 2

l

u u u

t t

t r t

r b t

c

W r

A

σ σ

λ    

′ = − + _ −  _{ }− + _

+ _ _{ } _. (3.6)

It is easy to check that W t

( )

is a T-periodic function. Indeed

(

)

( )

2 2 1 2 2 1 2

1 1 1

2 2

1 2

0

2

1 1 1

d d 2 2 d 2 2 0. t T t l

u u u

l T

u u u t T

T

t T t

t t

W W

W s s

c

r r s

r b A

c

r r s

r b A

σ σ σ σ + + − ′ =      − − +  _  _{ } _ +             − − +  _  _{ } _ +   + = − +      = 

∫

According to the periodicity of r ti

( ) ( )

,σi t i, =1, 2, W t

( )

is a T-periodic

function. To verify condition (2) of Lemma 3.1, we only need to show that

(_{, ,} )[_0, )

(

2_\

)

(

)

inf , ,

k

t x y∈ +∞ ×R U V t x y → ∞m as k→ ∞.

Here, Uk 1,k 1,k

k k

    =_ _{ }× _

   . All the coefficients of the quadratic term in

(

, ,

)

V t x y are positive, thus, condition (2) of Lemma 3.1 is satisfied.

Next we prove condition (1) of Lemma 3.1. By the Itô’s formula (3.2), we have

(

)

( )

(

)

( ) ( )

2

( ) ( )

_{( )}

1 22

( )

2 2 2

1 1

1

ln ln 1

1 2

p

j j

t t t

t t t c xA

L y r b A y

T xA t

σ β

=

− = − + + − +

+

∑

, (3.7)

( )

(

)

( ) ( )

( )

_{( )}

( )

(

)

(

)

( )

(

)

( )

(

( )

( ) ( )

)

( )

1 2

1 1 1

1 1 2 2 1 2 2 1

1 1 1 1

1

2 2

1 1 2

1 1 ln 1 1 1 ln 1 1 1 1 2 1 1 ln 1 2 1 1

2 1 1

p j j p j j x L x c yA

r b A x

x T xA

x x x

x

r r b A

T

t t

t t t

x x c yA

xA A

t t

t

t t t t

t t x t t t α σ σ α σ = = ₋    _ ₊ _     = − _ + + − − _ + __ + __ + − + +   ≤ − + + − + + +     − + + +

∑

( )

(

)

( )

(

)

2 1

1 1 2

1

1 1 1

1 1

ln 1

2

1

ln 1 .

1

p

j j

p u u u

j j

r t t c t yA t T

x

r b A

x T σ α α = =   ≤ − + + − +       + _ + + + _ +  

∑

(3.8)

(8)

( )

(

)

( ) ( )

( )

(

)

( )

(

)

( )

(

)

2 2

2 2 2

1 2 1 2 1 1

1 1 1

1 1 1 1 2 1 1 1 1 ln 1 2 1 ln 1 2 1 ln 1 1 1 , ln 1 p j j l p j u u u

j p j j l p j u u u

j

t

L r b t A y

T

c r

T

r b A

c xy Mq y T x c M T r

V x y M t t

t b t A t σ β σ α β α = = = = − + + +    − _ + + − _ + _ __   +  + +  +   ≤    + + + 

∑

(3.9)

where 2 1 2

1 1 1 2

l u u

u u u l

c c A q

r b A r

= ×

+ . From (3.5) and (3.8), we have

( )

(

)

( )

(

)

( )

(

)

( )

(

)

( )

(

)

( )

1 3

2 2

2 2 1

2 1

1 1 1 1 1

2

1 2 1

1

1 1 1 1 1

2 2 2

1

1 1

ln 1 ln 1

2 2

1 1

ln 1 ln 1

1 2 1 ln 1 2 l p p

j u u u j

j j

l

p p

j u u u j

j j p j j LV LV c

M r r

T r b A T

c xy c

Mq y M r

T x r b A T

t t

t

c

M M r M

T t t t t σ σ β α σ β α σ λ β = = = = = +    ≤  − + + − _ + + − _ +          + _ + + _+ _ + + − _ + +         − −  − + + +    

∑

2

(

)

1

1 1 1

1 1 ln 1 1 l p j u u u

j u

T

r b A

c xy

M Mq C

x α λ = + + ≤ − + + +

∑

(3.10)

Here,

(

)

2

(

)

1 1 1 1 1

1 1

ln 1 ln 1

l

p p

j u u u j

j j

c C Mq y M

T = β r b A T = α

= + + +

+

∑

, and

( )

(

)

( )

(

)

( ) ( )

(

)

( )

_{( )}

( )

( ) ( )

(

)

( )

2 2

1 1 1

1 1

1 2

2 2

2 2 1 2

1

1 2 2 2 2 2

1 2

,

1 1

ln 1 ln 1

1

2

p p

j j

LV x y

x py r x x b A x y

T T

c pc

p b y

t t t

t t

pr y xyA A

xA

x py x

t t p y t t t t t θ θ α β

θ _σ _σ

= = − +   = + _ + + − +  − + − − + + _   + + _ + _

∑

(

)

( )

(

)

(

)

(

)

( )

2

2 1 1 2 2 1 1

1 1 1 2 2 2

2 ₁ 1 1 1 2 1 2 1 1 1

2 2 1

2

1 1

ln 1 ln 1

2

2 2 2

u l l l l u

p p u j j j j l l l u

x py r x b A x p r y p b y p y

x py x y x

T T

b A x p

r y C

θ θ θ θ θ θ θ θ

θ θ

θ

θ _σ θ

α β σ

θ σ + + + + + + + + = = + + + + − − − +   + + _ + + + _+     ≤ − − _ − _ + ≤  

∑

(3.11)

where, ( )

( )

(

)

( )

(

)

(

)

(

)

2 2 1 2 1 1 1

1 _, 2 2 1

2 1 1

1 1

sup

2 2 2

1 1

ln 1 ln 1 ,

2

l l

l u u

x y R

p p

u

j j

b A x p

C r y x py r x

x x py x y

T T θ θ θ θ θ θ θ σ

θ σ α β

(9)

and in (3.11), we use

(

x+py

)

θ ≤xθ,

(

x+py

)

θ ≤p yθ θ in the second

inequali-ty. Then we have

(

)

( )

1 2 3

2 1

2 ₁

1 1 1

2 2 2

, ,

.

1 2 2 2

u l l

l u

LV t x y LV LV LV

c xy b A x p

M Mq r y C

x

θ θ

θ

λ + + σ +

= + +

 

≤ − + − − _ − _ +

+  

(3.12)

where C2 = +C C1.

Let _{( )} 2

( )

2 1

2 1 1 1

2 2 2

,

2

max 2, sup

4 4 2

l l

l u

x y R

b A x p

M r y C

θ θ

θ

θ σ

λ +

+ +

+ ∈

  

   

= _ _− − _ − _ + _

 

  

 .

To confirm the condition (1) of Lemma 3.2, we choose a sufficiently small con-stant ε_{such that:}

( )

1

2 1 1

2 2

0 , ,

4 4 2 4

l l

l u b A

p r

q Mq Mq

θ

λ θ

ε  +  σ   < ≤_ _ − _ _

 

 , (3.13)

1 1 3

2 1

4 l l

b A

Mλ _θ C

ε +

− − + ≤ − , (3.14)

( )

1

2

2 2 4

2 1

2 4

l u

p

M r C

θ

λ σ

ε

+

 

− − _ − _+ ≤ −

  . (3.15)

Define a bounded closed set as follows:

( )

2 1 1

, : ,

D x y R ε x ε y

ε ε

+

 

=_ ∈ ≤ ≤ ≤ ≤ _

 .

Denote

(

)

{

}

1 _, 2_{| 0}

D_ε = x y ∈R₊ < ≤x ε , _D2

{

(

_{x y}_,

)

_R2_{| 0} _y

}

ε = ∈ + < ≤ε ,

( )

3 2 1

, |

D_ε x y R x

ε

+

 

=_ ∈ ≥ _

 _,

( )

4 2 1

, |

D_ε x y R y

ε

+

 

=_ ∈ ≥ _

 _.

Clearly, C 1 2 3 4

D_ε =D_εD_ε D_εD_ε.

Now we prove LV t x y

(

, ,

)

≤ −1 on each domain.

Case 1. If

(

)

[

)

1

, , 0,

t x y ∈ +∞ ×D_ε, then it is easy to verify that

(

1

)

1 1

xy

xy y y

x

θ

ε ε +

≤ ≤ ≤ +

+ , and

(

)

( )

( ) 2

( )

2 1

2 ₁

1 1

2 2

2 1

2 ₁

1 1

2 2 2

, , ,

4 4 4 4 2

sup .

2 4 4 2

l l

l u

l l

l u

x y R

LV t x y b A x

M M p

Mq r Mq y

b A x

M p

r y C

θ θ

θ

θ θ

θ

λ λ _ε θ _σ _ε

λ _{θ σ}

+

+ +

+

+ +

+ ∈

 

   

≤ − − + −_ + _+ −_ _ − _+ _

  _   _

  _ _ 

+ − + − −  −  + 

 

  

 

Combining with the definition of 1

4

Mλ _≥

and (3.13), we have

(

)

1 1 2

, , 1

4 4 4

l l

b A x

M M

LV t x y

θ

λ + λ

≤ − − ≤ − ≤ − _,

(10)

Case 2. Similarly, for any

( )

2

,

x y ∈Dε , owing to

(

1 2

)

1

xy

xy x x

x

θ

ε ε +

≤ ≤ ≤ +

+ ,

we have that

(

)

( )

( ) 2

( )

1

2 ₁ ₂

1 1

2 2

1

2

2 1

1 1

2 2 2

, , ,

4 4 4 2 4

sup .

2 4 4 2

l l

l u

l l

l u

x y R

LV t x y

b A

M M p

Mq r y Mq x

b A

M p

x r y C

θ

θ θ

θ

θ θ

λ λ _ε θ _σ _ε

λ _{θ σ}

+ +

+

+ +

∈

 

   

≤ − + −_ + _− _ − _ + −_ + _

    _ _

    

+ −_ +_ − − _ − _ + __

 

  

 

Together with (3.13), we can also get

(

)

1

( )

2 ₁

2 2

, , 1

4 4 2 4

l u

M p M

LV t x y r y

θ

λ + θ _σ λ

+

 

≤ − − _ − _ ≤ − ≤ −

  .

Thus, LV t x y

(

, ,

)

≤ −1 for all

( )

x y, ∈D_ε2.

Case 3. For any

( )

3

,

x y ∈D_ε , since

1

xy y x≤

+ , it is easy to have

(

)

( )

1

2

2 1

1 1 1 1 1

2 2 2

2

1 1 3 2 , ,

4 2 2 1

4

. 4

l l l l u

l u

l l

LV t x y

b A b A p c xy

M x r y Mq C

x b A

M C

θ

θ θ

θ

λ σ

ε λ

ε

+

+ +

+

   

≤ − − + −_ _ − _ + _

+

  _

− +

 ≤ − − +

where _{( )} 2

( )

1

2

2 1

1 1 1

3 sup , ₄ ₂ 2 ₂ 2 ₁ 2

l l u

l u

x y R

b A p c xy

C x r y Mq C

x

θ

θ θ σ θ

+

+ +

∈

   

= − − _ − _ + ₊ + 

  ,

and we get LV t x y

(

, ,

)

≤ −1 in this domain from (3.14).

Case 4. Similarly, for any

( )

4

,

x y ∈D_ε , we have

(

)

( )

2 2

1 1

2 2 1

2 2

2

1 1 1

2

1

2

2 2 4

2

, ,

2 4 2

4

2 1

. 2

4

u u

l l

l l u

l u

LV t x y

p p

M r r y

b A c xy

x Mq C

x p

M r C

θ θ

θ

θ σ θ σ

λ ε

θ

λ σ

ε

+ +

+

   

   

≤ − − _ − _− _ − _

   

   

− + +

+

 

≤ − − _ − _+

 

where _{( )} 2

( )

1

2

2 1

1 1 1

4 sup _, 2 2 2

2 4 2 1

l l u

l u

x y R

b A p c xy

C x r y Mq C

x

θ

θ θ σ θ

+

+ +

∈

 _ _ 

= _− − _ − _ + + _

+

 

  .

It is clear that LV t x y

(

, ,

)

≤ −1 in this domain.

Therefore

(

, ,

)

1

LV t x y ≤ − , for all

(

t x y, ,

)

∈

[

0,+∞ ×

)

DC.

That is to say, Condition (1) of Lemma 3.1 is verified. Thus system (3.2) has a T-periodic solution. This completes the proof.

(11)

Case II. If 0< <m 1, then we have

( )

( ) ( )

( )

_{( ) ( )}

( ) ( ) ( )

( )

( ) ( )

( )

_{( ) ( )}

( )

( ) ( )

( )

1

1 1 1 1

2 1

2 2 2 2

d d d ,

1

d d d ,

1

, 1, 2, 3, . m

k m

k k k k

k

k k k k

c t

x t x t r t b t x t y t t t x t B t

x t

t t c t

y t y t r t b t y t x t y t t t y t B t

x t

x t x t x t

t t k

y t y t y t

σ

σ α

β

−

+

 

= _ − − _ +

+

 

≠

 

= _− − + _ +

 

+

 

 

 

 

 _

 _



 _

 − =  ₌ ₌

− = 



 _





(3.16) Theorem 3.3 Assume that the following assumption holds

( )

(

)

(

)

2 2

1 2

1 1

0 0

1 1

2

1 1

d d

2 2

1 1

ln 1 ln 1 0

T T

p p

j j

t t

r t t r t t

T MT

T T

σ σ

λ

α β

= =

   

= _ − _ − _ + _

   

+ + − + >

∫

∑

Then system (3.16) has a positive T-periodic solution.

Proof. We only need to prove the existence of a periodic solution of the equivalent system (2.2) without impulses. The global existence of the solution has been ensured by Theorem 2.1. Then, we only have to verify the conditions of Lemma 3.1.

Define a C2_-function

(

)

2

4 , , :

V t x y +→+

(

)

(

)

_{( )}

( )

4

1 ₁

1 2 ₂ ₂

1 1 1 1

2 2 1

5 6

, ,

ln ln

1

, .

l

l m _m _u _u

u

u l l

V t x y

b A _y _HA _c

M x y y y A x M W

m

c b c

V V

t

x y t

− ₋

 

 

= _− − + _− + + +

−

 

 

= +

(3.17)

Here, H>0 will be given in (3.18), and W t₁

( )

is T-periodic. Same as case I, so the condition (2) of Lemma 3.1 is satisfied.

Next, by the Itô’s formula, we show that the condition (2) of the Lemma 3.1.

(

)

_{( )}

₍

₎

( )

(

)

1 2

1 1

1 2 ₁

1 2

2

1

2 2

1 2

1

2

1 1 2

2

1 ln 1

1 2

. 1 l

l m

l m m p

m

j

u u

j

l l l l

l

m m

u u

b A m

b A y

L y r

m T

c c

b b b

t

A x

A y

c x

t

A

σ β

−

− −

−

=

− −

 

−  − + + 

 ₋  _ _

   

+ −

+

≤

_∑

(

)

1

( )

(

)

2

( )

1 1

( )

1

1 2

1

ln ln 1

2 p

u

u u u m m

j j

t

L x r t b A x c A y

T

σ α

=

 

− −_ + + − _+ +



≤ 



∑

.

Let

(

)

1 ₁

1 2

2

ln ˆ

1 l

l m _m

u

b A _y

x m c

V

− ₋

− −

−

(12)

( )

(

)

( )

(

)

( )

(

)

( )

(

)

( )

1 ₂

1 2 ₁

2 1 2 2 1 1 2 2 1 2

2 1 1 1 2

2 1 2 1 ln 1 2 1 ˆ . ln 1 2 l l m p m j u j p j j l l l u

m m u u u m m

u

b A m

L y r

T c

r T

b b

A y b A x c A y

c t V t t t σ β σ α − − = = − −   ≤ _ − + + _     −_ + + − _   + + +

∑

By Young inequality, there exists a positive constant H such that

( )

2 2

( )

(

)

2

( )

2 1 1 1

1

1 ln

2

ˆ l u u p ₁

j j

L H A y b A x r

T

t

V t α σ

=

 

≤ + −_ + + − _



∑

. (3.18)

Then

( )

(

)

( )

(

)

( )

2 2

1 1 1

1 2 2 2 1 1 2 1 2 2 1

ˆ _{ln 1}

2 1 . ln 1 u p u u j l j

u p _u

u u m m j

l l

j

HA

L V y b A x r T b

HA

y c A A xy t T b t H b σ α β = =     + ≤ − + + −    _ _     + + +

∑

Therefore,

( )

(

)

( )

(

)

( )

(

)

( )

(

)

5 2

1 1 1 1

1 2 1

2 1 2 2 2

2 1 2 2 2 1 2

2 1 2

1 1 1 1 2 2 2

1 2

,

1

ln 1 ln 1

2 1 ln 1 2 1 ln 1 u p p u u

j l j

j j

p u

u u m m u u

j l

j

l l p

l l u u l l l

j u j t t T t LV x y

HA

M b A x r y

T b

c A A xy r b A y T

b

c A b A x A x r y r

H

t

b A y

T c α β β σ σ α = = = =     ≤  −_ + + − _+ +          + +_ − + + _−       − + _ + + _− _ + _  

∑

.

(

)

( )

(

)

( )

(

)

( )

4 5 6 2 2

1 1 1 2 1 2 1 1

1

2 2

1 2

2 2 1 1 2 2 2 1 1

1 2

2

1 2

1 1

1 2 1 2 1

2 , , , 1 ln 1 1 ln 1 2

u p _u

u u u u m l l

j

l l

j

l l p

u u u u l l l

j u

j

l l l m

u u u m l

m

LV t x y

LV x y LV t HA

M b A x y c A A xy b A x T

b b

c A

b A y A x r y r b A y M

T c

c A b A x

M c A A xy

H H b β α λ λ = = = +   ≤ _ + + + _−       + + _ + + _− _ + _−     ≤ _ − _− −  

∑

( )

2

2 2 2 . 2 l l u b y Q

c + where ( )

(

)

( )

₍

₎

2 2 1 1

1 1 1 2 2 1 1

,

1

2

1 2 ₂ _{1 2} ₂ ₁ ₂

2

1

2 2 2

1

sup ln 1

2 1 ln 1 2 l l p

u u u u u u

j x y R

j

l l _l _l _l _u

p l

j

u u l

j

b A x Q M b A x b A y A x r

T c A _{c r A} _M _A

b y y H y

T

c c b

(13)

Let 1

1

2 Q M

λ +

= _{, then} −λ₁M₁+ = −Q 2. We choose a sufficiently small

con-stant δ _{such that}

( )

2

1 2 2

1 1 1

1

2 1

0 , ,

4 4 4

l l l _l _l

u

c A b _{b A} M c M

λ

δ  

< ≤  

 

 

, (3.19)

1 1

1 1 2 1 1

4 l l

b A

Mλ Q

δ

− − + ≤ − , (3.20)

( )

2

1 2 2

1 1 2 2

2

1 4

l l l

u

c A b

M Q

c

λ δ

− − + ≤ − _{. (3.21)}

Here

( )

2

_{( )}

2

1 2 ₂

1 1

1 2 1 2 1 2

2 2

4 2

l l l l

m

l u u u m

u l

c A b A x

b y M Hc A A y

Q x Q

c b

= − − + + _,

( )

2

_{( )}

2

1 2 ₂

1 1

2 2 1 2 1 2

2 2

2 4

l l l l

m

l u u u m

u l

c A b A x

b y M Hc A A y

Q x Q

c b

= − − + + _.

The following proof is similar to the proof of m=1 and is omitted. This

completes the proof.

4. Extinction and Permanence of (1.4)

In Section 3, we showed that under certain conditions, the system (1.4) has a pe-riodic solution. Because in system (1.4), when 0< <m 1 the predator birth rate

is of the form 2

( )

_{( )}

1

( )

1

m

c x y x

t

t t

−

+ . Therefore, the predator birth rate goes to infinity

when y→0, provided the prey population exists. So in this section, we will

show that if the noise is sufficiently large, the solutions to the associated

stochas-tic model will become extinct with probability one when m=1.

Definition 4.1 [15]. Let x t

( )

be a solution to system (1.4).

1) If lim

( )

0

t→∞x t = a.s., then species x t

( )

is said to be extinct;

2) If

( )

0 1

lim d 0

t

t→∞_t

∫

x s s> a.s., then species x t

( )

is said to be persistent in the

mean.

Theorem 4.1. For any initial value

(

)

2

0, 0

x y ∈R+, the solution

( )

(

( ) ( )

,

)

X t = x t y t of (3.2) obeys

( )

_{( )}

2

( )

₍

₎

1 1

1

ln 1

lim sup ln 1 0

2

p

j t

j

t t

x r

t t T

σ

α

→∞ ≤ − +

∑

₌ + < ,

( )

_{( )}

2

( )

₍

₎

2

2 2

1

ln 1

lim sup ln 1 0

2

p

j t

j

t t

y

r c

t T

σ

β

→∞ ≤ − + − +

∑

₌ + < .

that is, the solution of (3.2) is extinct exponentially with probability. Proof. For system (3.2), using the Itô’s formula, we have

( )

(

)

12

( )

( ) ( )

1 1 1

1

d ln ln 1 d d

2 p

j j

t

x r t t t B t

T

σ

α σ

=

 

≤_ + + − _ +

(14)

and

( )

(

)

( )

22

( )

2 2 2 2

1

d ln ln 1 d d

2 p

j j

t

y r t c t t t B t

T

σ

β σ

=

 

≤ −_ + + + − _ +



∑

 .

Integrate both sides from 0 to t, then we get

( )

_{( )}

₍

₎

2

( )

_{( ) ( )}

1

1 1 1

1

0 0

ln ln 1 1 1

ln 1 d d

2

0 t p t

j j

x x

r s B

t t T

t s

s t

t t

σ

α σ

=

 

−

≤ _ + + − _ +



∑



∫

,

( )

(

)

( )

22

( )

2 2 2 2

1

0 0

ln ln 0

1 1 1

ln 1 d d .

2

t p t

j j

t

s

s s

y y

t

r c s B

t T t t t

σ

β σ

=

−

 

≤ _− + + + − _ +



∑



∫

Let

( )

( ) ( )

0

d , 1, 2

t

i i i

P t =

∫

σ t B t i= , then P t_i

( )

is a local martingale, and

( )

2

( )

2

0 d t

u u

i s i t

σ ≤ σ

∫

.

By the strong law of martingale, we have

( )

lim i 0 . . t

t

t s

P

a

→∞ = .

Hence we can derive

( )

_{( )}

2

( )

₍

₎

1 1

1

ln 1

lim sup ln 1 0 . .

2

p

j t

j

t t

x

r a s

t t T

σ

α

→∞ ≤ − +

∑

₌ + <

( )

_{( )}

₍

₎

_{( )}

2

( )

2

2 2

1

ln 1

lim sup ln 1 0 . .

2 p

j t

j

t t

y

r c a s

t T

σ β

→∞ ≤ − +

∑

₌ + + − <

Thus, lim

( )

0

t→∞x t = , limt→∞y

( )

t =0 a.s..

Theorem 4.2. Assume

( )

12

( )

(

)

1

1 0

1

d ln 1 0

2

t p

j j

r

T s

s

s σ α

=

 

− + + >

 

 

∑

∫

,

( )

22

( )

(

)

2 2

1 0

1

d ln 1 0

2

t p

j j

s s

r c s s

T

σ

β

=

 

− + − + + <

 

 

∑

∫

.

then the predators of system (3.2) will eventually extinction, and prey popula-tions go to persistent.

Proof. By Theorem 4.1, we have

when

( )

(

)

2 2

1 0

1

d ln 1 0

2

t p

j j

s s

r c s s

T

σ

β

=

 

− + − + + <

 

 

∑

∫

, then lim

( )

0

t→∞y t = a.s.

So predators will eventually extinction. Next, we prove the persistence of prey quantity.