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# A stochastic ratio-dependent predator-prey model under regime switching

- Jingliang Lv
^{1}Email author and - Ke Wang
^{1, 2}Email author

**2011**:14

https://doi.org/10.1186/1029-242X-2011-14

© Lv and Wang; licensee Springer. 2011

**Received:**21 October 2010**Accepted:**22 June 2011**Published:**22 June 2011

## Abstract

This article presents an investigation of asymptotic properties of a stochastic ratio-dependent predator-prey model under regime switching. Both the white and color noises are taken into account in our model. We obtain the global existence of positive unique solution of the stochastic model. And we show the solution is bounded in mean. Moreover, the sufficient conditions for persistence in mean, extinction are obtained.

## Keywords

*Itô*formula- Irreducible
- Exponential martingale inequality
- Comparison theorem
- Persistent in mean
- Extinct

## 1. Introduction

where *x*(*t*) and *y*(*t*), respectively, denote population densities of prey and predator at time *t*. Here, *g* > 0 is the death rate of the predator, *a*, *c*, *m*, and *f* are positive constants that stand for prey intrinsic growth rate, capturing rate, half capturing saturation constant, and conversion rate, respectively.

where the perturbed terms *η*_{1}(*t*) and *η*_{2}(*t*) are uncorrelated Gaussian white noises. Maiti et al. [2] assumed the Stratonovich interpretation of stochastic differential equations, and discussed the properties of SDE (1.2) by using transformations.

where *a*, *d*, *c*, *f*, *g*, and *m* are positive constants.

*Itô*equation

where *B*_{
i
}(*t*), *i* = 1, 2, are independent standard Brownian motions.

*N*regimes, and the switching between these

*N*regimes is governed by a Markov chain

*r*(

*t*) on the state space

*S*= {1, 2,...,

*N*}. Therefore, when both the white and color noises are taken into account in the system (1.1). The population system under regime switching can be described by the stochastic model as follows:

where *B*_{
i
}(*t*), *i* = 1, 2, are independent standard Brownian motions.

When both the white and color noises are taken into account in our model (1.5), we obtain the global existence of positive unique solution of the stochastic model, that is, the solution of the system is positive and not to explode to infinity in a finite time in Section 3. Section 3 also shows that the solution is bounded in mean. Moreover, the sufficient conditions for persistence in mean, extinction are obtained in Section 4.

And throughout the article, we use *K* to denote a positive constant the exact value of which may be different in different appearances.

## 2. Stochastic differential equation under regime switching

*r*(

*t*),

*t*≥ 0, be a right-continuous Markov chain in the probability space tasking values in a finite state space

*S*= {1, 2,...,

*N*} with generator Γ = (

*γ*

_{ ij })

_{N×N}given by

where Δ > 0. Here, *γ*_{
ij
} ≥ 0 is the transition rate from *i* to *j* if *i* ≠ *j* while
. We assume that the Markov chain *r*(*t*) is independent of the Brownian motion. And almost every sample path of *r*(*t*) is a right-continuous step function with a finite number of simple jumps in any finite subinterval of *R*_{+}.

*N*-1. Under this condition, the Markov chain has a unique stationary distribution

*π*= (

*π*

_{1},

*π*

_{1},...,

*π*

_{ N }) ∈

*R*

^{1×N}which can be determined by solving the following linear equation

*k*= 1, 2,..., there is

*h*

_{ k }> 0 such that

for all (*x*, *t*, *i*) ∈ *R*^{
n
} × *R*_{+} ×*S*.

*C*

^{2,1}(

*R*

^{ n }×

*R*

_{+}×

*S*,

*R*

_{+}) denote the family of all non-negative functions

*V*(

*x*,

*t*,

*i*) on

*R*

^{ n }×

*R*

_{+}×

*S*which are continuously twice differentiable in

*x*and once differentiable in

*t*. If

*V*∈

*C*

^{2,1}(

*R*

^{ n }×

*R*

_{+}×

*S*,

*R*

_{+}), define an operator

*LV*from

*R*

^{ n }×

*R*

_{+}×

*S*to

*R*by

## 3. Positive, global and bounded solutions

As *x*(*t*), *y*(*t*) of the SDE (1.5) are sizes of the species in the system at time *t*, it is obvious that the positive solution are of interest. The coefficients of (1.5) are locally Lipschitz continuous and do not satisfy the linear growth condition, so the solution of (1.5) may explode at a finite time. The following theorem shows that the solution will not explode at a finite time.

**Theorem 3.1**. *For given initial value*
, *there is a unique positive solution X*(*t*) = (*x*(*t*), *y*(*t*)) *to (1.5) on t* ≥ 0, *and the solution will remain in*
*with probability one*, *namely*
*for all t* ≥ 0 *almost surely*.

*Proof*The proof is similar to [10, 11]. Since the coefficients of the equation are locally Lipschitz continuous, for given initial value , there is a unique local solution

*X*(

*t*) on

*t*∈ [0,

*τ*

_{ e }), where

*τ*

_{ e }is the explosion time. To show this solution is global, we need to show that

*τ*

_{ e }= +∞ a.s. Let

*k*

_{0}> 0 be sufficiently large for every component of

*x*(

*t*) and

*y*(

*t*) all lying within the interval . For each integer

*k*≥

*k*

_{0}, define the stopping time

*τ*

_{ k }is increasing as

*k*→ ∞. Let

*τ*

_{∞}= lim

_{k→∞}τ

_{ k }, whence

*τ*

_{ ∞ }≤

*τ*

_{ e }a.s. If we can show that

*τ*

_{ ∞ }= ∞ a.s., then

*τ*

_{ e }= ∞ a.s. and a.s. for all

*t*≥ 0. So we just to prove that

*τ*

_{ ∞ }= ∞ a.s. If not, there is

*ε*∈ (0, 1) and

*T*> 0 such that

*k*

_{1}≥

*k*

_{0}such that

*P*{

*τ*

_{ k }≤

*T*} ≥

*ε*for all

*k*≥

*k*

_{1}. Define a function by

*V*(

*x*,

*y*) = (

*x*-1 - ln

*x*) + (

*y*-1 - ln

*y*). The non-negativity of this function can be seen from

_{ k }=

*τ*

_{ k }≤

*T*for

*k*≥

*k*

_{1}then

*P*(Ω

_{ k }) ≥

*ε*. Note that for every

*ω*∈ Ω, there is

*x*(

*τ*

_{ k },

*ω*) or

*y*(

*τ*

_{ k },

*ω*) equals either

*k*or , and hence,

*V*(

*x*(

*τ*

_{ k },

*ω*)) is no less than either

So, we have that *τ*_{
∞
} = ∞ a.s. The proof is complete.

So, we complete the proof.

## 4. Asymptotic behavior

### 4.1. Limit results

To demonstrate asymptotic properties of the stochastic system (1.5), we discuss the long time behavior of ln *x*(*t*)/*t* and ln *y*(*t*)/*t*.

Here we impose the following assumption:

**(H)**
.

*X*

_{1}as the solution of stochastic differential equation

*V*(

*t*,

*X*

_{2}) =

*e*

^{ t }ln

*X*

_{2}, using the generalized

*Itô*formula, we obtain

Choose *T* = *kγ* , *δ*= *nεe* ^{-kδ}, and
, where *k* ∈ *Z*^{+}, 0 < *ε* < 1, *θ* > 1 and *γ* > 0 above.

_{ i }⊂ Ω with

*P*(Ω

_{ i }) = 1 such that for any

*ω*∈ Ω

_{ i }, an integer

*k*

_{ i }=

*k*

_{ i }(

*ω*) such that for any

*k*>

*k*

_{ i }, we get

as desired.

as required.

*Proof* Under the condition
, by the same way of Lemma 4.1, we can imply the desired assertion.

**Theorem 4.3**.

*Assume the conditions (H) hold. Then for any initial value x*

_{0}> 0,

*the solution x(t) to (1.5) satisfies*

*Proof* By (4.3), Lemmas 4.1 and 4.2, we can conclude the assertion.

*y*(

*t*). By the comparison theorem of stochastic equations, we have

*Y*

_{1}as the solution of stochastic equation

**Lemma 4.4**.

*Under Assumption (H), for any initial value y*

_{0}> 0,

*the solutions Y*

_{1}(

*t*)

*and Y*

_{2}(

*t*)

*satisfy*

*Proof*It is clear that the quadratic variation of the stochastic integral is . hence, the strong law of large numbers of local martingales yields that

as required.

**Theorem 4.5**.

*Under Assumption (H), for any initial value y*

_{0}> 0,

*the solution y*(

*t*)

*to (1.5) has the property*

The proof is complete.

### 4.2. Persistent in mean

As we know, the property of persistence is more desirable since it represents the long-term survival to a population dynamics. Now, we present the definition of persistence in mean proposed in Ji et al. [4] and [12].

**Theorem 4.7**. *Assume the condition (H) hold. Then system (1.5) is persistent in mean*.

*t*, letting

*t*→ ∞ and by the strong law of large numbers and Theorem 4.3, we obtain

*t*, letting t → ∞ and by the strong law of large numbers and Theorem 4.5, we have

So, the system is persistent in mean.

### 4.3. Extinction

In Section 4.2, under the condition (H), we show that the system is persistent in mean. To a large extent, (H) as the condition that stands for small environmental noises. That is, small stochastic perturbation does not change the persistence of the system. Here, we will consider that large noises may make the system extinct.

**Theorem 4.7**. *Assume the condition*
,
*hold. Then system (1.5) will become extinct exponentially with probability one*.

The proof is complete.

## 5. Conclusions

Both the white and color noises are taken into account in our model in this article. It tells us that when the intensities of environmental noises are not too big, some nice properties such as non-explosion, boundedness, and permanence are desired. However, Theorem 4.7 reveals that a large white noise will force the population to become extinct while the population may be persistent under a relatively small noises.

## Declarations

### Acknowledgements

This research is supported by the National Natural Science Foundation of P.R. China (No. 10701020)

## Authors’ Affiliations

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