# Asymptotic properties and simulations of a stochastic single-species dispersal model under regime switching

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DOI: 10.1007/s12190-013-0669-x

- Cite this article as:
- Zu, L., Jiang, D. & O’Regan, D. J. Appl. Math. Comput. (2013) 43: 387. doi:10.1007/s12190-013-0669-x

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## Abstract

Taking both white noise and colored environmental noise into account, a single-species logistic model with population’s nonlinear diffusion among two patches is proposed and investigated. The sufficient conditions of the existence of positive solutions, stochastic permanence, persistence in mean and extinction are established. Moreover, we use an example and simulation figures to illustrate our main results.

### Keywords

Stochastic permanencePersistent in meanExtinctionEnvironment noise### Mathematics Subject Classification

34F0534E1060H1060H20## 1 Introduction

*x*

_{i}denotes the density dependent growth rate in patch

*i*. The constants

*d*

_{ij}(

*i*,

*j*=1,2,

*j*≠

*i*) are the dispersal rate from

*j*-th patch to

*i*-th patch, and the nonnegative constant

*α*

_{ij}can be selected to represent different boundary conditions in the continuous diffusion case [4]. Allen proved that initial value problem of (1.1) has a unique positive solution on a maximal interval (see [5]), the system is strongly persistent, and the population size can be unbounded or bounded under reversed conditions (see [3]). The authors in [4] extended Allen’s results and obtained the following necessary and sufficient conditions:

- (i)
The system (1.1) possesses a globally stable positive equilibrium point \((x_{1}^{*},x_{2}^{*})\), if the largest eigenvalue of matrix

*A*is less than 0; - (ii)
Every solution of the system is unbounded, if the above condition does not hold.

*A*=(

*a*

_{ij})

_{2×2}, and

*a*

_{ij}=

*d*

_{ij}for

*i*≠

*j*,

*a*

_{11}=−

*b*

_{1}−

*d*

_{12}

*α*

_{12},

*a*

_{22}=−

*b*

_{2}−

*d*

_{21}

*α*

_{21}. That is to say, if (

*b*

_{1}+

*d*

_{12}

*α*

_{12})(

*b*

_{2}+

*d*

_{21}

*α*

_{21})>

*d*

_{12}

*d*

_{21}, then system (1.1) has a globally stable positive equilibrium point.

*a*

_{i}. Suppose where

*B*

_{i}(

*t*) is mutually independent Brownian motion,

*σ*

_{i}is a positive constant representing the intensity of the white noise. Then the stochastic system takes the following form

*N*regimes and the switching between these

*N*regimes is governed by a continuous-time Markov chain

*r*(

*t*),

*t*≥0 on the probability space, taking values in a finite state space

*S*={1,2,…,

*N*} with the generator

*Γ*=(

*γ*

_{uv})

_{n×n}given by where Δ

*t*>0. Here

*γ*

_{uv}is the transition rate from

*u*to

*v*and

*γ*

_{uv}≥0 if

*u*≠

*v*, while

*Γ*always has an eigenvalue 0. The algebraic interpretation of irreducibility is that rank(

*Γ*)=

*N*−1. Under this condition, the Markov chain has a unique stationary (probability) distribution

*π*=(

*π*

_{1},

*π*

_{2},…,

*π*

_{N})∈

*R*

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

*r*(⋅) is independent of the Brownian motion

*B*

_{1}and

*B*

_{2}.

In this paper, in order to obtain better dynamic properties of the SDE (1.4), we arrange the content as follows. In Sect. 2, we show that there exists a positive global solution with any initial positive value under some conditions. We investigate the persistence under two different meanings: stochastic permanence and persistence in mean in Sects. 3 and 4, respectively. Section 5 studies extinction and we get the result that a large intensity of white noise can cause the extinction of populations. Finally we illustrate the main results in Sect. 6.

Throughout this paper, unless otherwise specified, let be a complete probability space with a filtration satisfying the usual conditions (i.e. it is right continuous and contains all *P*-null sets). Let \(R_{+}^{2}\) denote the positive cone of *R*^{2}, namely . For convenience and simplicity in the following discussion, denote *x*(*t*)=(*x*_{1}(*t*),*x*_{2}(*t*)). If *A* is a vector or matrix, its transpose is denoted by *A*^{T}. If *A* is a matrix, its trace norm is denoted by \(|A|=\sqrt{\hbox{trace}(A^{T} A)}\) whilst its operator norm is denoted by ∥*A*∥=sup{|*Ax*|:|*x*|=1}. Let \(\hat{\bar {b}}_{i}=\min_{k\in S}\{\bar{b}_{i}(k)\}\) (*i*=1,2), \(\check{d}_{12}=\max_{k\in S}\{ d_{12}(k)\}\) and \(\check{d}_{21}=\max_{k\in S}\{d_{21}(k)\}\) and we impose the following assumptions:

### Assumption 1

\(\hat{\bar{b}}_{1}\hat {\bar {b}}_{2}>\check{d}_{12}\check{d}_{21}\).

### Assumption 2

For each *k*∈*S*, \(a_{i}(k)-\frac{1}{2}\sigma_{i}^{2}(k) >0\), *i*=1,2.

## 2 Positive and global solutions

Now *x*(*t*) of system (1.4) denotes population densities at time *t*, so we are only interested in the positive solutions. The coefficients of SDE (1.4) do not satisfy a linear growth condition, though they are locally Lipschitz continuous. However, in order for a stochastic differential equation to have a unique global (i.e. no explosion in a finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy a linear growth condition and a local Lipschitz condition (cf. Mao [15]). To solve this problem, we will use a method similar to [16, Theorem 2.1] to prove the solution of (1.4) is nonnegative and global.

### Theorem 2.1

*Let Assumption *1 *hold*. *For any given initial value*\(x(0)\in R_{+}^{2}\), *there is a unique positive solution**x*(*t*) *of system* (1.4), *and the solution will remain in*\(R_{+}^{2}\)*with probability* 1.

### Proof

*C*

^{2}-function \(V : {R}_{+}^{2} \times S\rightarrow{R}_{+}\) by

*c*

_{1}(

*k*) and

*c*

_{2}(

*k*) are positive constants to be determined. The nonnegativity of this function can be observed from

*a*−1−log

*a*≥0 on

*a*>0 with equality holding iff

*a*=1. If \(x\in R_{+}^{2}\), we see that In fact, in order to ensure

*LV*is bounded, we only need

*c*

_{1}(

*k*),

*c*

_{2}(

*k*) satisfying the inequality (2.4). The coefficients of the quadratic terms of

*LV*are negative and we have Making use of the inequality

*a*≤2(

*a*−1−log

*a*)+2 on

*a*>0, we see that Let . By the definition of

*V*(

*x*,

*k*), for any

*k*,

*l*∈

*S*, we have

## 3 Stochastic permanence

From Theorem 2.1 we know that the solution of SDE (1.4) will remain in the positive cone \(R_{+}^{2}\) with probability 1 if Assumption 1 holds. This nice property provides a great opportunity for us to discuss how the solution varies in \(R_{+}^{2}\) in detail. We will first give the definitions of stochastically ultimate boundedness and stochastic permanence.

### Definition 3.1

*ϵ*∈(0,1), there exist positive constants

*χ*

_{1}(=

*χ*

_{1}(

*ϵ*)),

*χ*

_{2}(=

*χ*

_{2}(

*ϵ*)), such that for any initial value \(x(0)\in R_{+}^{2}\), the solution of the SDE (1.4) has the property that

*x*

_{1}(

*t*),

*x*

_{2}(

*t*)) is the solution of SDE (1.4) with any initial value \(x(0)\in R_{+}^{2}\).

### Definition 3.2

*ϵ*∈(0,1), there are positive constants

*χ*

_{1}(=

*χ*

_{1}(

*ϵ*)),

*χ*

_{2}(=

*χ*

_{2}(

*ϵ*)) and

*δ*

_{1}(=

*δ*

_{1}(

*ϵ*)), \(\delta'_{1} (=\delta'_{1} (\epsilon))\) such that

It is clear that if the system is stochastically permanent, it must be stochastically ultimately bounded.

### Lemma 3.1

*Under Assumption*1,

*for any given initial value*\(x(0)\in R_{+}^{2} \),

*there exists a positive constant*

*ι*(

*p*)

*such that the solution*

*x*(

*t*)

*of SDE*(1.4)

*has the following property*:

### Proof

*x*(

*t*) with initial value \(x(0)\in R_{+}^{2}\) will remain in \(R_{+}^{2}\) with probability 1. For any given value \(x(0)\in R_{+}^{2}\) and any given positive constant

*p*>1 and positive constants

*c*

_{1},

*c*

_{2}to be determined, define

*ε*

_{1},

*ε*

_{2}are positive constants to be determined. Then we have We can find

*ε*

_{1},

*ε*

_{2}and

*c*

_{1},

*c*

_{2}, such that and note the inequalities can be turned into

*T*>0, such that In addition, \(E {[} x_{1}^{p}(t)+ x_{2}^{p}(t) {]}\) is continuous and there exists a

*C*(

*p*)>0 such that Let \(\iota(p)=\max\{\frac{2L(p)}{\min\{c_{1}, c_{2}\}}, C(p)\}\), and then The proof is complete. □

### Theorem 3.1

*Under Assumption *1, *solutions of SDE* (1.4) *are stochastically ultimately bounded*.

The proof of Theorem 3.1 is a simple application of the Chebyshev inequality and Lemma 3.1.

Since the solution of SDE (1.4) is positive, we have the following lemma.

### Lemma 3.2

*Let Assumption*1

*hold*,

*x*(

*t*)

*is the solution of SDE*(1.4)

*with initial value*\(x(0)\in R_{+}^{2}\).

*Then*

*x*(

*t*)

*has the property that*

*where*

*ϕ*

_{1}(

*t*)

*and*

*ϕ*

_{2}(

*t*)

*are the solutions of equations*:

### Proof

*x*

_{2}(

*t*)≥

*ϕ*

_{2}(

*t*). This completes the proof. □

*H*

_{1},

*H*

_{2}and

*θ*such that \(a_{i}(k)-\frac{\theta+1}{2}\sigma_{i}^{2}(k) >0\) (

*i*=1,2) satisfying the following inequalities

### Lemma 3.3

*Under Assumptions*1

*and*2

*the solution*

*x*(

*t*)

*of SDE*(1.4)

*with any initial value*\(x(0)\in R_{+}^{2}\)

*has the property that*

*and*

*where*

*H*

_{1},

*H*

_{2}

*are positive constants and*

*θ*>0

*such that*\(a_{i}(k)-\frac{\theta+1}{2}\sigma_{i}^{2}(k) >0\),

*i*=1,2,

*k*∈

*S*.

### Theorem 3.2

*Under Assumptions *1 *and* 2, *SDE* (1.4) *is stochastically permanent*.

### Proof

*x*(

*t*) be the solution of SDE (1.4) with any given positive initial value \(x(0)\in R_{+}^{2}\). By (3.10) of Lemma 3.3, we have

*ϵ*>0, let \(\delta_{1}=(\frac {\epsilon}{H_{1}})^{\frac{1}{\theta}}\), \(\delta'_{1}=(\frac{\epsilon }{H_{2}})^{\frac{1}{\theta}}\) and we have

## 4 Persistence in mean

In this section, we will investigate persistence in mean. First we introduce the definition.

### Definition 4.1

*m*

_{i},

*M*

_{i}(

*i*=1,2) such that the solution

*x*(

*t*) of SDE (1.4) has the following property:

### Lemma 4.1

*Let Assumption*1

*hold*.

*For any given initial value*\(x(0)\in R_{+}^{2}\),

*the solution*

*x*(

*t*)

*of SDE*(1.4)

*has the property that*

### Proof

*V*(

*x*(

*t*))=

*x*

_{1}(

*t*)+

*x*

_{2}(

*t*) and applying the generalized Itô’s formula, one can see that here \(\check{a}=\max_{k\in S} \{a_{1}(k),a_{2}(k) \}\), \(\check {b}_{0}=\max_{k\in S} \{|d_{21}(k)-\bar{b}_{1}(k)|,|d_{12}(k)-\bar {b}_{2}(k)| \}\). From (3.1) of Lemma 3.1, we have

*ϵ*>0 be arbitrary. Then, by the well-known Chebyshev inequality, we have Applying the Borel-Cantelli lemma (see [18]), for almost all

*ω*∈

*Ω*, we obtain that

*m*. Hence, we have that

*m*

_{0}(

*ω*), for almost all

*ω*∈

*Ω*, such that (4.7) holds whenever

*m*≥

*m*

_{0}. Consequently, for almost all

*ω*∈

*Ω*, if

*m*≥

*m*

_{0}and

*m*≤

*t*≤

*m*+1, results in Therefore Letting

*ϵ*→0 we obtain the desired assertion (4.3). □

### Theorem 4.1

*Under Assumptions *1 *and* 2, *for any initial value*\(x(0)\in R_{+} ^{2}\), *the solution**x*(*t*) *of SDE* (1.4) *is persistent in mean*.

### Proof

*c*

_{1},

*c*

_{2}satisfy the following inequality

*c*

_{1},

*c*

_{2}satisfying the inequality (4.9). From the inequality (4.3) of Lemma 4.1 and (3.11) of Lemma 3.3, one can derive that

*t*gives where

*M*(

*t*) is a martingale defined by

*M*(0)=0. The quadratic variation of this martingale is By the strong law of large numbers for martingales (see [17, 18]), we therefore have We can therefore divide both sides of (4.15) by

*t*and take the superior limit to obtain which means that

*i*=1,2. These, together with Lemma 3.2, yields

## 5 Extinction

In the previous sections we have showed that under the conditions of Assumption 1 and \(a_{i}(k) >\frac{\sigma_{i}^{2}(k)}{2}\) (*i*=1,2), that is, the white noise intensity is smaller, the species will be stochastically permanent and persistent in mean, so the population will not become extinct. However, we will show in this section that if the noise is sufficiently large, the solution to the associated SDE (1.4) will become extinct with probability 1.

### Theorem 5.1

*Let Assumption*1

*hold*.

*Let*\(\check{a}(k)=\max\{a_{1}(k),a_{2}(k)\}\), \(\frac{\hat{\sigma}^{2}(k)}{ 2}=\frac{1}{2(1/\sigma_{1}^{2}(k)+1/\sigma_{2}^{2}(k))}\)

*and*

*c*

_{1},

*c*

_{2}

*be positive constants satisfying inequality*(4.9).

*For any given initial value*\(x(0)\in R_{+}^{2}\),

*the solution of the SDE*(1.4)

*has the property that*

*In particular if*\(\sum_{k=1}^{N}\pi_{k}[\check{a}(k)-\frac{\hat {\sigma }^{2}(k)}{2}]<0\),

*then*lim

_{t→∞}

*x*(

*t*)=0

*a*.

*s*.

### Proof

*t*gives

*M*(

*t*) is a martingale defined in the proof of Theorem 4.1. By the strong law of large numbers for martingales and dividing

*t*on the both sides of (5.2) and then letting

*t*→∞ yields We obtain the desired assertion. □

## 6 Example and numerical simulations

*α*

_{ij}=1, and then \(\bar{b}_{1}=b_{1} +d_{12}\), \(\bar{b}_{2}=b_{2} +d_{21}\). Let the state space

*S*={1,2}, so the SDE (1.4) with regime switching takes the following form

*t*≥0. We numerically simulate the solution of (6.1). By the method mentioned in [19], for

*k*∈

*S*, we consider the discretized equation

*x*

_{1}(0),

*x*

_{2}(0))=(0.8,0.7), time step

*h*=0.01 and illustrate our main conclusions through the following example and figures.

### Example 6.1

*r*(

*t*) on the state space

*S*={1,2} with the generator

*r*(

*t*) directly because

*π*can be obtained by solving the simple linear equations

### Case 1

*i*=1,2, then Assumptions 1 and 2 hold. Making use of Theorems 2.1 and 3.2, we know that SDE (6.1) has a unique positive solution

*x*(

*t*) for any positive initial condition and it is stochastically persistent.

*c*

_{1}and

*c*

_{2}to satisfy the inequality (4.9), so we only take the two numbers

*c*

_{1},

*c*

_{2}equal to 1. We compute Then it follows from Theorem 4.1 that SDE (6.1) is permanent in mean.

*x*

_{1}(

*t*) and

*x*

_{2}(

*t*), they are similar to the straight lines. This means that the distribution is approximately a standard normal distribution (see the middle histograms). In the left pictures of Fig. 3, we can see that the red □ represents the phase portrait of

*x*

_{1}(

*t*) and

*x*

_{2}(

*t*) when there is only one state

*k*=1. Similarly, the blue ∘ represents the phase portrait of

*k*=2, while, the black + describes the switching back and forth from one state

*k*=1 to another state

*k*=2 according to the movement of

*r*(

*t*). The black area is located between the red region and the blue region, and the red area and the blue area is similar to the two limit state of the black region. Corresponding to the left graph, the right picture in Fig. 3 describes the state with no random disturbance. From Fig. 3, we can clearly see the impact of white noise and colored environment noise on populations. From Figs. 1–3, we know that, with starting from the initial point (0.8,0.7), the process

*x*(

*t*) is positive recurrent with respect to the rectangle {(

*x*

_{1},

*x*

_{2}):1.5<

*x*

_{1}<2.7, 1.3<

*x*

_{2}<2.5}, which also verifies the results correctly in Case 1.

### Case 2

*c*

_{1},

*c*

_{2}equal to 3 and 4, respectively. The condition \(\hat{\bar{b}}_{1}\hat{\bar{b}}_{2}=6>\check {d}_{12}\check {d}_{21}=3\) holds, but, \(\sum_{k=1}^{2}\pi_{k}[\check{a}(k)-\frac{\hat{\sigma }^{2}(k)}{2}]\doteq -0.0817<0\), so the conditions of Theorems 3.2 and 4.1 are not satisfied and the extinction conditions in Theorem 5.1 are satisfied, as the result of Markovian switching, the overall behavior, i.e. the SDE (6.1) will be extinctive by Theorem 5.1.

*x*

_{1},

*x*

_{2}suffer large white noise. By comparing the images of random disturbance and no disturbance in the left pictures of Fig. 4 (blue lines and black lines, respectively), we can see that the fluctuations of random model is more violent and the species will die out, which can also be seen by the histogram and Fig. 5 and the left pictures in Fig. 6. From Fig. 6, we know that large white noise will lead to population extinction, even though the corresponding deterministic model is persistent (see the right picture in Fig. 6).