PERSISTENCE AND STATIONARY DISTRIBUTION OF A STOCHASTIC PREDATOR-PREY MODEL UNDER REGIME SWITCHING

. Taking both white noise and colored environment noise into ac-count, a predator-prey model is proposed. In this paper, our main aim is to study the stationary distribution of the solution and obtain the threshold between persistence in mean and the extinction of the stochastic system with regime switching. Some simulation ﬁgures are presented to support the ana-lytical ﬁndings.


1.
Introduction. The dynamic of predator and prey models is a dominant theme in ecology. The classical deterministic predator-prey Lotka-Volterra model with density-dependent logistic growth of the prey takes the form ẋ(t) = x(t) a − bx(t) − cy(t) , where x(t) and y(t) are the sizes of prey population and predator population on time t, respectively. a, b, c, h and f are positive constants. The prey carrying capacity of the environment is a b , the feeding efficiency in turning predation into new predators is f . System (1) was formulated by Pielou [18]. Nisbet and Gurney pointed out that if the initial value x(0) > 0 and af c bh ≤ 1 or x(0) > 0 and y(0) = 0, then x → a dy(t) = [−hy(t) + f cx(t)y(t)]dt + βy(t)dB 2 (t), where B 1 (t) and B 2 (t) are one-dimensional standard Brownian motion, α 2 and β 2 stand for the intensities of the white noises. Under the premise of the system (1) having a positive equilibrium (x * , y * ), Li et al. [6] gave sufficient conditions for permanence, extinction and the existence of the stationary distribution in (2). We will also consider colored noise in this paper. Let {r(t), t ≥ 0} be a rightcontinuous Markov chain on the probability space (Ω, F, {F t } t≥0 , P) with finitestate space S = {1, . . . , m}. The generator Γ = (γ ij ) 1≤i,j≤m is given, for δ > 0, by P {r(t + δ) = j|r(t) = i} = γ ij δ + o(δ), if i = j, Here γ ij is the transition rate from i to j and γ ij ≥ 0 if i = j, while γ ii = − i =j γ ij .
The aim of this paper is to study system (3) and find the threshold between persistence in mean and the extinction and obtain sufficient conditions for system (3) being positive recurrent (and the existence of a unique ergodic stationary distribution). Settati et al. [19] and Liu et al. [11], using the theory of [24], investigated a stochastic Lotka-Volterra mutualistic system with regime switching, respectively, and obtained the existence of the stationary distribution. Zu et al. [25] studied a predator-prey model with density dependence of predator and obtained the results about ergodic property. Liu et al. [13] investigated a stochastic logistic model with regime switching and obtained the threshold between weak persistence and extinction. To our knowledge, there are very few results on the model (3) and other stochastic predator-prey systems under regime switching. First we apply the theory and methods of [24], in this paper, to a stochastic Lotka-Volterra predatorprey model with regime switching to study the ergodic stationary distribution by constructing appropriate Lyapunov functions. The persistence in mean-extinction threshold we obtain and sufficient conditions about the stationary distribution fully reflect the effect of white noise and the probability distribution of the Markov chain.
The rest of the paper is arranged as follows. In Section 2, we introduce some preliminaries which will be used in the next sections and give two theorems concerning the existence of a global positive solution and we give the (θ + 1)-moment estimation (0 < θ < min k∈S 2h(k) β 2 (k) ). Some desired population dynamical properties of model (3) are investigated in Section 3 and 4. In Section 3, we establish sufficient conditions for persistence in mean and the extinction of system (3) and we give the threshold between them. We study the stationary distribution and ergodicity in Section 4. We present the figures to illustrate the main results in Section 5 and close the paper with conclusions and future directions in Section 6.
In investigating the ergodic stationary distribution, we will use the following lemma which gives a criterion for positive recurrence in terms of Lyapunov function (see e.g. Theorem 3.13 in [24], Lemma 2.1 in [11] and Lemma 3.1 in [19]). Let (z(t), r(t)) be the diffusion process described by the equation: dz(t) = ρ(z(t), r(t))dt + g(z(t), r(t))dB(t), where B(·) and r(·) are the d-dimensional Brownian motion and the right-continuous Markov chain in the above discussion, respectively, and ρ(·, ·) : R n × S → R n , g(·, ·) : R n × S → R n×d satisfying g(z, k)g T (z, k) = D(z, k). For each k ∈ S, and for any twice continuously differentiable function V (·, k), we define L by (i).f or i = j, γ ij > 0, (ii).f or each k ∈ S, D(z, k) is symmetric and satisf ies with some constant ∈ (0, 1] f or all z ∈ R n , (iii).there exists a bounded open subset D of R n with a regular (i.e.smooth) boundary satisf ying that, f or each k ∈ S there exists a nonnegative f unction is twice continuously dif f erentiable and that f or some ς > 0, LV (z, k) ≤ −ς, f or any (z, k) ∈ D C × S, then system (4) is ergodic and positive recurrent. T hat is to say, there exists a unique stationary density µ(·, ·) and, f or any Borel measurable f unction If there are positive constants λ 0 , T and λ ≥ 0 such that Now we give two theorems concerning the existence and uniqueness of positive solutions and moment boundedness, and based on these we present our main results in the next sections. Theorem 2.3. For any given initial value (x(0), y(0), r(0)) ∈ R 2 + × S, system (3) has a unique positive solution. Moreover, this solution remains in R 2 + × S with probability 1.

The proof of Theorem 2.3 is similar to that in Example 4.2 in [22] so we omit it.
Theorem 2.4. For any 0 < θ < min k∈S 2h(k) β 2 (k) , there exists a positive constant ι(θ), such that for any given initial value (x(0), y(0), r(0)) ∈ R 2 + × S, the solution (x(t), y(t), r(t)) of system (3) has the property Proof. Define a C 2 -function U : here p =ĉ fč . According to the generalized Itô's formula, we have From the condition 0 < θ < min k∈S , for sufficiently small positive constant w, we can see that the coefficient of y 2 is negative. Using the inequality xy ≤ , and putting it into (7), yields where in which we put 1 in order to make K positive. By applying Theorem 3.1 in [21], we can get EU (x(t), y(t)) ≤ EU (x(0), y(0))e −ωt + K and U (x(t), y(t)) is continuous. Obviously, and then The required assertion (5) follows immediately.
3. The threshold between persistence in mean and the extinction. In the study of population dynamic, extinction and persistence are two important issues.
In this section, we will find out the persistence in mean-extinction threshold of system (3).
Furthermore, by taking into consideration the results of [19], we can determine that if k∈S π k a(k)− α 2 (k) 2 > 0, for any Borel measurable function ρ(·, ·) : R×S → R, then system (9) has a unique stationary distribution ν(·, ·) with ergodic property: Use the generalized Itô's formula on (9) and integrate, and one can obtain Let which is the threshold we want between persistence in mean and the extinction of the predator population.
Proof. Applying the generalized Itô's formula to the second equation of (3) results in where M 2 (t) = t 0 β(r(s))dB 2 (s). Similar to the previous proof we get In the case when λ < 0, y(t) tends to zero a.s.. This completes the proof of assertion.
Proof. First, by the generalized Itô's formula, we get from the first equation of (3) that Combining (11) and (16) yields which implies

LI ZU, DAQING JIANG AND DONAL O'REGAN
Now, we will focus on finding 1 t t 0 X(s) − x(s) ds from system (3). Let us first calculate, by (14) and (17), that It follows from the property of inferior limits and lim t→∞ M2(t) t = 0 a.s., that for arbitrary ε > 0, there exists t > 0 such that − 1 and by virtue of the arbitrariness of ε and Lemma 2.2, we have lim inf We therefore obtain the desired assertion from (20).
4. Ergodic property of positive recurrence. In this section, we will investigate the ergodic property of system (3) by using the Lyapunov function method. Now we impose the condition For the purpose of proving our theorem, we will first introduce a transformation of system (3). Let u(t) = log x(t) and v(t) = log y(t), for t ≥ 0. Applying the generalized Itô's formula, yields dv(t) = − h(r(t)) − β 2 (r(t)) 2 + f (r(t))c(r(t))e u(t) dt + β(r(t))dB 2 (t).
By Lemma 3.2 in [15], we know that the ergodic property and positive recurrence of system (3) are equivalent to those of system (21). The proof of the following theorem will be to verify that system (21) satisfy the three conditions of Lemma 2.1.
Define a C 2 -function here, p =ĉ fč and M = (2/λ) max 2,  2 k and k (k ∈ S) will be determined in the rest of the proof. Note that we put | | in order to make k + | | non-negative. By Itô's formula, we have, respectively and

LI ZU, DAQING JIANG AND DONAL O'REGAN
Let us define the vector Λ = (Λ 1 , . . . , . Since the generator matrix Γ is irreducible, then for Λ k , there exists = ( 1 , . . . , m ) T a solution of the Poisson system (see Lemma 2.3 in [5] and Remark 2 for detailed description), such that where − → 1 denotes the column vector with all its entries equal to 1. Thus, we have Thereby, combining (24), (25) and (27), yields L( Then we have In the set D C × S, we choose sufficiently small such that Remark 2. We introduce Lemma 2.3 in [5]: Suppose that Γ, an m × m constant matrix, is the generator of a continuous-time Markov chain r(t) and that Γ is irreducible. Then Γπ = η has a solution if and only if πη = 0, where η ∈ R m and π ∈ R 1×m denotes the associated stationary distribution. We will verify πη = 0 and choose the vector Λ = (Λ 1 , . . . , Theorem 4.1 provides us with the good property of system (3) having a unique stationary distribution in R 2 + ×S, expressed as µ(·, ·). Thus, we can get the following theorem.
Proof. By (5) of Theorem 2.4, for any 0 < θ < min k∈S 2h(k) . Then, by the ergodic property of (x(t), y(t), r(t)) and for any n > 0, we have P lim Thus, by the dominated convergence theorem, we get which implies the function f (x) = x is integrable with respect to the measure µ. Similarly, we can get also k∈S R+ yµ(y, k)dy ≤ lim sup t→∞ E[y(t)] < +∞.
Return to (16), from the ergodic property of (x(t), y(t), r(t)), we know that lim t→∞ log x(t) t = 0 a.s. Otherwise, if lim t→∞ log x(t) t > 0 or < 0, we know that, when t → ∞, x(t) → +∞ or 0, which contradicts the fact that its stationary density lies in R + . Therefore, we have which implies the first result (37). The second result also holds by the same way.
The density function images of stationary distribution of x(t) and y(t) in k = 1 and k = 2 will be seen in Fig.1. In Fig.2 (a), the red , blue • and black + represent the phase portrait of x(t) and y(t) when there is only one state k = 1, k = 2 and switching back and forth from one state k = 1 to another state k = 2 according to the movement of r(t), respectively. Clearly, 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. Correspondingly, the subgraph Fig.2 (b) describes the state with no random disturbance. From Fig.2, we can clearly see the impact of white noise and colored environment noise on populations.

Case 2.
Assume that (α(1), α(2)) = (1.2, 1.7), (β(1), β(2)) = (1.5, 1.4). Predator and prey are affected by bigger white noise. The subgraphs (a) and (b) denote sample phase portrait of stochastic system and the corresponding deterministic system. The red, blue and black area denote x(t), y(t) in k = 1, k = 2 and converting between the above two states, respectively. The white noise intensity on x(t) and y(t) are all relatively small.  The subgraphs (a) and (b) have the same definitions as in Fig.2. The white noise intensity on x(t) and y(t) are all relatively big.
6. Conclusions and future directions. This paper studies a stochastic predatorprey Lotka-Volterra model with density-dependent logistic growth of the prey under regime switching. Sufficient conditions for persistence in mean and the extinction of system (3) are established and we obtain the threshold between them. We prove the stochastic system with regime switching has a unique stationary distribution which is the first attempt to solve this problem by using the Lyapunov function method.
The conclusions we obtained on system (3) in this paper can be a good description of the biological significance. From λ < 0, we can see that, the predator can die out when the parameters c(k) and feeding efficiency f (k) or the intrinsic growth rate −h(k) is too small or the white noise intensity α 2 (k) is too big, which takes place also in the case without noise. If the predator-prey model wants to have a stationary distribution, that isλ =fĉ k∈S π k a(k) − α 2 (k) 2 −b k∈S π k h(k) + β 2 (k) 2 > 0, the intrinsic growth rate a(k) of prey, c(k) and f (k) will not be too small, and the density-dependent b(k) and h(k) will not be too big. Of course, the white noise intensity α 2 (k) and β 2 (k) can not be too big. Therefore, the main difference between the deterministic and stochastic model is that large noise can also cause extinction. Obviously, these important factors play a key role in the predator extinction or not, that gives its threshold λ between persistence in mean and extinction. Furthermore, the role of the colored noise is clear in the examples. The presence of the Markov chain in this stochastic system can contribute to the survival of the system, and reduce the risk of extinction.
In future studies, we will try to promote this approach to other predator-prey models.