ASYMPTOTIC BEHAVIOR OF A DELAYED STOCHASTIC LOGISTIC MODEL WITH IMPULSIVE PERTURBATIONS

. In this paper, we investigate the dynamics of a delayed logistic model with both impulsive and stochastic perturbations. The impulse is intro- duced at ﬁxed moments and the stochastic perturbation is of white noise type which is assumed to be proportional to the population density. We start with the existence and uniqueness of the positive solution of the model, then establish suﬃcient conditions ensuring its global attractivity. By using the theory of integral Markov semigroups, we further derive suﬃcient conditions for the existence of the stationary distribution of the system. Finally, we perform the extinction analysis of the model. Numerical simulations illustrate the obtained theoretical results.


1.
Introduction. This paper considers the long-term behavior of a system that results from impulsive and stochastic perturbations of a deterministic dynamical system with delay. This type of dynamics occurs naturally for example in the modeling of biological systems such as a growing bacterial colony, modeled deterministically, from which samples of random size are drawn regularly in an experiment or a marine ecosystem from which fish are harvested with a net [2], or the modeling of stochasticity in gene regulatory networks with feedback through a cell's signaling system [19], etc.
The associated mathematical model with stochastic perturbations where x(t) is the population size, B(t) is a standard Brownian motion, r(t), a(t) and σ(t) are continuous bounded functions on [0, ∞), has been considered by many 1478 SANLING YUAN, XUEHUI JI AND HUAIPING ZHU authors [3,6,11,12,15,16]. Recently, Liu and Wang [17] studied a stochastic logistic system with impulsive effects which takes the following form where N denotes the set of positive integers. The time sequence {t k } is strictly increasing such that lim k→∞ t k = +∞. The term b k is impulsive perturbations at the moments of time t k , and satisfies that 1 + b k > 0 for all k ∈ N . The authors obtained the sufficient conditions for extinction, non-persistence in the mean, weak persistence, persistence in the mean and stochastic permanence of the system. As is well known, time delay always exists in the evolutionary processes of the population (e.g., resource regeneration times, maturation periods, feeding times, reaction times, etc) and it can cause the oscillatory or even unstable phenomena (see monographs [8,14]). Therefore, it seems more realistic to consider model (2) with delay. As far as we know, few investigations have been made on such type system. Based on model (2), we propose the following model: where c stands for the effect power of the past history. f (s), called the delay kernel, is a weighting factor which indicates how much emphasis should be given to the size of the population at earlier times to determine the present effect on resource availability [24]. Usually, we use the Gamma distribution delay kernel f n α (s) = α n s n−1 e −αs (n − 1)! , where α > 0 is a constant, n an integer, with the average delay T = n/α. The corresponding version of model (3) with a Gamma distribution delay kernel can be written as To perform a thorough analysis on model (3) with a general delay kernel function is very difficult. So in this paper we mainly devote our attention to the investigation on the dynamics of its special case model (4). Notice that in the absence of impulsive and random perturbations, it is obvious that model (4) has a stable positive equilibrium x * = r a+c . However, under impulsive and random perturbations, the equilibrium does not exist. That is, the steady state of model (4) can no longer be represented by a single point. Therefore, we turn our attention to the study of the stationary distribution of the system. To the best of our knowledge, though some significant progress has been made in the techniques and methods of determining the existence of the stationary distribution and the stability of the density for stochastic differential equations [9,20], unfortunately, few works have been performed for the corresponding problems of impulsive stochastic differential equations up to now. So, our work in this paper may be considered as a first attempt to the investigation on the stationary distribution of the impulsive stochastic differential equations.
The organization of this paper is as follows. In the next section, we show the existence and uniqueness of a global positive solution of model (4). Then, in section 3, we present our main results: we first carry out the global attractivity analysis of the model; then using the Hasminskii's method and constructing Lyapunov function, we prove the existence of the stationary distribution of the model; we also perform an extinction analysis of the model. Finally, some discussions and numerical simulations are presented in section 4.

2.
Existence and uniqueness of a global positive solution. In this section, we show the uniqueness and global existence of a positive solution of model (4) for any positive initial value. Using the linear chain trick, if we define then model (4) is equivalent to the following system Therefore, to study the dynamics of model (4), we need only to consider system (5).
To begin with, we consider a general d-dimensional impulsive stochastic differential equation: with initial condition X(0) = X 0 ∈ R d . B(t) denotes m-dimensional standard Brownian motions defined on the probability space (Ω, F , {F t } t≥0 , P ), which is a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e. it is right continuous and F 0 contains all P-null sets). Denote the differential operator L associated with Eq. (6) by for t ∈ [t k−1 , t k ). Let P C 1,2 ([t k−1 , t k )×R d ; R + ) denote the family of all nonnegative functions V (t, X) on [t k−1 , t k ) × R d which are continuous once differentiable in t and twice differentiable in X. If L acts on such V (t, X), then and X(t k ) = X(t − k ) with probability one; (iii) for almost all t ∈ [0, t 1 ], X(t) obeys the integral equation

SANLING YUAN, XUEHUI JI AND HUAIPING ZHU
and for almost all t ∈ (t k , t k+1 ], k ∈ N, X(t) obeys the integral equation Moreover, X(t) satisfies the impulsive conditions at each t = t k , k ∈ N with probability one. We now establish a fundamental lemma to reduce the dynamics of nonlinear stochastic differential system under impulsive perturbation to the corresponding problems of a nonlinear stochastic differential system without impulses.
On the other hand, for each k ∈ N and t k ∈ [0, ∞), and which implies that X(t) is the solution of Eq. (6). We now prove (ii). Since X(t) is a solution of Eq. (6) and is continuous on each . What's more, we have that, for any k ∈ N and t k ∈ [0, +∞), and It should be emphasized that we consider Eqs. (6) and (8) on the same probability space. For Eq. (8), the associated differential operator L is the same as that defined in (7) for t ∈ [0, ∞) rather than for t ∈ [t k−1,t k ). Now we turn to show the existence and uniqueness of the solution of system (5) by employing Lemma 2.1. Let x(t) = 0<t k <t (1 + b k )u(t) and y i (t) = v i (t), i = 1, 2, . . . , n, then system (5) becomes ) is a solution of system (9), then (x(t), y 1 (t), . . . , y n (t)) is a solution of system (5). So, in the following, we only need to show the global existence of the positive solution of system (9).
There is a unique positive global solution (u(t), v 1 (t), . . . , v n (t)) of system (9) a.s. for any initial value (u 0 , v 10 , . . . , v n0 ) ∈ R n+1 + . Proof. Consider the system with initial valueū(0) = ln u 0 ,v i (0) = ln v i0 , i = 1, 2, . . . , n. Obviously, the coefficients of system (10) satisfy the local Lipschitz condition, then there is a unique local solution (ū(t),v 1 (t), . . . ,v n (t)) on t ∈ [0, τ e ), where τ e is the explosion time. By Itô's formula, it is easy to see is the unique positive local solution of system (9) with initial value (u 0 , v 10 , . . . , v n0 ) ∈ R n+1 + . We now show the solution of system (9) is global, i.e., τ e = ∞. Since the solution is positive for t ∈ [0, τ e ), we have is the unique solution of the equation and by the comparison theorem for the stochastic equation, yields Besides, we can get Obviously, is the solution to the equation On the other hand, It follows that Arguing as above, we can get To sum up, we have that This completes the proof of the lemma.
By Lemma 2.2, we have the following result on the global existence of the positive solution to system (5).
3. Asymptotic behavior of system (5). In this section, we will first investigate the global attractivity of system (5), then prove the existence of its stationary distribution, and finally, we perform an extinction analysis of the system. We first give the following fundamental assumptions on the impulsive perturbations b k .
• Assumption 3.2. There are two positive constants θ 1 and θ 2 such that for all t > 0, 3.1. Global attractivity of system (5). Before proving the global attractivity of system (5), we first prepare some lemmas.

SANLING YUAN, XUEHUI JI AND HUAIPING ZHU
where Therefore, On the other hand, one can compute that Using Hölder inequality, one can then derive that From (13) we know that for any ε > 0, there exists a T > 0 such that when t > T , It then follows from (14) that Using comparison theorem and noting also that ε is arbitrary, we obtain Similarly, for i = 2, 3, . . . , n, we compute that Then we can deduce that This completes the proof of Lemma 3.1.  (See [13]) Suppose that a stochastic process X(t) on t ≥ 0 satisfies the condition for some positive constants α 1 , β and c 1 . Then there exists a continuous modificatioñ X(t) of X(t) which has the property that for every γ ∈ (0, β/α 1 ), there is a positive random variable h(ω) such that In other words, almost every sample path ofX(t) is locally but uniformly Hölder continuous with exponent γ.
Proof. From (13) and the continuity of E(u q (t)) we have that there is a K * 1 (q) such that for t ≥ 0, Similarly, by (15) and the continuity of . The first equation of system (9) is equivalent to the following stochastic integral equation By the moment inequality for stochastic integrals, we know that for 0 ≤ t 1 ≤ t 2 and q > 2, E|u(s)| q ds.

SANLING YUAN, XUEHUI JI AND HUAIPING ZHU
Then it follows from Lemma 3.2 that almost every sample path of u(t) is locally but uniformly Hölder-continuous with exponent γ 1 for every γ 1 ∈ (0, q−2 2q ). From the second equation of system (9), we have Then for 0 < t 1 < t 2 < ∞ and t 2 − t 1 ≤ 1, we compute that Similarly, for i = 2, . . . , n, we can compute that . In view of Lemma 3.2, almost every sample path of v i (t)(i = 1, 2, . . . , n) is locally but uniformly Hölder-continuous with exponent γ 2 for every γ 2 ∈ (0, q−1 q ). Therefore, almost every sample path of (u(t), v 1 (t), . . . , v n (t)) is uniformly continuous on t ≥ 0.  Proof. Let (x(t), y 1 (t), . . . , y n (t)) and (x(t),ȳ 1 (t), . . . ,ȳ n (t)) be two arbitrary solutions of system (5) with positive initial values, and suppose by directly calculating the right differential D + V 2 (t) of V 2 (t) and then making use of Itô's formula, we have Integrating both sides of (16) from 0 to t yields 3.2. Stationary distribution of system (5). In this subsection, we will prove the existence of a stationary distribution of system (5). Due to the complexity of system (5) and the lake of effective mathematical techniques available, we only consider the special case for Gamma distribution delay kernel with n = 1, i.e., f (s) = αe −αs , which is called weak delay kernel, indicating that the maximum weighted response of the growth rate is due to current population density while past densities have (exponentially) decreasing influence [24].
Notice that system (18) is the limit system of (9), then by the global attractivity of (9), we only need to study the existence of stationary distribution of system (18).
Remark 1. By the support of a measurable function f we simply mean the set Besides, note that the Fokker-Planck equation corresponding to system (19) is of a degenerate type, thus the asymptotic stability of the system can't follow directly from the known results from Hasminskii [9]. We will show it by using the theory of integral Markov semigroups (see Appendix A). Now we introduce an integral Markov semigroup connected with system (19).
Denote by k(t,x,ỹ;x 0 ,ỹ 0 ) the density of P(t,x 0 ,ỹ 0 , ·). Then and consequently {P(t)} t≥0 is an integral Markov semigroup. Thus the asymptotic stability of the semigroup {P(t)} t≥0 implies that all the densities of the process (ξ(t),η(t)) convergence to an invariant density in L 1 . Therefore, for Theorem 3.2, we only need to show the asymptotic stability of the semigroup {P(t)} t≥0 . The outline of our proof is as follows: (i) First, using the Hörmander condition [21], we show that the transition function of the process (ξ(t),η(t)) is absolutely continuous; (ii) Then according to support theorems [1,5], we find a set E on which the density of the transition function is positive; (iii) Next we show that the set E is an attractor and the semigroup satisfies the Foguel alternative [22,23,25]; (iv) Finally, we exclude sweeping by showing that there exists a Khasminskiȋ function.
In the following, we give the proof of Theorem 3.2 through four lemmas in succession, which correspond respectively to (i)-(iv) above.
Proof. The proof is based on the Hörmander theorem for the existence of smooth densities of the transition probability function for degenerate diffusion processes. If a(x) and b(x) are vector fields on R d , then the Lie bracket [a, b] is a vector field given by Let Thus for every (x,ỹ) ∈ R 2 , vectors b(x,ỹ) and [a, b](x,ỹ) span the space R 2 . This implies that the transition probability function P(t,x 0 ,ỹ 0 , A) has a density k(t,x,ỹ;x 0 ,ỹ 0 ) and k ∈ C ∞ (0, ∞) × R 2 × R 2 . This completes the proof of lemma 3.5.
Notice that for every density f , one has Lemma 3.7. The semigroup {P(t)} t≥0 is asymptotically stable or is sweeping with respect to compact sets.
Proof. By Lemma 3.5, the semigroup {P(t)} t≥0 is an integral Markov semigroup with a continuous kernel k(t,x,ỹ) for t > 0. Let E = R 2 , then it is sufficient to investigate the restriction of the semigroup {P(t)} t≥0 to the space L 1 (Ē), whereĒ denotes the closure of the set E. In view of Lemma 3.6, for every f ∈ D, we have where D is defined in Appendix A. Therefore, according to Lemma A.1 in Appendix A, it follows that the semigroup {P(t)} t≥0 is asymptotically stable or is sweeping with respect to compact sets. This completes the proof of Lemma 3.7.
Proof. In order to exclude sweeping, we now construct a nonnegative C 2 −function V and a closed set Γ ∈ Σ such that sup x,ỹ / ∈Γ where A * is the adjoint operator of the infinitesimal generator A of the semigroup {P(t)} t≥0 , which is of the form where f i (x,ỹ), i = 1, 2 are defined in (21). Define wherex * = ln r aθ + cθ ,ỹ * = ln r a + c .

By (29), one has
Thus, the ellipsoid lies entirely in R 2 . Thus, there exist a closed set Γ ∈ Σ which contains this ellipsoid and c > 0 such that sup The function V is called a Khasminskiȋ function. By means of standard arguments similar to those in [22], one can check that the semigroup is not sweeping from the set Γ due to the existence of a Khasminskiȋ function. Therefore, according to Lemma A.1 in Appendix A, we conclude that the semigroup {P(t)} t≥0 is asymptotically stable.
In view of Theorem 3.2, for system (18) there exists a unique positive invariant density with the support set R 2 + since it is equivalent to system (19). Notice that system (18) is the limit system of (9) as lim t→∞ 0<t k <t (1 + b k ) = θ, then by the global attractivity of (9), we conclude that there exists a unique positive invariant density for system (9). Further notice that system (5) is equivalent to system (9). Therefore, we have the following result for system (5). Theorem 3.3. Assume that condition (17) holds. If r > 1 2 σ 2 and c < a, then for system (5) there exists a unique positive invariant density with the support set R 2 + . 3.3. Extinction of system (5). In this subsection, we show that large noise can lead to the extinction of system (5).
It then follows from (31) that Remark 2. From Theorems 3.3 and 3.4, one can see that if the intensity of the noise is small, there exists an invariant and asymptotically stable density of the system, while large noise will make the population extinct eventually. Noting further that if c < a and condition (17) hold, then r − 1 2 σ 2 can be considered as a threshold determining the asymptotical stability and the extinction of system.

4.
Simulations and discussions. Impulsive and uncertain variability together with time delay are always present in a natural system, which should be accounted for in its mathematical model. The research performed in this paper is an attempt in this direction, using an impulsive stochastic model with delay. More specifically, the impulse is introduced at fixed moments, the stochastic perturbation is of white noise type and is assumed to be proportional to the population density, and the delay takes the distributed type with a weak delay kernel. To perform a detailed analysis on the dynamics of model (4), we first transform it to an equivalent stochastic system (5) with impulsive effects using the linear chain trick. Then based on Lemma 2.1, it can be further reduced to the problem of a nonlinear stochastic differential system without impulses, i.e., system (9).
For system (5), we first carry out the analysis of its global attractivity. Theorem 3.1 shows that the system is globally attractive provided that c < a, Assumptions 3.1 and 3.2 hold. Note that Assumptions 3.1 and 3.2 are only relative to the impulse, so we see that the noise and the delay do not affect the global attractivity of the system. Then we study the existence of the stationary distribution of system (5). It is difficult to directly study its distribution, so we turn to study its equivalent system (9). Notice that (9) is non-autonomous and it has been showed to be globally attractive in Theorem 3.1. So we only need to study the limit system (18) of (9). Due to the Fokker-Planck equation corresponding to system (18) is of a degenerate type, thus the existence of the stationary distribution can't follow directly from the known result from Hasminskii [9]. We show it based on the theory of integral Markov semigroups. Theorem 3.3 shows that for system (5) there exists a unique invariant density with the support R 2 + provided that r > 1 2 σ 2 , c < a and condition (17) hold. Moreover, it is shown in Theorem 3.4 that if r < 1 2 σ 2 , the population will be extinct eventually. Thus, under c < a and condition (17), r − 1 2 σ 2 can be considered as a threshold determining the asymptotical stability and the extinction of system (5).
By Theorem 3.3, system (5) is asymptotical stability. Our simulation supports this conclusion as shown in Fig. 1, where we show the effect from different noise intensities σ and different impulse intensities b k , respectively.
It is seen in Fig. 1 (a) that the steady state of the system is point E * (x * , y * ) = (1.0909, 1.8182) in the absence of impulsive and random perturbations (i.e. σ = b k = 0). Increasing the value of σ and keeping b k the same, we can see that the steady state of the system can no longer be represented by a single point; instead, it is represented by a region around E * . Shown in Fig. 1 (a)-(c), the larger the noise intensity is, the more diffusive of the system state is. When b k changes and σ stays 0, we can see in Fig. 1 (d)-(e) that there is an attractor for the system, but E * is not included in the attraction, which is above the attraction in the case of b k < 0 and is below the attraction in the case of b k > 0. When we choose σ = 0.03 and b k = 0.3/k, the system is still stable, please see Fig. 1 (f). Fig. 1 shows the trajectories of system (5) under different values of parameters. However, we should point out that the trajectory in each subgraph is drawn for a single sample, which is stochastic for each sample under the same parameters, that is the outcome for a single trajectory is not predictable. But, the probability distribution of all possible outcomes can be determined. Our simulation supports this conclusion as shown in Fig. 2, where σ = 0.03, b k = 0.3/k and the densities are drawn based on 10000 sample pathes, computed with differential initial value and different iterative times, respectively. We can see from the figure the distribution is stationary.
To sum up, this paper presents an investigation on the dynamics of a impulsive stochastic system with delay. Our findings are useful for better understanding of the effects of impulses, stochastic perturbations and delay on the dynamics of a system. We should point out there are still some other interesting topics meriting further investigation, for example, the long term behavior of multi-population system with impulsive and stochastic perturbations. We leave these for future considerations.   for every density f .  A family {P(t)} t≥0 of Markov operators which satisfies conditions: (i) P(0) = Id, (ii) P(t + s) = P(t)P(s) for s, t ≥ 0, (iii) for each f ∈ L 1 the function t → P(t)f is continuous with respect to the L 1 norm is called a Markov semigroup. A Markov semigroup {P(t)} t≥0 is called integral, if for each t > 0, the operator P(t) is an integral Markov operator.
A density f * is called invariant if P(t)f * = f * for each t > 0. The Markov semigroup {P(t)} t≥0 is called asymptotically stable if there is an invariant density f * such that lim t→∞ P(t)f − f * = 0 for f ∈ D.
A Markov semigroup {P(t)} t≥0 is called sweeping with respect to a set A ∈ Σ if for every f ∈ D We need some result concerning asymptotic stability and sweeping which can be found in [7].
Lemma A.1. Let X be a metric space and Σ be the σ−algebra of Borel sets. Let {P(t)} t≥0 be an integral Markov semigroup with a continuous kernel k(t,x,ỹ) for t > 0, which satisfies (A.1) for allỹ ∈ X. We assume that for every f ∈ D, we have ∞ 0 P(t)f dt > 0 a.e.
Then this semigroup is asymptotically stable or is sweeping with respect to compact sets.
The property that a Markov semigroup {P(t)} t≥0 is asymptotically stable or sweeping for a sufficiently large family of sets (e.g. for all compact sets) is called the Foguel alternative.