On hybrid stochastic population models with impulsive perturbations

Abstract This paper considers the dynamic behaviours of a hybrid stochastic population model with impulsive perturbations. The existence of the global positive solution is studied in this paper. Moreover, under some conditions on the noises and impulsive perturbations, the properties of the persistence and extinction, stochastic permanence, global attractivity and stability in distribution are presented. Our results illustrate that impulsive perturbations play a crucial role in these properties. The bounded impulse term will not affect these properties, however, when the impulse term is unbounded, some of the properties, such as the persistence and extinction may be changed significantly. As a part of this paper, a couple of examples and numerical simulations are provided to illustrate our results.


Introduction
The famous logistic growth model is often described by the ordinary differential equation where a be the intrinsic growth rate. In ecosystems, the population dynamics are inevitably affected by various environmental noises. First of all, let us consider one type of environmental noise, namely white noise. When we incorporate white noise in the intrinsic growth rate a of (1), then it becomes a + σḂ(t), whereḂ(t) is white noise and σ is a positive number representing the intensity of the noise. As a result, (1) becomes the following Itô stochastic differential equation (see e.g. [5,27]) which exhibits an approximate solution for the transient partial differential equations of x(t). The dynamic behaviours of (2) have been studied by many authors (see e.g. [2,6,10,11,15,20,21]). Now, let us take a further step by considering coloured noise, say telegraph noise (see e.g. [16,24,28]). This kind of noise can be illustrated as a switching between finite regimes of the environment. The switching is memoryless and the waiting time for the next switch has an exponential distribution. Hence, we assume there are N regimes and (2) becomes the following hybrid stochastic logistic model dx(t) = [a(r(t))x(t) − b(r(t))x 2 (t)] dt + σ (r(t))x(t) dB(t), where r(t) is a right-continuous Markov chain which takes values in a finite state space S = {1, 2, . . . , N}. The model (3) can be regarded as the results of the following N stochastic differential equations switching from one to the other according to the movement of the Markov chain. One can understand the hybrid stochastic model (3) as: if r(0) = j ∈ S, the system satisfies (3) with r(t) = j until t = τ 1 , when r(t) jumps to another state r(t) = l, the system will obey (3) with r(t) = l. Recently, the author in [18] studied the dynamics of a stochastic predator-prey model under regime-switching, modified Leslie-Gower Holling-type II and prey harvesting. Considering Lèvy and Markov chain noises, the authors in [22] gave the existence of the stationary distribution, ergodicity of a stochastic hybrid competition model. Moreover, the impulsive process is often studied in various areas of science and technology such as biological systems, population dynamics, physics, chemistry, and optimal control theories (see e.g. [3,14]). There are many important and interesting results about the behaviours of the population dynamic systems of impulsive differential equations. For example, S. Ahmad and I. Stamova in [1] investigated competitive systems with delays and impulsive perturbations, they also gave the conditions for asymptotic stability of competitive systems. The global dynamics for the periodic logistic equation and periodic multi-species predator-prey system under the impulsive perturbations were studied in [8,19], respectively. The permanence and global stability for nonautonomous N-species impulsive Lotka-Volterra competitive system were considered by J. Hou, Z. Teng and S. Gao in [9]. Persistence and extinction of a stochastic predator-prey model with impulsive toxicant input were considered in [23].
Motivated by the work of persistence and extinction, stochastic permanence, the existence of stationary distribution, ergodicity and stability in distribution for stochastic population model. In this paper, we will study the following hybrid stochastic logistic model with impulsive perturbations, which is defined as follows: where A k (r(t k−1 )) is the impulse gain coefficient, N is the set of positive integers, and for k ∈ N, t k satisfies 0 < t 1 < t 2 < · · · such that lim k→+∞ t k = +∞, is the impulse at the moment and x(t k ) right-continuous at t k , i.e. x(t k ) = x(t + k ). For any fixed i ∈ S, a(i), b(i) and σ (i) are continuous bounded functions on R + . In the biological sense, it is reasonable to assume that If A k (i) > 1, we say that the species are planting, while A k (i) < 1 stands for harvesting. Remark 1.1: Our motivations for studying the properties of hybrid stochastic model (4) stem from the emerging and existing applications of the biological systems under the influence of random environments and impulse. The model (4) is different from the model given by Liu and Wang in [21]. The impulses occur at a fixed sequence of times, t k , which are dependent on the environmental state in (4). These results in the underlying problems challenging to handle due to the interaction of the impulse times, continuous processes, the tangled and hybrid information pattern.
In our paper, the coloured noises (Markov chain), the white noises (Brownian noises), and impulsive perturbations are considered in the classical logistic model. We investigate how these noises and impulses affect the dynamic behaviours of (4). As far as we know, there was no much contribution to this kind of stochastic differential equation. The authors in [30,31] studied the impulsive perturbations at the switching points of a hybrid stochastic differential equation with impulsive perturbations. This motivated us to investigate some properties of (4).
We consider the regime switching in the impulsive term. Our results show how the impulse and the environmental noises affect the properties of the logistic model, such as the existence and uniqueness of the positive solution, persistence and extinction, and global attractivity. More importantly, we give the result of asymptotic stability in distribution for this hybrid stochastic model with impulsive perturbations. Some simulation examples are given in the last part of this paper to illustrate our theoretical results.

The existence of the global positive solution
Throughout this paper, let ( , F, F t≥0 , P) be a complete probability space with a filtration F t≥0 , B(t) be a given one-dimensional standard Brownian motion defined on this probability space and r(t) be a right-continuous Markov chain which is independent of B(t). The generator = (γ ij ) N×N of r(t) is given by where δ > 0 and γ ij ≥ 0 be the transition rate from i to j satisfying Since r(t) is right-continuous, almost every sample path of r(t) is a right-continuous step function with a finite number of sample jumps in the finite subinterval of R + . That is to say, there exists a sequence of stopping times, 0 = t 0 < t 1 < · · · < t k , and t k → ∞ almost surely such that r(t) is a constant in every interval [t k−1 , t k ) for any k ≥ 1, i.e.
Moreover, we assume that Markov chain r(t) is irreducible which can guarantee that the hybrid system controlled by r(t) will always switch from one regime to another. Under this condition, Markov chain r(t) 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 Consider the following hybrid stochastic differential equation without impulsive perturbations We can check that if y(t) is a solution of (6), then is a solution of (4). Conversely, if x(t) is a solution of (4), then is a solution of (6), so we get the following result.
Theorem 2.1: For any initial value x(0) = x > 0, r(0) = i ∈ S, the model (4) has a unique continuous solution (x(t), r(t)) on t ≥ 0 which can be represented by and the solution will remain in R + × S almost surely.
Details of the proof are omitted to conserve space and given in Appendix 1.
Remark 2.1: From Theorem 2.1, one can find that if 0<t k−1 <s A k (r(t k−1 ))b(r(s)) < 0 the solution may explode at a finite time, and inf i∈S b(i) > 0, A k (i) > 0, ∀ i ∈ S is a necessary condition for the global existence of solutions.

Remark 2.2:
Theorem 2.1 indicates that the solution of (4) with the positive initial value will remain positive. This property provides us with an opportunity to check persistence and extinction, stochastic permanence, the existence of the stationary distribution and stability in distribution.

The dynamics behaviours of the model (4)
In this section, we will discuss the dynamic behaviours of the model (4). The properties, such as stochastically permanence, persistence and extinction, global attractivity, all have great biological meanings. These proofs and conclusions can be found in Appendix 2, and here we give some necessary explanations for our results.
In Section 2, we showed that (4) has a positive solution for any initial values. In this section, we will investigate how the solution varies in R + . For convenience and simplicity in the following discussion, for any constant sequence c(i), i ∈ S, leť

Definition 3.1:
The solution x(t) to (4) is said to be extinctive with probability one if lim t→∞ x(t) = 0 a.s.

Remark 3.1:
If there are no impulsive perturbations, then Theorem 3.2 indicates that the model (4) is extinctive with probability one when πβ < 0, which is a result in [17].

Remark 3.2:
Notice that, by the definition of extinction, the population system will be extinctive with probability one if A * < 0. Ma and his co-authors in [7,25,29] proposed the concepts of the persistence, including non-persistence in mean, weakly persistent, persistence in mean, for some deterministic models. For stochastic systems, the concept of persistence is complex, we first discuss some kinds of persistence, such as nonpersistent in mean, weakly persistent and persistent in mean, which are the analogous concepts to the deterministic population models.
The following conditions for persistence and weakly persistent of (4) are obtained. For the corresponding definitions to deterministic population models, the definitions of persistence and weakly persistent for the stochastic system is defined as the following.
, that is, the population represented by the model (4) is nonpersistent in mean.
Theorem 3.6: it is alternative to to say that the model (4) is persistent in mean almost surely. Remark 3.3: Theorem 3.4-3.6 indicate that the intensity of impulsive perturbations and the Markov chain will affect the persistence of (4) to some extent.
In the following, we provide the result for stochastic permanence, which shows that the population system will survive in the future with large probability. The proof is omitted here as it is only a direct promotion for stochastic permanence of (4) without Markovian switching, the results for the model without impulsive perturbations can be found in [17].     (4) is said to be globally attractive if any two solutions x 1 (t) and x 2 (t) with the initial values x 1 (0) and

Asymptotic stability in distribution
The concept of stability in distribution means that the solution of stochastic differential equation does not converge to 0 or the equilibrium state, but to some a stationary distribution. Please see [26] for the accurate definition. In this section, we will prove that the model (4) is stability in distribution under some conditions. Let P(R + × S) be the space of all probability measures on R + × S, we define the metric d L in P(R + × S) as  (4) is said to be asymptotically stable in distribution if transition probability density function (p 1 (t, x, ·), p 1 (t, x, ·), . . . , p N (t, x, ·)) of the solution (x(t), r(t)) converges weakly to a probability measure (π 1 (·), π 2 (·), . . . , π N (·)) of the probability measures space as t → ∞.
Moreover, it is easy to check that x(t) satisfies (9). For t = t k , because , so x(t) also satisfies (9) immediately. Proof: Let K ∈ R + be any compact set, from the global attractivity of (4) (Theorem 3.13), for any x, y ∈ K, i, j ∈ S and f ∈ L, we can compute that as the proof of Lemma 5.6 in [26], where (x(t), r i (t)) and (y(t), r j (t)) are the two solutions of (4) with the initial value (x, i) and (y, j), respectively. By the definition of d L , for any x, y ∈ K and i, j ∈ S, we have lim t→∞ d L (p i (t, x, ·), p j (t, y, ·)) = 0.
Therefore, the Equations (10) and (11) imply that By the definition of the weak convergence of probability measures, for any x ∈ R + , i ∈ S the transition probabilities {p i (t, x, ·) : t ≥ 0} converge weakly to the probability measure π(·). So the model (4) is asymptotically stable in distribution. According to our results, we know that this equilibrium solution becomes a stationary distribution that is stable (stability in distribution) under the perturbations of the white noise, the Markov chain and the impulse.
For the unique stationary distribution μ(·) of the solution process (x(t), r(t)) to the model (4), we have the following result according to [13].

Concluding remarks and examples
The hybrid system is often used to describe the sudden changes in the process that appeared in various fields of science and technology. This paper investigates the dynamic behaviours of the hybrid stochastic logistic model with impulsive perturbations. The switching may occur at the impulse points in our model. We first give the existence and uniqueness of the global positive solution to the model (4), and then obtain sufficient conditions for the persistence and extinction, stochastic permanence, global attractivity and stability in distribution. Also, we will illustrate some of our results given above in the following two examples.

Example 5.1:
Consider where the Markov chain r(t) taking values in S = {1, 2}. Then the model (12) can be regarded as a switching between and according to the movement of the Markov chain r(t), where We consider no impulsive perturbations and constant coefficient impulsive perturbations respectively in the model (12).
In the following, we will consider two cases of the Markov chain r(t) as defined in Case 1 and Case 2 of (1). A more interesting fact is that under the regime switching of the Markov chain r(t) and constant coefficient impulsive perturbations, the population is persistent in mean although it is stochastic permanent and extinctive under the regime switching of Markov chain r(t) without impulsive perturbations. Case 1. Let the Markov chain r(t) as in Case 1 of (1), we can get that By Theorem 3.6, the model (12) is persistent in mean; see Figure 4(a). Therefore, Theorem 3.6 implies that the model (12) is persistent in mean; Figure 4(b).
For the Markov chain r(t) as in Case 2 of (1), we get that Then Theorem 3.8 implies that the model (12) is stochastically permanent; see Figure 5(a) and Figure 5(b). This paper devotes to studying the dynamic behaviours of a hybrid stochastic single population logistic model, the methods developed here may also applicable to other models, such as stochastic Gompertz model, Lotka-Volterra systems with more species, and we leave these for our future work. The results and some methods given in this paper provide a fundamental work for the further research.

Funding
This research work was supported by the Fundamental Research Funds for the Universities of Henan Province (NSFRF140139) and the Research start-up funds SDUFE 1701037.
is a solution to (4) and the proof is similar to Theorem 1 of [32]. In fact, For every t k ∈ [0, +∞), we have Similarly, So from the definition of the solution for ISDE, we have proved that (x(t), r(t)) is a solution to (4) with the initial value (x, i). Following we will prove the uniqueness of the solution to (4). For t ∈ (0, t 1 ) and t ∈ (t k , t k+1 ), k ∈ N, the Equation (4) becomes the following classical stochastic differential equation Note that the coefficients of (A1) are local Lipschitz continuous, by uniqueness theorem for SDE (see [26]), the solution to (A1) is unique. For t = t k , k ∈ N, exists and unique. So the uniqueness of the solution to (4) is obviously. Moreover, if x(t) is a solution to (4), then y(t) = 0<t k−1 <t A −1 k (r(t k−1 ))x(t) is a solution to (6). In fact, since x(t) is right-continuous on each interval [t k , t k+1 ), so for any k = 1, 2, . . . , = y(t k ), k = 1, 2, . . . , which implies that y(t) is continuous on R + , and then we can get that y(t) is a solution to (6). The proof is complete.
Particularly, if A * < 0, then On the other hand, Substituting (A7) into (A8), we can get that for all t ≥ T Integrating both sides of this inequality from T to t, we have Therefore As a consequence lim sup Then the L'Hospital's rule implies that lim sup The required assertion follows from the arbitrariness of ε.
Proof of Theorem 3.6: We suppose that A * > 0 in the following proof, then for ∀ ε > 0, there exists a positive T such that for all t > T 1 t Substituting these into (A8) and noting that x(s) ds.

E|y(s)| q ds
For any t ≥ 0, as Theorem 6 of [21], we have E|y(s)| q ≤ G(q), G(q) is related to q. Then for 0 < t 1 < t 2 < ∞, one can derive that where K(q) is related to G(q). Therefore from Lemma 3.10 that almost every sample path of y(t) is uniformly continuous on t ≥ 0.