Dynamics of a stochastic Gilpin–Ayala population model with Markovian switching and impulsive perturbations

In this paper, we consider the dynamics of a stochastic Gilpin–Ayala model with regime switching and impulsive perturbations. The Gilpin–Ayala parameter is also allowed to switch. Sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence are provided. The critical number among the extinction, nonpersistence in the mean, and weak persistence is obtained. Our results demonstrate that the dynamics of the model have close relations with the impulses and the Markov switching.


Introduction
Population ecology is an important branch of ecology; people often use some basic models in mathematical ecology and the mathematical analysis methods to study the living environment conditions of organisms. In many cases, mathematical models are used as a tool to study related problems in biology and ecology to make the internal laws of organisms more vividly displayed. One of the hot spots in the field of mathematical ecology today is the study of population size. The most basic mathematical model for a single species population growth is the logistic equation, which has been studied by many scholars. The classical logistic model is described by an ordinary differential equation: where x(t) is the population size at time t, r > 0 stands for the intrinsic growth rate of species, k > 0 denotes the intraspecific competition coefficient, and r k is the carrying capacity. However, the logistic model assumes that the exponential growth of species in an ecosystem is a linear. This simplifies the model, but is very different from reality. Therefore, in order to better fit the actual data, Gilpin-Ayala [1] considered the following model: where θ is a positive parameter to modify the classical logistic model. Many important research works about various forms of Gilpin-Ayala model have appeared in the literature, see [2][3][4] and the references therein.
In the real world, any individual organism in nature is inevitably affected by its own changes in quantity and environmental noises, which are important components of the ecosystem [5]. The effects of demographic stochasticity can be neglected when the population sizes are large enough. However, the environmental noise affects the population growth rate, carrying capacity, competition coefficient, and so on. Then it follows that a deterministic system cannot reflect the authenticity of population growth, an also ignores the impact of random fluctuations of various internal and external factors on population growth. Random external perturbations need to be considered. In recent years, stochastic Gilpin-Ayala models have been studied extensively [6][7][8][9][10].
Generally speaking, there are two types of environmental noise, namely, white noise and colored noise. White noise, which describes the small perturbations around the equilibrium state, is common in the real world. Considering the white noise in the growth rate in Gilpin-Ayala model, Mao et al. [11] presented the following stochastic Gilpin-Ayala model: The authors showed that this equation is stochastically permanent and the solution is globally attractive under some basic assumptions.
The colored noise can be explained as a switching between two or more regimes of environment [12][13][14][15][16]. It should be pointed out that colored noise has a great influence on the properties of the system. The switching among different environments is memoryless, and the waiting time for the next switch is exponentially distributed [17]. So the regime switching can be modeled by a right-continuous Markov chain (γ (t)) t≥0 taking values in a finite state space S = {1, 2, . . . , m}. It has been noted [18] that population models may experience random changes in their structure and parameters, by factors such as nutrition or as rain falls. These random changes cannot be described only by the white noise [13,19]. Inspired by this, Liu et al. [17,20] considered the following stochastic Gilpin-Ayala model under regime switching: They studied the existence of global positive solution, extinction, nonpersistence, and weak persistence of the species, and obtained the stochastic permanence of the species under the conditions that 0 < θ ≤ 1 and 0 < λ ≤ 1+θ . Settati et al. [21] considered a Gilpin-Ayala model with the Gilpin-Ayala parameter θ under regime switching: (1.1) They presented global stability of the trivial solution and sufficient conditions for the extinction, persistence, and existence of stationary distribution. Later, in [22], they presented a stochastic Gilpin-Ayala model under regime switching on patches, studied the effect of environmental noises in the asymptotic properties of the model on patches under regime switching, and established some sufficient conditions for extinction and persistence of species.
It is important to point out that the Markov chain may have significant impacts on the properties of the population dynamics. Takeuchi et al. [23] have investigated a predatorprey deterministic system described by Lotka-Volterra equations in a random environment and revealed a very interesting and surprising result of Markovian switching on the population system: both its subsystems develop periodically, but switching makes them become neither permanent nor dissipative. Now let us take a further step. There is no independent living organism in nature, and any organism is subject to a variety of transient effects that cause sudden changes in system variables or growth patterns. The common transient effects include external factors, internal laws, and human factors. Impulsive differential or difference equation describes the rapid changes or jumps of some moving states at fixed or nonfixed moments. Hence, impulsive equation models are more realistic responses to some natural phenomena and biological processes, and have more abundant properties and contents than differential equations [24][25][26][27].
However, because of the jump in the state of the system, the solution of the continuous system is discontinuous, which makes the study of the impulse equation more difficult. In recent years, the stability of impulsive differential equations and the existence and stability of periodic solutions of periodic systems have been studied systematically. Wu [28] considered the Gilpin-Ayala model under regime switching with impulsive effects: (1.2) He focused on exploring the effects of Markovian switching and impulse on the extinction, the threshold value of persistence and extinction. The author also established sufficient criteria for stochastic permanence by using the theory of M-matrices. Motivated by the above discussion, this paper considers the following general stochastic hybrid system with Markovian switching and impulsive effects: where (ξ (t)) t≥0 is a right-continuous Markov chain taking values in a finite state space S = {1, 2, . . . , m}; θ (i) > 0 for any i ∈ S. In regime i ∈ S, the system obeys (1.4) The model is presented for the following two improvements:

Global positivity
Throughout this paper, we suppose that there is a complete probability space (Ω, F, where t > 0, q ij ≥ 0 for all i = j is the transition rate from i to j while m j=1 q ij = 0. As a standing hypothesis, we assume in this paper that the Markov chain (ξ (t)) t≥0 is irreducible. Under this assumption, the Markov chain has a unique stationary distribution π = (π 1 , π 2 , . . . , π m ) satisfying πQ = 0 subject to m i=1 π i = 1 and π i > 0, ∀i ∈ S [29]. Firstly, we show the global positivity of system (1.3), which is a necessary condition among biological models. For convenience and simplicity, we definê Throughout the paper, we always assume that Proof Consider the following Itô equation without impulse: with initial value y(0) = x(0). Here for simplicity, we drop t from r(ξ ) and r(ξ ) k(ξ ) , etc. By a similar proof as that of [17, Theorem 1], (2.1) has a unique global positive solution y(t) for And for t = t k , Further, for every t k ∈ N, and This completes the proof.

Persistence and extinction
The persistence or extinction, as a measure of population size in the long run, is of importance in the study of a population model. This section will study the persistence and extinction of the system. Firstly, we introduce some concepts.
Now, we are ready to give our main results of this section.
Proof It follows from the ergodicity of ξ (t) that Applying generalized Itô formula to Eq. (2.1), one can see that Then we have Making use of the strong law of large numbers for martingales results in Recalling that for all t > 0. Hence, we have Taking limit superior of both sides and then making use of (3.1) and (3.2), we can conclude lim sup t→∞ t -1 ln x(t) ≤b. That is, ifb < 0, then lim t→∞ x(t) = 0. The proof is completed.
If for all t > 0, where v 0 ≥ 0 and v > 0 are two real numbers.
Theorem 3.2 Ifb = 0, then species x is nonpersistent in the mean.
Proof For arbitrarily fixed ε > 0, there exists a constant T > 0 such that for all t ≥ T. By the above inequalities, (3.3) becomes Using Lemma 3.1, we obtain Hence, conditionb = 0 and the arbitrariness of ε will lead to This completes the proof.
If for all ω ∈ U, lim sup t→+∞ x(t, ω) = 0, then Substituting the above into (3.6) results in which is a contradiction. This completes the proof.
Ifb > 0, the population x(t) is weakly persistent. Ifb < 0, the population x(t) becomes extinct. These also imply that the long time behaviors of system (1.3) have close relations with impulse, white noise, and stationary distribution of the Markovian switching.
Remark 3.2 Let us look at the effects of impulse on the extinction and persistence of species. The result in no-impulse system [17,Theorem 2] has shown that condition m ξ =1 π ξ (r(ξ ) -1 2 σ 2 (ξ )) < 0 leads to the species extinction. However, Theorem 3.3 tells us that if the impulsive effects h k (k ∈ N) are large enough such that lim sup t→∞ t -1 × 0<t k <t ln(1 + h k ) > | m ξ =1 π ξ (r(ξ ) -1 2 σ 2 (ξ ))|, i.e.,b > 0, then species will be weakly persistent. This means that a positive impulse is always advantageous for the existence of the species, e.g., planting. Furthermore, we can see that the positive impulse can resist the impact of the white noise, which in accord with reality and will not happen in the model without impulses. Remark 3.3 Let us consider the effects of white noise on the species. From the expression of the critical valueb, we can see that the white noise σ (ξ (t)) imposed on the intrinsic growth rate r(ξ (t)) leads to the extinction of the species and is disadvantageous to the survival of the species.
Remark 3.4 Let us turn to the impact of the Markovian switching on the system. For subsystem (1.4), we can easily prove that (see [31]) ifb(i) = r(i) -0.5σ 2 (i) + lim sup t→∞ t -1 × 0<t k <t ln(1 + h k ) < 0, then species x(t) of (1.4) will go extinct. And ifb(i) > 0, the species x(t) will be weakly persistent. Now, let us consider a case that the impulsive effect is nonpositive. In this case, if r(i) -0.5σ 2 (i) < 0 for every i ∈ S, then every subsystem of (1.4) will become extinct. From Theorem 3.1 we can see that the ultimate behavior of (1.3) is also extinction even if Markovian switching exists. However, if there is a subset S 0 ⊂ S such that r(i) -0.5σ 2 (i) > 0 for all i ∈ S 0 , the irreducibility of the Markov chain will guarantee that if the species spends enough time in these "good" states (i.e., the states with r(i)-0.5σ 2 (i) > 0, i ∈ S 0 can contribute to the resultb > 0), the species can still be persistent under regime switching.

Theorem 3.4 If
then the solution of system (1.3) obeys Proof Applying the inequality (see [21]) By the ergodic theory of Markov chains, we have, for any ε > 0, there is a T > 0 such that for all t ≥ T. Substituting this inequality into (3.8), we deduce that for all t ≥ T. From (3.7) and using Lemma 3.1, we get Letting ε → 0, we finally obtain the desired result.

Theorem 3.5 Ifd > 0, then
and Λ is the unique positive solution of the equation Proof Consider the function Obviously, f (z) is a continuous, strictly decreasing function on (0, ∞) and Hence the existence, uniqueness, and positivity of Λ can be ensured by the intermediate value theorem. First we prove the right side of (3.9). If the assertion is not true, we suppose that Then there is A ∈ ( 1 2 , 1) > 0 such that P(Ω 1 ) > 0, where So, for every ω ∈ Ω 1 , there is a T(ω) > 0 such that for all t ≥ T(ω), From (3.3) and using (3.11), we deduce that Based on the ergodic theory of Markov chains and (3.2), we can easily show from (3.12) that there exists Ω 1 ⊂ Ω 1 such that P(Ω 1 ) = 1 and, for every ω ∈ Ω 1 , we get It follows from the Λ-equation (3.10) and Aθ < 1 that Therefore lim t→∞ x(t) = ∞. This contradicts (3.11). The required assertion must therefore hold. Similarly, we can prove the left side of (3.9).
Theorem 3.5 tells us that solution of system (1.3) oscillates infinitely often about Λ with probability one. Therefore, it is helpful to know more information about Λ. For this, we have the following corollary. where Proof Denote the left-hand side of Eq. (3.10) by f (Λ). The condition of claim (i) indicates that f (1) = 0, which means that Λ = 1. Hence, the result of (i) holds. The condition of (ii) indicates that f (1) > 0. Applying the intermediate value theorem, we have Λ > 1. It is easy to calculate that for all i ∈ S, Λθ ≤ Λ θ(i) ≤ Λθ . Then, from (3.8) we obtain that Λ 1 ≤ Λ ≤ Λ 2 , which implies that x(t) lies in the interval [Λ 1 , Λ 2 ] infinitely often with probability one.

Stochastic permanence
Stochastic permanence describes the persistence of a population in the presence of random disturbances. In the following, we will further study the stochastic permanence of the system. We first give the definition of stochastic permanence.   Now we present our main result of this section. In the following, we write V (y(t), r(t)) with V (y, k) for simplicity. Proof First let us demonstrate that for arbitrary given ε > 0, there exists a constant β 2 > 0 such that lim inf t→∞ P{|x(t)| ≥ β 2 } ≥ 1 -. Define where λ is any positive number. Applying generalized Itô formula to Eq. (2.1), one can see that For α given in Lemma 4.2, there exists a vector p = (p 1 , p 2 , . . . , p m ) T 0 (where p 0 means p i > 0 for every i = 1, 2, . . . , m) such that A(α) p 0, which is equivalent to Making use of Itô formula leads to where Now, choose η > 0 sufficiently small to satisfy Define V 3 (y, k) = e ηt V 2 (y, k) and let r K = ψ. By virtue of the generalized Itô formula, dV 3 (y, k) = ηe ηt V 2 (y, k) dt + e ηt dV 2 (y, k).
Integrating this inequality and then taking expectations of both sides, we can see that where LV 2 (y, k) + ηV 2 (y, k) We claim that L(y, k) is upper-bounded. Without loss of generality: (1) If y ≥ 1, since λ > 0 is arbitrarily, we can choose λ to satisfyθ ≤ 1 + λ. Then by the definition of V 1 , we can check that L(y, k) is upper-bounded. That is to say, there exists a positive number L 1k such that sup y≥1 L(y, k) < L 1k .
Remark 4.1 The stochastic permanence of the system means that the population will remain in a positive and bounded range with probability one. From the conditions of Theorem 4.1, we can see that the stochastic permanence has close relations with the impulse and stationary distribution of the Markov chain. In order to make the population survive forever, the impulse must be in a certain range, which is consistent with reality.

Conclusions and further remarks
This paper is concerned with the dynamics of a stochastic Gilpin-Ayala model with regime switching and impulsive perturbations. The Gilpin-Ayala parameter is also allowed to switch. We established some criteria on the permanence and extinction of the system. Further, we obtained some sufficient conditions on the extinction and stochastic permanence. The critical value among the extinction, nonpersistence in the mean and weak persistence has been given. Our results demonstrate that the dynamics of the model have close relations with the impulse and the Markov switching. Some interesting topics deserve further investigating. It is an interesting question to investigate the ergodicity and stationary distribution of the system. It is also interesting to study the asymptotic properties of stochastic Lotka-Volterra systems under regime switching.