Analysis of a Stochastic Ratio-Dependent Predator-Prey System with Markovian Switching and Lévy Jumps

In this paper, the dynamics of a stochastic ratio-dependent predator-prey system with markovian switching and Lévy noise is studied. Firstly, we show the existence condition of global positive solution under the given positive initial value. Secondly, sufficient conditions for system extinction and persistence are obtained through some assumptions. Then, the sufficient conditions of stochastically persistence are obtained by combining stochastic analysis technique and M-matrix analysis. In addition, under appropriate conditions, we demonstrate the existence of a unique stationary distribution for a system without Lévy jumps. Finally, the empirical and Mlistein methods are used to verify the theoretical results through numerical simulation.


Introduction
As far as we know, rate-dependent predator-prey system models have become the focus of mathematical ecology and have been extensively studied in recent years (see e.g., [1] [2] [3]). The dynamic relationship between predator and prey is ubiquitous in ecology and mathematical ecology [4]. The relationship between two species is usually thought of as competition, predation and cooperation. Here is the Lotka-Volterra model of a predator-predator with intra-species competition: represent the effect of one species on the other. It is well known that the solution of system (1.1) is asymptotically stable.
In the study of biological phenomena, there are many factors affecting the dynamic properties of biological and mathematical models, and functional reaction is one of the common nonlinear factors [4]. System (1.1) assumes the prey biomass is enough and a individual predator consume the prey with functional response of type 1 c x . When predators face an increase in the density of their local prey, they usually change their consumption rate. The concept of functional response was first proposed by Solomon and later discussed in detail by Holling. Holling [5] proposed three types of functional responses, i.e., Hollings type . Proportional-dependent functional responses are a better description of how predators must find food and therefore must share or compete for it. Based on Holling type II function, Arditi and Ginzburg [2] first proposed a rate-dependent functional response model: x ey c xy y a y t x ey where parameters 1 2 , , c c e are all positive constants, representing capturing rate, conversion rate and half capturing saturation constant, respectively.
These papers are all deterministic models, which do not consider the impact of environmental fluctuations, nor the impact of population randomness. Population dynamics in nature will inevitably be affected by environmental noise in the ecosystem. More recently, a number of authors have looked at stochastic predator-prey models with white noise and revealed how white noise affects population systems, such as [4] [6] [7] [8]. Considering that the environmental fluctuation is mainly manifested as the fluctuation of the internal growth rate of the predator population and the mortality rate of the predator population [9], they supposed parameters 1 a and 2 a were perturbed with ( ) ( ) Nguyen and Ta [7] introduced intra-specific competition into the stochastic rate-dependent model to obtain the model (4), and considered the corresponding non-autonomous stochastic system, and estimated the high population growth rate and exponential mortality rate. The stochastic predator-prey system ( ) ( ) Furthermore, from a biological point of view, population dynamics may encounter sudden environmental disturbances that cannot be described by white noise, such as earthquakes, epidemics, floods and hurricanes. In this case, there are many references to stochastic differential equations with jumps (see e.g., [9] [10] [11] [12] [13]), and Lévy jumping into a potential population system may be a reasonable method to describe these phenomena. Therefore, this paper considers the random ratio-dependent model with jumps:  and characteristic measure λ on a measurable subset  of ( )  is bounded and continuous with respect to λ and is -measurable, 1, 2 l = . Now let's go one step further and add another environmental noise that is Telegraph noise. Telegraph noise can be described as a random switch between two or more environmental states that differ in factors such as nutrition or rainfall [14] [15]. The stochastic differential equations driven by a continuous-time Markov chain have been used to model the population system [16] [17] [18] [19] with this type of noise. Suppose the Markov chain on the state space controls the switching between the environmental regimes. Then the prey-predator model with two types of noise can be described by the following stochastic differential equation τ when Markov chain jumps to 1 i from 0 i ; the system will then obey (8) with τ to 2 τ when the Markov chain jumps to 2 i from 1 i . The system will continue to switch as long as the Markov chain jumps. In other words, system (8) can be regarded as system (6) [25]. The key methods used in this paper are m-matrix analysis (e.g., [18]) and stochastic analysis of Lyapunov functions developed by Khasminskii (see e.g., [18] [26]). This paper is organized as follows: In Section 2, we give the global existence and positive properties of the solutions of system (1.8). In section 3 and section 4, we give sufficient conditions for the non-persistence, weak persistence and extinction and stochastic permanence respectively. In Section 5, we demonstrate that system (5.1) has a unique stationary distribution under some appropriate conditions when Lévy jumps are not present. Finally, we illustrate our main results with two examples.

Preliminaries
Throughout this paper, let where 0 ζ > . Here ij γ represents the transition rate from i to j and We assume that the Markov chain ( ) x t ∈  be a solution of the stochastic differential equation with regime-switching jump-diffusion processes taking the form x t Then the generalized Itô's formula with jumps is given by For convenience and simplicity in the following discussion, define In this paper, we impose the following assumptions: Proof. The proof is standard (see e.g., [27] [28]) and hence is omitted. 2) the species ( )

The Persistence and Extinction of Populations
x t is said to be weakly persistent in the mean a.s., if [29]). Suppose that is Meyer's angle bracket process.
In view of the exponential martingale inequality, for any positive number , ,T α β , Applying the Borel-Cantelli Lemma, we can obtain that for all ω ∈ Ω , there is a random integer Substituting the above inequalities into (3.2) leads to It is easy to find that for any 0 s k γ ≤ ≤ and 0, 0 x y > > , there exists two In other words, for any  ( ) x t and ( ) y t of system (1.6) will tend to be extinct.
Proof. By the generalized Itô's formula, we derive from (1.6) that r t x t a r t r t r t c r t y t b r t x t t x t e r t y t r t B t r t N t r t y t a r t r t r t c r t x t b r t y t t x t e r t y t     Dividing by t on both sides of (18) and then taking the superior limit, we obtain Integrating the above inequality from T to t results in Taking the logarithm on both sides, we have  x t of system (1.6) will be weakly persistent in the mean.
Proof. By the generalized Itô's formula, and dividing by t on both sides, we

Stochastic Permanence
In the study of population dynamics, stochastic permanence is one of the most important properties. We first introduce the definition of stochastic permanence [6], which is widely used in the field of population dynamics (see e.g., [22] [28]). In this section, we shall discuss this property.
dropping t from ( ) x t and ( ) y t . We notice that both coefficients of the higher 1 1 x θ + , According to (4.5)  To state our main result, we give some notations. Let C be vector or matrix. , then the following statements are equivalent: 1) C is nonsingular M-matrix.
2) All of the principal minors of are positive; that is 3) C is semi-positive; that is, there exists The proof of the stochastic permanence is rather technical, so we prepare sev- Proof. The proof is rather standard and hence is omitted (see [15]).
In view of (4.7), (4.8) and (4.9), there exists Integrating (4.10) from 0 to t and taking expectation of both sides, we have Base on (4.11), we obtain which is the required assertion.
Proof of ( ) y t using the same method, namely, Finally, (4.2) is obtained by combining lemma 4.1 and Chebyshev inequality. Therefore, the system (1.6) is stochastically permanent.
In the same spirit as in the proof of Theorem 4.1, we yield the result of the subsystem (1.8) as follows.

Stationary Distribution
As far as we know, the existence of the ergodic stationary distribution of a stochastic competition model with high order stochastic perturbations has not been c r t y t x t x t a r t b r t x t t x t e r t y t r t x t B t c r t x t y t y t a r t b r t y t t x t e r t y t Next, from the theorem in [30] we have the following lemma, we will use this lemma to prove ergodic stationary distribution.
Lemma 5.1. If the following conditions are satisfied: 2) for each k ∈  ,

( )
, D x k is symmetric and satisfies ( ) ⋅ is twice continuously differential and that for some 0 ς > , such that for any 0 θ θ , , ,

Conclusions and Example
The stochastic persistence and extinction of a stochastic ratio-dependent predator-prey system with Markovian switching and Lévy noise are studied. Our main results are as follows: Theorem 3.1 gives sufficient conditions on extinction and non-persistent in mean of each population. Theorem 3.3 gives sufficient conditions for each population to be weakly persistent. Further, Corollaries 3.1 tells us that for some i S ∈ , if ( ) ( )  From (6.4) and Corollary 4.2, it is easy to find that the subsystem (6.1) is extinct, but subsystem (6.2) is persistent. According to Theorem 4.1, system (1.6) is stochastically permanent. This example shows that although some subsystems are impermanent, the overall behavior of system (1.6) is stochastically permanent as a result of Markovian switching. Thus, Markovian switching may contribute to permanence to some extent. The numerical result is shown in Figure  1