Long-time behavior of a non-autonomous stochastic predator-prey model with jumps

It is proved the existence and uniqueness of the global positive solution to the system of stochastic differential equations describing a non-autonomous stochastic predator-prey model with a modified version of Leslie-Gower and Holling-type II functional response disturbed by white noise, centered and non-centered Poisson noises. We obtain sufficient conditions of stochastic ultimate boundedness, stochastic permanence, non-persistence in the mean, weak persistence in the mean, and extinction of the solution to the considered system.


Introduction
The deterministic predator-prey model with modified version of Leslie-Gower and Holling-type II functional response is studied in [1]. This model has a form dx = x a − bx − cy m 1 + x dt, where x(t) and y(t) are the prey and predator population densities at time t, respectively. Positive constants a, b, c, r, f, m 1 , m 2 defined as follows: a is the growth rate of prey x; b measures the strength of competition among individuals of species x; c is the maximum value of the per capita reduction rate of x due to y; m 1 and m 2 measure the extent to which the environment provides protection to prey x and to the predator y, respectively; r is the growth rate of predator y, and f has a similar meaning to c. In [1] the authors study boundedness and global stability of the positive equilibrium of the model (1).
In the papers [2], [3], [4] it is considered the stochastic version of model (1) in the following form dt + αx(t)dw 1 (t), dy = y r − f y m 2 + x dt + βy(t)dw 2 (t), where w 1 (t) and w 2 (t) are mutually independent Wiener processes in [2], [3], and processes w 1 (t), w 2 (t) are correlated in [4]. In [2] the authors proved that there is a unique positive solution to the system (2), obtained the sufficient conditions for extinction and persistence in the mean of predator and prey. In [3] it is shown, that under appropriate conditions there is a stationary distribution of the solution to the system (2) which is ergodic. In [4] the authors prove that the densities of the distributions of the solution to the system (2) can converges in L 1 to an invariant density or can converge weakly to a singular measure under appropriate conditions. Population systems may suffer abrupt environmental perturbations, such as epidemics, fires, earthquakes, etc. So it is natural to introduce Poisson noises into the population model for describing such discontinuous systems.
In this paper, we consider the non-autonomous predator-prey model with modified version of Leslie-Gower and Holling-type II functional response, disturbed by white noise and jumps generated by centered and non-centered Poisson measures. So, we take into account not only "small" jumps, corresponding to the centered Poisson measure, but also the "large" jumps, corresponding to the non-centered Poisson measure. This model is driven by the system of stochastic differential equations where x 1 (t) and x 2 (t) are the prey and predator population densities at time t, To the best of our knowledge, there have been no papers devoted to the dynamical properties of the stochastic predator-prey model (3), even in the case of centered Poisson noise. It is worth noting that the impact of centered and non-centered Poisson noises to the stochastic non-autonomous logistic model and to the stochastic two-species mutualism model is studied in the papers [5] - [7].
In the following we will use the notations For the bounded, continuous functions f i (t), t ∈ [0, +∞), i = 1, 2, let us denote We prove that system (3) has a unique, positive, global (no explosion in a finite time) solution for any positive initial value, and that this solution is stochastically ultimate bounded. The sufficient conditions for stochastic permanence, non-persistence in the mean, weak persistence in the mean and extinction of solution are derived.
The rest of this paper is organized as follows. In Section 2, we prove the existence of the unique global positive solution to the system (3) and derive some auxiliary results. In Section 3, we prove the stochastic ultimate boundedness of the solution to the system (3), obtain conditions under which the solution is stochastically permanent. The sufficient conditions for non-persistence in the mean, weak persistence in the mean and extinction of the solution are derived.

Existence of global solution and some auxiliary lemmas
Let (Ω, F , P) be a probability space, w i (t), i = 1, 2, t ≥ 0 are independent standard one-dimensional Wiener processes on (Ω, F , P), and ν i (t, A), i = 1, 2 are independent Poisson measures defined on (Ω, F , P) independent on w i (t), , i = 1, 2 are finite measures on the Borel sets in R. On the probability space (Ω, F , P) we consider an increasing, right continuous family of complete sub-σ- We need the following assumption.
In what follows we will assume that Assumption 1 holds.
Proof. Let τ n be the stopping time defined in the Theorem 1. Applying the Itô formula to the process V (t, x 1 (t)) = e t x p 1 (t), p > 0 we obtain Under Assumption 1 there is constant K 1 (p) > 0, such that Letting n → ∞ leads to the estimate So from (22) we derive (18). Let us prove the estimate (19). Applying the Itô formula to the process For the function For From (23) and (24) by integrating and taking expectation, we derive Letting n → ∞ leads to the estimate From (25) we have (19).

The long time behaviour
Definition 1. ( [9]) The solution X(t) to the system (3) are said to be stochastically ultimately bounded, if for any ε ∈ (0, 1), there is a positive constant χ = χ(ε) > 0, such that for any initial value X 0 ∈ R 2 + , the solution to the system (3) has the property that In what follows in this section we will assume that Assumption 1 holds.
The solution X(t) to the system (3) is stochastically ultimately bounded for any initial value X 0 ∈ R 2 + . Proof. From the Lemma 3 we have estimate (32) Let χ > L/ε, ∀ε ∈ (0, 1). Then applying the Chebyshev inequality yields The property of stochastic permanence is important since it means the long-time survival in a population dynamics. Definition 2. The population density x(t) is said to be stochastically permanent if for any ε > 0, there are positive constants H = H(ε), h = h(ε) such that for any inial value x 20 > 0. Theorem 3. If p 2 inf > 0, where p 2 (t) = a 2 (t)−β 2 (t), then for any initial value x 20 > 0, the predator population density x 2 (t) is stochastically permanent.
Remark 1. If the predator is absent, i.e. x 2 (t) = 0 a.s., then the equation for the prey x 1 (t) has the logistic form. So Theorem 4 gives us the sufficient conditions for the stochastic permanence of the solution to the stochastic non-autonomous logistic equation disturbed by white noise, centered and non-centered Poisson noises.
Theorem 5. If then the solution X(t) to the equation (3) with initial condition X 0 ∈ R 2 + will be extinct.