Analysis of a stochastic single species model with Allee effect and jump-diffusion

In this paper, we consider the effects of the small and abrupt random perturbations in the environment, and formulate a stochastic single species model with Allee effect and jump-diffusion. We first prove that the model admits a unique solution which is global and positive. Then we study the stochastic permanence and extinction of the model. In addition, we estimate the growth rate of the solution. Our results reveal that the properties of the model have close relationships with the jump-diffusion. Finally, we work out several numerical simulations to validate the theoretical results.

On the other hand, the growth of populations often encounters environmental per- turbations. Therefore, one should introduce stochastic factors into population models [8,9,12,13,18,26]. In [9,26] the authors researched the following stochastic single species model with Allee effect driven by the Brownian motion: where B(t) stands for a standard Brownian motion defined on a completed probability space (Ω, F , P), ξ represents the intensity of the stochastic perturbations. The authors of [26] investigated the persistence, extinction, stochastic permanence and the ergodicity of model (2). Nevertheless, there are lots of abrupt perturbations in the environment which cannot be characterized by the Brownian motion, for instance, drought, epidemic, pesticides. Several scholars (see [2,3,8,19,20]) have suggested that one might utilize a Lévy jump process to characterize these abrupt perturbations. Hence, Eq. (2) is replaced by where ψ(t -) is the left limit of ψ(t), X ⊂ (0, +∞), Υ (dt, dx) = Υ (dt, dx)π(dx) dt, Υ represents a Poisson counting measure, π stands for the characteristic measure of Υ with π(X) < +∞. To the best of our current knowledge, there are no studies of Eq. (3).
In this paper we explore model (3). We will prove in Sect. 2 that the model admits a unique solution which is global and positive. Then we study the stochastic permanence and extinction of the model in Sect. 3. In Sect. 4, we estimate the growth rate of the solution. In Sect. 5, we work out several simulations to numerically illustrate that the properties of the model have close relationships with the jump-diffusion.

The existence and uniqueness of the solution
Let C 1 , C 2 , . . . , C 7 be positive constants. Throughout this article, we assume that Υ and B(t) are independent, 1 + θ (x) > 0, ∀u ∈ X, and (H) X [ln(1 + θ (x))] 2 π(dx) < C 1 . This hypothesis illustrates that the jumps are tempered. Equation (3) is a population system, we must illustrate that it admits a unique global positive solution for any given positive initial value. To prove this, we need to recall a lemma. Let Y (t, y) represent the solution of the following scalar equation with initial value Y (0) = y = 0: where G 1 (0) = G 2 (0) = 0, and G 3 (0, x) = 0 for ∀x ∈ X; G 1 , G 2 , G 3 are locally Lipschitz continuous and measurable functions. Define

Permanence and extinction
Now we explore the permanence and extinction of model (3). and then Eq. (3) is said to be stochastically permanent (SP).
Proof By Itô's formula, one has where In view of Hypothesis (H), we can derive that Then the strong law of large numbers for local martingales (see [16]) indicates that lim t→+∞ t -1 Q(t) = 0 a.s.

Conclusions and simulations
In this paper, we utilized a Lévy jump process to characterize the abrupt perturbations in the environment, and formulated a stochastic single species model with Allee effect and jump-diffusion. We proved that the model admits a unique solution which is global and positive, and investigated the stochastic permanence, extinction and the growth rate of the solution.