DYNAMICS OF A STOCHASTIC POPULATION MODEL WITH ALLEE EFFECT AND JUMPS

. This paper is concerned with a stochastic population model with Allee eﬀect and jumps. First, we show the global existence of almost surely positive solution to the model. Next, exponential extinction and persistence in mean are discussed. Then, we investigated the global attractivity and stability in distribution. At last, some numerical results are given. The results show that if attack rate a is in the intermediate range or very large, the population will go extinct. Under the premise that attack rate a is less than growth rate r , if the noise intensity or jump is relatively large, the population will become extinct; on the contrary, the population will be persistent in mean. The results in this paper generalize and improve the previous related results.


Introduction
The Allee effect represents the relationship between population growth and population density. That is, for a population with Allee effect, if its population density is too sparse, it is so difficult to find a mate that reproduction does not compensate for mortality, then its population number will be reduced (see [1,10,34]). Allee effects mainly classified into two broad categories : strong Allee effect and weak Allee effect (see [4]). There is a threshold population level for the strong Allee effect such that the species become extinct below this threshold population density. On the other hand, the weak Allee effect occurs when the growth rate reduces but remains positive at low population density (see [4]). Recently, many literatures have studied the population dynamics with Allee effects (see [5][6][7]14]). In [7], the author used Poincaré-Bendixson theory to study the following single population model with Allee effect (1) Mots Clés. Lévy noise, Allee effect, extinction, attractivity, stability. with x(0) = x 0 . Here, r represents the intrinsic growth rate, c stands for the strength of intra-competition of the population. Under predation satiation circumstance, h and a represent the handling time and the attack rate of predator, respectively. Here, ax 1+hax is the Allee effect (see [10] for more details). In addition, populations are actually subject to the environmental noise. From [21], the extra noise can be divided into white noise and color noise. The white noise is uncorrelated, that is, each and every randomly drawn noise value is totally independent on previous value. However, the colored noise is temporally autocorrelated, that is, the values of random numbers used in the noise process will depend on the previous ones. Thus, it is more objective to modeling stochastic population models with white noise in mathematical biology (to name a few, see [8,9,11,22,25,30]). In these papers the authors revealed how noise affects the population dynamics. Especially, [8,9,11,25,30] investigated the dynamics of stochastic population models with Allee effect. There are many different ways to introduce environmental noise into the model. For example, due to the environment's continuous fluctuation, the intrinsic growth rate r always fluctuates around some average value. In this sense r → r + σḂ(t). Here σ 2 represents the intensity of white noiseḂ(t). Thus, one can get the following stochastic population model with Allee effect dt + σx(t)dB(t) (2) with x(0) = x 0 > 0. As we all know, Brownian motion is a stochastic process whose paths are continuous. However, natural populations are actually subject to sudden environmental shocks (earthquakes, hurricanes, epidemics, etc.), which have to be taken into account when the population dynamics model is established. Here, a sudden environmental shock will cause a sudden shift on the size of biological population, and the mathematical explanation is that sample paths are not continuous almost surely. It is recognized that stochastic differential equations with Lévy process are quite suitable to describe such discontinuous system. From [32], Lévy processes are stochastic processes with stationary and independent increments, in which they can form some special classes of both semi-martingales and Markov processes. Moreover, Lévy processes are not continuous, but their sample paths are right-continuous and have a number of random jump discontinuities occurring at random times, on each finite time interval. Thus, in order to depict these sudden changes in the environment, as in [17,18,[27][28][29], we propose to introduce Lévy process into the stochastic model (2) and obtain the following stochastic population model with Lévy noise and Allee effect with x(0) = x 0 > 0. Here x(t−) means the left limit of x(t), the parameters r, c, a and h are defined as in model (1). B(t) is a standard Brownian motion defined on a filtered compete probability space (Ω, F, {F t } t≥0 , P) satisfying usual hypotheses. N (·, ·) is a Poisson process with parameter λ on the measurable subset Γ ⊂ [0, ∞) with λ(Γ) < ∞. The compensated Poisson random measure N (dt, dz) = N (dt, dz) − λ(dz) dt is a martingale, which is independent ofḂ(t). σ 2 represents the intensity ofḂ(t). γ is the effect of jumps on the population. The remainder of this paper is organized as follows. In Section 1, we show that the model has a unique global positive solution by the comparison theorem of stochastic differential equations. In Section 2, the asymptotic pathwise behavior and stochastically ultimate boundedness of the model are investigated. In Section 3, We discuss the effect of Allee effect and stochastic perturbation on population dynamic behavior. Further, in Section 4, we first show that the solution of the model is globally attractive in mean. Then, the stability in distribution of the model is investigated. Section 5 contains numerical results, which are used to demonstrate the effectiveness of the theoretical results in this paper. The paper ends with a conclusion. The results in this paper generalize and improve the previous related results.

Existence and uniqueness of the positive solution
Note that x(t) is the size of the population. Thus, we should consider positive global solutions of (3).
Démonstration. Consider the following stochastic differential equation with v(0) = ln x 0 . Note that the coefficients of (5) satisfy the local Lipschitz condition. Hence, (5) has a unique maximal local solution v(t) on [0, τ e ), where τ e is the explosion time. Let x(t) = e v(t) . From the generalized Itô formula, model (3) has a unique positive local solution Next, we show that v(t) is a global solution to equation (5), i.e. τ e = ∞. Consider the following two stochastic auxiliary systems From Lemma 4.2 in [2], for any ϕ(0) = x 0 , equation (6) has a unique solution ϕ(t) > 0 on R + and can be explicitly solved as follow Similarly, for any ψ(0) = x 0 , equation (7) has a unique solution ψ(t) > 0 on R + and From stochastic comparison theorem (Theorem 3.1, [20]), it follows that Further, we have Since ln ϕ(t) and ln ψ(t) exist on [0, ∞), it follows that τ e = ∞. Thus, for any initial value v(0) = ln x 0 , equation (5) has a unique solution v(t) on R + a.s. Therefore, for any initial value x 0 ∈ R + , model (3) has a unique solution x(t) > 0 on R + a.s. The proof is therefore complete.

Asymptotic properties of the solution
In this section, by using the generalized exponential martingale inequality and Borel-Cantelli lemma, we investigate the asymptotic pathwise behavior of the model. Then, we continue to examine the stochastically ultimate boundedness of the model. Lemme 2.1 (see [16,24]). Let f : [0, ∞) → R and h : [0, ∞) × Γ → R be both predictable {F t }-adapted processes such that for any T > 0, Then, for any positive constants ϑ, ϱ, Here Démonstration. Applying Itô's formula to e t ln x and using inequality ln x ≤ x − 1 for x > 0, we have Note that ∫ T 0 σ 2 e 2s ds = σ 2 2 (e 2T − 1) < ∞ for any T > 0. Moreover, from (4), it follows that for any T > 0 By virtue of Lemma2.1, for any ϑ, ϱ, T > 0, we have Choose T = nγ, ϑ = εe −nγ and ϱ = (θe nγ ln n)/ε, where n ∈ N, γ > 0, θ > 1 and 0 < ε < 1 in above equation. Since ∑ ∞ n=0 1 n θ < ∞ for θ > 1, the Borel-Cantelli lemma (see Lemma 1.2.4 in [19]) implies that there exists a set Ω 0 ∈ F with P(Ω 0 ) = 1 and an integer-valued random variable n 0 = n 0 (ω) such that for every ω ∈ Ω 0 holds for all 0 ≤ t ≤ nγ, n ≥ n 0 . Substituting the above inequality into (8), we see that where in the second inequality, we use the inequality Let us consider function q(x) = ln It is easy to show that q has maximum value for x = 1 c > 0 and maximum value of q is q max = ln 1 Letting n → ∞ (and so t → ∞), we obtain lim sup t→∞ The proof is complete. Now, we show that model (3) is stochastically ultimate bounded. Définition 2.3 (see [12,13]). Model (3) is called stochastically ultimate bounded, if for any ε ∈ (0, 1), there exist a positive constant H 2 = H 2 (ε) such that the solution x(t) of model (3) with any initial value x 0 ∈ R + has the property that

Further, model (3) is stochastically ultimate bounded.
Démonstration. For p > 1, applying Itô's formula to e t x p and taking expectations, we obtain ds.

Extinction and persistence
In this part, we will investigate how Allee effect and stochastic perturbation affect the population dynamics behavior. For simplicity, denote ⟨u(t)⟩ = 1 t ∫ t 0 u(s)ds.

Extinction
If one of the following two conditions holds : then the population becomes extinct exponentially with probability one.
Démonstration. Applying the generalized Itô formula, we obtain Using the strong law of large numbers (see [19]), we have In addition, M (t) is a local martingale with M (0) = 0 and the quadratic variation of From the strong law of large numbers for local martingales (see [15]), Therefore, from (12) and (13), it follows that Consider By a simple calculation, we can obtain solutions of quadratic equation g(x) = 0 are Now, we differ two cases : which, together with (14), yields lim sup (11), it follows that which, together with (14), yields From Theorem 3.1, one can get the following corollary.

Corollaire 3.2. For any
This means that the high intensity noise and jump can lead to the extinction of the population.
Hence, one can get the conclusion.
Moreover, for models (1) and (2), from the proof of Theorem 3.1, one can get the following two corollaries.

Corollaire 3.3.
For any x 0 ∈ R + , let x(t) be the solution of model (1) corresponding to x(0) = x 0 . If one of the following conditions holds :

Remarque 3.5.
Kang and Udiani [10] investigated the following single species model with Allee effects , with . For model (15), from Lemma 2.1 in [10], we know that (1) If a < r, then model (15) has the following two equilibria : x 0 = 0 and x K , where x 0 = 0 is unstable while x K is locally asymptotically stable.
(2) If r < a < 1 hK or max < a and rhK < 4, then model (15) has only the extinction equilibrium x 0 = 0 which is globally stable.
(3) If 4hK < r < a hold or max and r < 4 hK hold, then (15) has three equilibria : x 0 , x K and x θ , where both x 0 and x K are locally asymptotically stable while x θ is unstable.
If we take c = r K , then model (15) can be reduced to model (1). From Lemma 2.1 (2) in [10], for any initial value x 0 ∈ R + , if one of the following conditions holds :  Remarque 3.6. In [26], the authors discussed the persistence and ergodicity of the following stochastic model under regime switching where r(i), K(i), a(i) and h(i) are all positive for any i ∈ M = {1, 2, · · · , N }, and ξ(t) is F t adapted but independent of the Brownian motion   (16) can be reduced to model (2). From Theorem 3.1 in [26], for any initial value x 0 ∈ R + , if one of the following conditions holds :  show that if attack rate a is less than growth rate r and the noise intensity or jump is relatively large, the populations in models (2) and (3) will become extinct (see Fig.3).

That is, population x(t) in model (3) is persistent in mean.
Démonstration. It follows from (11) that Thus, we have From (14), r + β − a > 0 and Lemma 3.8, we have The proof is complete.
From Theorem 3.9, one can get the following result.

Corollaire 3.10.
For any x 0 ∈ R + , let x(t) be the solution of model (2) Remarque 3.11. From Theorem 3.9, if attack rate a is less than growth rate r and the noise intensity or jump is relatively small such that a < r + β, the population in model (3) will be persistent in mean (see Fig. 4).

Remarque 3.12.
If there is no Allee effect, then stochastic model (3) can be reduced to For stochastic model (17), from Theorems 3.1 and 3.9, one can get the following results (i) if r + β < 0, then lim t→∞ x(t) = 0 a.s. ; (ii) if r + β > 0, then lim t→∞ ⟨x(t)⟩ = r+β c a.s. Wu and Wang [24] discussed non-autonomous model corresponding to model (17). From [24], we know that the jump noise and the general noise can make the population extinct. [31] discussed the dynamics of a stochastic predator-prey model with habitat complexity and prey aggregation. The results show that environmental noise is disadvantage for the survival of biological population.

Théorème 4.5. If a < √ c h , then the solution of model (3) is globally attractive in mean. Démonstration.
For any x i0 ∈ R + , let x i (t) be the solution of model (3) corresponding to x i (0) = x i0 (i = 1, 2). From the generalized Itô formula, it follows that dt.

Numerical simulations
In this part, using the theory and method (stationary Poisson point processes) mentioned in [33], we give some numerical simulations. Numerical experiments are made by using Γ = (0, +∞) and λ(Γ) = 1. Denote Exemple 5.1. From Theorem 3.1, it follows that no matter how large the noise intensity or jump is, as long as r < a ≤ √ c h , or a > √ c h and a > A > 0, the population will be extinct with probability 1. Here we choose x 0 = 2, r = 0.6, c = 0.2, a = 0.8. In Fig. 1 3 shows that white noise and jump may affect the survival of the population. Moreover, the results show that if the attack rate a is less than the growth rate r and the noise intensity or jump is relatively large, the populations in models (2) and (3) will become extinct.  As can be seen from Figs. 4(a)-4(c), populations described by models (1), (2) and (3) will be persistent in mean. The results show that if the attack rate a is less than the average growth rate r and the noise intensity or jump is relatively small, populations in models (1), (2) and (3) will be persistent in mean. Moreover, from Fig. 4(d), one can see that Allee effect would have an adverse effect on the survival of the population.       PDF of x at t=18000 PDF of x at t=19000 PDF of x at t=20000 (b) PDFs of x(t) at different times. (c) Trajectories of (3) with x 0 = 3, 4, 5.  then r + β − a > 0. This means that in the cade of σ = 0.2, the population described by (3) will be persistent in mean for different jumps (see Figure 7(b)).  such that r − σ 2 2 − a > 0, then the population in (2) will be persistent in mean. For model (3), if noise intensity or jump is large such that r + β + c ha − 2 √ c h < 0, then the population in (3) will become extinct. However, if noise intensity or jump is relatively small such that r + β − a > 0, then the population in (3) will be persistent in mean. According to the above analysis, if the population x(t) in (2) is extinct, the population x(t) in (3) will become extinct. Conversely, if the population x(t) in (3) is persistent in mean, the population x(t) will be persistent in mean. Based on this, the dynamic behavior of models (2) and (3) can be summarized as the following Table 2.
• if the population in model (2) is extinct, the population in model (3) must go to extinction. If the population in model (3) is persistent in mean, the population in model (3) must also be persistent in mean ; • the Allee effect, white noise and jump may have great influence on the survival of the population. As a result, it is more nature and realistic to use the stochastic model especially with Allee effect and jump to describing the population dynamics. From Remark 3.5, we know that Theorem 3.1 generalizes and improves Lemma 2.1 in [10]. Moreover, from Remark 3.6, if there is only one state in the state space M and c = r K for model in [26], then extinction conditions in Theorem 3.1 are more general than those in [26].
Note that the population's environment may be randomly switched between two or more environmental regimes. As done in [26], one can introduce the Markovian switching into the stochastic population model (3). This is a significant problem. We leave it to future consideration.