Persistence and extinction of a modified Leslie–Gower Holling-type II two-predator one-prey model with Lévy jumps

This paper is concerned with a modified Leslie–Gower and Holling-type II two-predator one-prey model with Lévy jumps. First, we use an Ornstein–Uhlenbeck process to describe the environmental stochasticity and prove that there is a unique positive solution to the system. Moreover, sufficient conditions for persistence in the mean and extinction of each species are established. Finally, we give some numerical simulations to support the main results.


Introduction
The relationship between prey and predator is one of the most important and interesting topics in biomathematics. Functional response is a significant component of the predator-prey relationship. The famous predator-prey framework with modified Leslie-Gower and Holling-type II schemes proposed by Aziz-Alaoui and Okiye [4] can be denoted as follows: ), where x(t) and y(t) represent the population sizes of the prey and the predator, respectively. r 1 , r 2 , a, c, f and h are positive constants. r 1 and r 2 are the growth rates of the prey and the predator, respectively, a represents the competitive strength among individuals of the prey, c stands for the per capita reduction rate of prey x, the meaning of f is similar to c, and h describes the protection of the environment. Aziz-Alaoui and Okiye [4] studied the boundedness and global stability of model (1). From then on, many authors have paid attention to model (1) and its generalized forms (see, e.g. [1, 2, 5, 10-14, 26, 27, 30, 31, 33, 35]). The above studies have focused on two-species models. However, it is a common phenomenon that several predators compete for a prey in the natural world. At the same time, the growth of the population is inevitably affected by environmental fluctuations in real situations. Suppose that the growth rate r i is affected by white noise (see, e.g. [8, 15-17, 19, 24, 32, 37]), with r i → r i + σ iẆi (t), Xu et al. [32] proposed a stochastic two-predator one-prey system with modified Leslie-Gower and Holling-type II schemes: dt + σ 2 y 1 (t)dW 2 (t), where W i (t) is a standard Brownian motion defined on a complete probability space ( , F, P) with a filtration {F t } t>0 satisfying the usual conditions and σ 2 i stands for the intensity of the white noise.
However, the growth of species in the real world is often affected by sudden random perturbations, such as epidemics, harvesting, earthquakes, and so on; and these phenomena cannot be described by white noise. Bao and Yuan [6] and Bao et al. [7] suggested that these phenomena can be described by a Lévy jump process. Therefore, we can obtain the following two-predator one-prey model with white noise and Lévy jumps, which introduced Lévy noise into the population model in the same way as [22]: with initial data x(0) > 0, y 1 (0) > 0 and y 2 (0) > 0, where x(t − ), y 1 (t − ) and y 2 (t − ) are the left limit of x(t), y 1 (t) and y 2 (t), respectively. N is a Poisson counting measure with characteristic measure η on a measurable subset Y of (0, Model (2) assumes that the growth rate is linearly dependent on the Gaussian white noise in the random environments Integrating on the interval [0, T], we can see that Hence, the variance of the average per capita growth rater i over an interval of length T tends to ∞ as T → 0. According to this point, we can see that model (2) cannot accurately describe the real situation. Therefore, many authors (see [9,34]) have proposed that using the mean-reverting Ornstein-Uhlenbeck process is a more appropriate way to incorporate the environmental perturbations. On account of this approach, one has i means the intensity of stochastic perturbations and α i > 0 characterizes the speed of reversion. As a result, Zhou et al. [36] considered the following stochastic model: Motivated by these, according to model (2), we can derive the following stochastic twopredator one-prey model with modified Leslie-Gower and Holling-type II schemes with Lévy jumps: To the best of our knowledge, there are few studies related to model (4), so we mainly study the properties of model (4) in this paper. The rest of this paper is organized as follows. In Section 2, we give some lemmas for our main results and obtain sufficient conditions for persistence in the mean and extinction for each species. In Section 3, we introduce some simulation figures to illustrate our main theoretical results. Some concluding remarks are given in Section 4.

Main results
For convenience and simplicity, we define some notations as follows: First, we give the following assumption and definition.

Assumption 2.1:
There exists a constant m > 0 such that which means that the jump noise is not too strong.
is said to be persistent in the mean if lim inf t→+∞ t −1 t 0 x(s)ds > 0 a.s. Before we state and prove our main results, we recall some lemmas which will be used later.

Proof:
Here we only prove the case b 2 > β 2 , the proof of b 3 > β 3 is similar.
For sufficiently small ε > 0, there is sufficiently large T such that, for t ≥ T, Then when t ≥ T, by (12), where Moreover, for t ≥ T 1 , substituting the above inequalities into (14) leads to where For this reason, According to Assumption 2.1, we get In view of Lemma 2.2, then We then deduce from lim t→+∞ t −1 t Substituting the above identities into (17) That is to say For arbitrary given ε > 0, there exists T > 0 such that, for t ≥ T, We then deduce from (18) that, for t ≥ T, where ε is sufficiently small such that 0 < ε < 1 2 (b 2 − β 2 ). According to Lemma 2.1, we can obtain We then deduce from the arbitrariness of ε that which indicates that lim t→+∞ t −1 ln 1 (t) = 0 a.s. In accordance with (11), The proof of Lemma 2.4 is completed. Now we are in the position to give our main result. (4), the following conclusions hold:
(v). Sinceb 2 < β 2 ,b 3 < β 3 , (i) implies lim t→+∞ y i (t) = 0 a.s., i = 1, 2. Besides,b 1 > β 1 , for sufficiently large t, by (22), we obtain Making use of Lemma 2.1 to (28) and (29) results in On the basis of Lemma 2.4, for arbitrary ε > 0, there is a constant T > 0 such that For this reason, for t ≥ T. Substituting the above inequalities into (30) yields for all t ≥ T almost surely. Let ε be sufficiently small such that 0 < ε < , thus lim t→+∞ x(t) = 0 a.s. Then similar to the proof of (ii), we can prove We then deduce from Lemmas 2.2, 2.4 and lim t→+∞ t −1 t 0 σ 3 (s)dW 3 (s) = 0 that As a result, for any ε > 0, we can find out T > 0 such that, for t ≥ T, Substituting (34) into (22), one can derive that for sufficiently large t, where ε > 0 obeys 1 2 Then the arbitrariness of ε means The proof of (vii) is analogous to that of (vi) and hence is left out.

Discussions and numerical simulations
Now we test the functions of the mean-reverting Ornstein-Uhlenbeck process on the persistence and extinction of Model (4). We note that the speed of reversion α i and the intensity of the perturbation ξ 2 i are two key parameters in the Ornstein-Uhlenbeck process. Theorem 2.1 shows that the persistence and extinction of system (4) are entirely dominated by the signs ofb Hence, as α i (respectively, ξ 2 i ) increases, species i tends to be persistent (respectively, extinct), i = 1, 2, 3. Moreover, due to the fact that > 0), so sufficiently large α 2 (respectively, ξ 2 2 ) could make x extinct (respectively, persistent) ifb 1 > β 1 andb 2 > β 2 . Similarly, sufficiently large α 3 (respectively, ξ 2 3 ) could make x extinct (respectively, persistent) ifb 1 > β 1 and b 3 > β 3 . Now we use the Euler scheme offered in [29] to prove our theoretical results numerically (here we only provide the functions of α i since the functions of ξ 2 i can be proffered analogously). Consider the following model:   Figure 5 confirms these. Comparing Figure 1 with Figure 2, we can see that with the rise of α 3 , y 2 tends to be persistent. Similarly, comparing Figure 1 with Figure 3 (respectively, Figure 1 with Figure 5), we can see that with the rise of α 2 (respectively, α 1 ), y 1 (respectively, x) tends to be persistent.     Figure 7 confirms these. Comparing Figure 6 with Figure 7, we can see that with the rise of α 3 , the prey population tends to become extinct.      Comparing Figure 8 with Figure 9, we can see that with the rise of α 2 , the prey population tends to become extinct.  In the following, we discuss the effect of Lévy jumps on model (4).  • In Figure 12, we choose λ 2 (u) = 1.358 (i.e. β 2 = 0.5) and assume that all other parameters are the same as those in Figure 4. It follows from (ii) in Theorem 2.1 that both x and y 1 become extinct and lim t→+∞ t −1 t 0 y 2 (s)ds = h 2 (b 3 −β 3 ) f 2 = 0.465. Comparing Figure 12 with Figure 4, we can see that with the rise of λ 2 (u), y 1 becomes extinct.

extinct, and lim
• In Figure 13   Comparing Figure 13 with Figure 4, we can see that with the rise of λ 3 (u), y 2 becomes extinct.
• In Figure 14, we choose λ 1 (u) = 1.527 (i.e. β 1 = 0.6) and assume that all other parameters are the same as those in Figure 5. It follows from (i) in Theorem 2.1 that all the species become extinct. Comparing Figure 14 with Figure 5, we can see that with the rise of λ 1 (u), x becomes extinct.
By analysing Figures 12-14, we can see that Lévy noise can change the properties of the population systems, and it can force the population to become extinct when λ i (u) is sufficiently large.

Concluding remarks
In this paper, we take advantage of a mean-reverting Ornstein-Uhlenbeck process to describe the random perturbations in the environment and formulate a stochastic threespecies predator-prey system with Lévy jumps, which might be more appropriate to depict reality than model (2). We obtain sharp sufficient conditions for persistence in the mean and extinction for each species of model (4) and uncovered some significant functions of Ornstein-Uhlenbeck process: sufficiently large α i (the speed of reversion) could make species i persistent, i = 1, 2, 3; moreover, in some situations, sufficiently large α 2 and α 3 could make x become extinct.
Some interesting questions deserve further investigation. The present article probed into the white noises and Lévy noise, one could examine other random noises such as the telephone noise (see [21]), etc. Besides, one could consider and investigate model (4) in higher dimensions. All these considerations are left for future study.