Optimal harvesting strategies of a stochastic competitive model with S-type distributed time delays and Lévy jumps

The aim of this paper is to investigate the optimal harvesting strategies of a stochastic competitive Lotka–Volterra model with S-type distributed time delays and Lévy jumps by using ergodic method. Firstly, the sufficient conditions for extinction and stable in the time average of each species are established under some suitable assumptions. Secondly, under a technical assumption, the stability in distribution of this model is proved. Then the sufficient and necessary criteria for the existence of optimal harvesting policy are established under the condition that all species are persistent. Moreover, the explicit expression of the optimal harvesting effort and the maximum of sustainable yield are given.

where x 1 and x 2 stand for the population size of two species, respectively, r i > 0 is the growth rate of x i , i = 1, 2. h i > 0 represents the harvesting effort of x i , i = 1, 2. c ii > 0 is for the intraspecific competition coefficients of x i , i = 1, 2; c ij (i = j; i, j = 1, 2) denotes the interspecific competition rate.
In fact, the dynamics and optimal harvesting of population are inevitably affected by some environmental perturbations in virtually all ecosystems, which mainly include two types: white noise and jumping noise. White noise describes the continuous noise, such as light, drought, cold wave and so on (see, e.g., [6][7][8][9]). Jumping noise describes sudden environment shocks, such as earthquakes, floods, and epidemics (see, e.g., [10,11]). In recent years, several authors have studied some systems both with white noise and jumping noise and published a number of successful articles (see, e.g., [12][13][14][15][16][17]).
On the other hand, Gopalsamy [18] have pointed out that "the current growth of a population should also be influenced by the past history of the species". So it is necessary to take time delay into consideration. To the best of our knowledge to date, "systems with discrete time delays and those with continuously distributed time delays do not contain each other. However, systems with S-type distributed time delays contain both. "(see Wang, Wang and Wei [19,20]). And stochastic systems with distributed delays were considered in several publications (see e.g. [21][22][23]). Qiu and Deng [24] have discussed the optimal harvesting problem of a stochastic delay competitive model with Lévy jumps, which is about discrete time delays, and obtained the result that discrete time delays have no impact on the optimal harvesting policy in some cases. Therefore, an interesting and significant problem arises: how does the S-type distributed time delays affect the population dynamics and the optimal harvesting policy? It is more natural and practical for us to consider. In this paper, we consider the following model: with initial data 12 (θ ) and 0 -τ 21 x 1 (t + θ ) dF 21 (θ ) are Lebesgue-Stieltjes integrals. F 12 (θ ) and F 21 (θ ) are nondecreasing bounded variation functions defined on [-τ , 0]. N(dt, dv) = N(dt, dv)μ(dv)dt, N is a Poisson counting measure, μ is the characteristic measure of N on a measurable subset Z of (0, +∞) with μ(Z) < +∞. γ i is the effect of Lévy noises on species i, if γ i (v) > 0, the jumps represent the increasing of the species; if γ i (v) < 0, the jumps represent the decreasing of the species; Therefore, it is reasonable to assume that 1 + γ i (v) > 0, v ∈ Z, i = 1, 2.
We wish to solve the problem above and get the optimal harvesting effort (OHE) H * = (h * 1 , h * 2 ) such that the expectation of sustainable yield (ESY) ) is maximum (all species are persistent). Firstly, we establish the sufficient criteria for the extinction and persistence of each species in Sect. 2. Then in Sect. 3, we prove the stability in distribution of this model. Finally, we establish the sufficient and necessary conditions for the existence of the optimal harvesting policy and obtain the explicit expression of OHE and the maximum of ESY (MESY) in Sect. 4.

Extinction and persistence
At first, we define some notations for the sake of convenience, Before we state our results, we make some assumptions.
which means that the jump noise is not too strong.
This implies that Similarly, one can also derive that First, we prove (I). From the first equality in (4), we can get In the same way, we can derive that if b 2 < 0, then lim t→+∞ x 2 (t) = 0, a.s. by (5). Second, we prove (II). Since b 2 < 0, from (I), we can note that lim t→+∞ x 2 (t) = 0, a.s. Thus, for arbitrary ε > 0, there is a random time T 1 > 0 such that, for t ≥ T 1 , The above inequality can be applied to the first equality in (4), we can get Because of the arbitrariness of ε, we can choose ε sufficiently small such that b 1ε > 0 (b 1 > 0). Applying Lemma 2.2 to (6) and (7), respectively, we can obtain Letting ε → 0, we get lim t→+∞ x 1 (t) = b 1 /c 11 , a.s. Third, we prove (III). The proof of (III) is similar to that of (II) by symmetry and hence is omitted.
(ii): The proof of (ii) is similar to that of (i) by symmetry and hence is omitted. We let ε → 0 so that , a.s.
In the same way, using (12), (A) in Lemma 2.2 and the arbitrariness of ε we see that , a.s.
The proof is complete.

Stability in distribution
In this section, we study the stability in distribution of model (1). Firstly, we state an assumption and a lemma.  Remark 3.1 We omit the proof for Lemma 3.1. For details, please refer to Bao and Yuan [28]. (1) is said to be stable in distribution, i.e., there is a unique probability measure ϕ(·) such that, for every initial data x(θ ) ∈ C([-τ , 0], R 2 + ), the transition probability p(t, x(θ ), ·) of x(t) converges weakly to ϕ(·) as t → +∞.

Optimal harvesting
In this section, we will state and prove our main results. For the sake of making the proof work, we introduce the following technical assumption.     On the other hand, thanks to H * being OHE, then H * is the solution of (24). Note that the solution of (24) is unique and H * is also the solution of (24). Hence, H * = H * , i.e., b 1 | h 1 =h * 1 ,h 2 =h * Remark 4.1 From Theorem 4.1 we can note that the existence of an optimal harvesting policy has a close relationship with S-type distributed time delays, white noises and Lévy jumps.

Conclusions
In this paper, we considered the optimal harvesting of a stochastic competitive Lotka-Volterra model with S-type distributed time delays and Lévy jumps. We established the sufficient and necessary conditions for the existence of optimal harvesting policy, and we also obtained the explicit expression of the optimal harvesting effort and maximum yield by using the ergodic method. Theorem 4.1 indicates that the existence of an optimal harvesting policy has a close relationship with S-type distributed time delays, white noises and Lévy jumps. Some interesting problems can be further investigated, such as the optimal harvesting problem for N -dimensional stochastic competitive Lotka-Volterra model with S-type distributed time delays and Lévy jumps. We can also study the optimal harvesting problem for some stochastic model with infinite time delays and Lévy jumps. It is necessary for us to work hard on these investigations.