Abstract

We consider a stochastic one-predator-two-prey harvesting model with time delays and Lévy jumps in this paper. Using the comparison theorem of stochastic differential equations and asymptotic approaches, sufficient conditions for persistence in mean and extinction of three species are derived. By analyzing the asymptotic invariant distribution, we study the variation of the persistent level of a population. Then we obtain the conditions of global attractivity and stability in distribution. Furthermore, making use of Hessian matrix method and optimal harvesting theory of differential equations, the explicit forms of optimal harvesting effort and maximum expectation of sustainable yield are obtained. Some numerical simulations are given to illustrate the theoretical results.

1. Introduction

Many researchers are widely focused on the complex dynamics of biological systems such as delay population systems [13], stochastic population systems [412], and impulsive population systems [1315]. Recently, many scholars have investigated two-species models and studied the extinction and persistence [1618]. However, in the real world, it is a common phenomenon that one predator catches two or more kinds of preys [19]. Consequently, models with three or more species which can explain the dynamical behaviors of the population accurately are investigated [20, 21]. On the other hand, it is necessary and important to consider time delay caused by the competition and predation of species. Delayed differential equations can exhibit much more complex dynamics than differential equations without delay, and stable equilibrium can become unstable with the effects of a time delay. Therefore, many researchers have studied the Lotka-Volterra time delay models with two competitive preys and one predator [22, 23]. Notice that the composite population systems with stochastic effects and time delays present some complex dynamics; thus this causes widespread researchers concern [2430].

In order to rationalize the model, some coefficients should be modified into the existing models. The one-predator-two-prey model with time delays is described by with initial valuewhere denotes the size of the -th species of the prey at time (=1, 2), denotes the size of the predator at time , is the time delay, , , . (=1, 2, 3), and (=1, 2) stands for the growth rates of two preys, and stands for the death rate of the predator, respectively. is the intraspecific competition rate of species , . and are the efficiency of food conversion. and are the interspecific competition rates between 1 and 2, and and are the capture rates.

Many systems may suffer environment perturbation, the growth rate is affected by the white noise [31, 32], is a real-valued Brownian motion defined on a complete probability space with . () denotes the coefficients of the environmental stochastic effects on the preys and the predator populations, respectively.

Generally, the dynamical behavior of the species may suffer from the sudden environmental change significantly. Nevertheless, white noise cannot explain huge, occasionally catastrophic disturbances. Therefore, applying the discontinuous stochastic process as Lévy jump to model the abrupt nature phenomenon in ecosystem is necessary [3335]. In many cases, populations suffer sudden distribution, and this causes changes in the productivity of marine and freshwater species. How to use the discontinuous stochastic process to study these abrupt nature phenomena has been an interesting topic.

Recently, some scholars have applied Lévy jump into their models and showed that Lévy jump could describe sudden random environmental perturbations. According to the Lévy decomposition theorem [36], we get that is a compensated Poisson process and is a Poisson counting measure with characteristic measure on a measurable subset of with . The distribution of Lévy jump can be completely parameterized by and satisfies the property of infinite divisibility. It is characterized by its characteristic function ; we can get a detailed explanation from the following Lévy-Khintchine formula [37]. There are many other papers about stochastic models with Lévy jump; the readers could refer to [38, 39] and references therein. Considering the inevitable situations in the real world, we assume that the intrinsic growth rates and and the death rate of the model are perturbed by the Lévy jump to signify the sudden climate change, so we introduce the Lévy jump into the underlying stochastic model (1). Taking the economic factors into account, reasonable natural resources management can increase sustainable production and profits. Therefore, harvesting models have been already used to exploited the optimal harvesting policies of renewable resources [4043]. We only consider to harvest preys and and and are the harvesting effort rates of and , respectively [42, 43]. From system (1), we can obtain the following stochastic harvesting model with Lévy jump:

This paper is organized as follows. In Section 1, we formulate the model. We show the solution of model (5) is global and positive. We give the main theorems for persistence in mean and extinction under the model (5) and its proof in Section 3. In Section 4, we prove the global attractivity and stability in distribution. Our main aim of this paper is to investigate the optimal strategy of the proposed model; we give the conclusions in Section 5. Finally, we carry out numerical simulations and some figures to support the main conclusions in Section 6.

2. Global Positive Solution

In order to explore the dynamical behaviors of ecological population, we first study the positivity of the solutions of system (5).

Lemma 1. There exists an positive constant ; we have

Lemma 2 (see [44]). We assume that, for the each , there exists an satisfying
, and , with .

Lemma 3. For any given initial value , model (5) has a unique global positive solution on Moreover, there is a positive constant K such that

Proof. For any given initial value , there is a unique positive for , where is the explosion time. We verify that the positive solution is global, that is as. Let be sufficiently large so , , and all lie within the interval . For each integer , we define the stopping time: Evidently, is strictly increasing when . Let ; thus as. If this statement is not true, then there exist pairs of constants , , and such that Define a -function by letting be positive constants. Using Itô’s formula, we get where where is a positive constant, Thus, Taking expectation, yieldsSet for and we can obtain . For each , there are , , equaling either or and yields where stands for the indicator function of .
Let , which implies is a contradiction. So, we have that . This completes the proof of Lemma 1.

3. Persistence in Mean and Extinction of the Model

For the sake of convenience, we introduce some following notations. Let and stand for all continuous functions from to . Additionally, we give some notations with initial value . It is not difficult to know that the coefficients of model (5) satisfy the local Lipschitz condition; therefore, model (5) has a unique local solution on , where is the explosion time. Applying Itô’s formula, we can get the following solution: It is the unique positive local solution to model (5). Now let us prove . Thus, we introduce the following auxiliary model: with initial value Obviously Before starting proving, we state several hypotheses. We assume that which implies that the persistent ability of species 1 is better than that of species 2.

Assumption 4. , express that all the population could coexist under the condition that the model frees from stochastic noises. , , which imply that the two prey populations could coexist when there is no environmental noises and the predators are absent, where is the complement minor of in the determinant .

Assumption 5. , , , .

Assumption 6. , , .

Lemma 7 (see [26, 33]). Let .
(i) If there exist three constants , , and , such that for all , and as.
Then, for all , where and are constants, then (ii) If there exist three constants , , and , such that for all , and as.
Then, for all , then , a.s.

Lemma 8. For arbitrarily ,

Proof. Therefore, if , then If , then

Lemma 9. For model (20), we have the following.
() If , then () If , and , then () If , and , then () If , and , then () If , and , then () If , and , then () If , and , then

Proof. First, let us prove . Applying Itô’s formula to model (20), we can get that Dividing both sides of (38), (39), and (40) by , we can obtain that Note that Firstly, we prove (a). Since , , using Lemma 7, yields and Substituting (45) and (46) into (40) gives where is small enough satisfying . Applying (i) in Lemma 7 gives Secondly, we prove (b) and (c). Using Lemma 7, since and , it is easy to obtain that and As a consequence, we can study and discuss the following model: We know that It is not difficult to obtain the system The proofs of (d) and (e) are similar to these of (b) and (c).
Then, we give the proofs of (f) and (g). Using Lemma 7, since and , it is easy to obtain that and Using (54) and (55), then combining (44) leads to andMultiplying (41), (42), and (43) by , , and , respectively, and adding them, we can derive that An application of (56), (57), (58), Lemma 7, and Lemma 8 gives that This completes the proof.

Lemma 10. The solution of model (5) obeys

Proof. From Lemma 9, we can obtain that either or , is a constant, . Since , we only need to prove the following results.
If , thenIf constant, then and hence

Theorem 11. For system (5), defineFrom Assumptions 4, 5, and 6, we can obtain the following results.(I)If , then ; moreover,(i)if , then species i, , go to extinction a.s.; i.e., (ii)If , then the predator goes to extinction a.s., and one prey is persistent in mean, and another goes to extinction; i.e., (iii)If , then then species 3 goes to extinction and species 1,2 are persistent in mean; i.e., (iv)If , then three species are persistent in mean; i.e., (II)If , then ; moreover,(v)if , then species i, () go to extinction, a.s;(vi)if , then the predator goes to extinction and one prey is persistent in mean, and another goes to extinction, a.s.;(vii)if , then the species 2 goes to extinction and species 1,3 are persistent in mean, a.s;(viii)if , then three species are persistent in mean, a.s.

Proof. Now we will give the proof of (I), and the proof of (II) is parallel to (I). We can get and, as a consequence, we can also compute that So is established.
Applying Itô’s formula to model (5) yieldsFirstly, we prove (i). Since , and are positive, we can get Note that , . By Lemma 9, we get In other words, all the populations go to extinction a.s.
Now we give the proof of (ii). Multiplying both sides of (75) and (76) by and , respectively, and then adding these two equations, we can get that From Lemma 3.4 of [26], we obtain that either or c, c is a constant; it follows thatIn view of (22) and Lemma 8, we can obtain that Putting (60) and (81) into (79) leads to where is small enough such that . By Lemma 7, we can get Taking advantage of the above identity, we will get the following two-species models: It is easy to get the result with the similar proof of that in [4042]Thirdly, we prove (iii). Denote , as the solution of the following equations: Consequently, According to Lemma 7, for arbitrarily given , there exists a , for all , Multiplying (75), (76), and (77) by , , and , respectively, and adding them, one can observe that for sufficiently large such that , Substituting (60) and (81) into (89) yields For sufficiently large , by , and choosing to be sufficiently small, then we have Similarly, denote as the solution of the following equations: Then we have In the same way, we can choose a , for arbitrarily given , such that It follows that, for any sufficiently small , there exists and such that multiplying (75), (76), and (77) by , (−1) and , respectively, and adding them, we can obtain that Applying (60) and (81) into (95) yields that Note that . According to the arbitrariness of and Lemma 7, we have Substituting (81), (91), and (97) into (77), For sufficiently large , then according to , Lemma 7, and the arbitrariness of , it is not difficult to show Consequently, model (5) reduces to the following model: which has already been investigated in [5]. Then similarly to the proof of Theorem 5.1 in [5], the following identities can be derived: Fourthly, we prove (iv). Since , for arbitrariness of , the application of Lemma 7 to (98) yields Substituting (103) into (75), we can obtain that, for sufficiently large , According to the arbitrariness of and Lemma 7, we have Analogously, we can prove that is established.
Substituting (103) and (105) into (77) yields that when is large enough, According to the arbitrariness of and Lemma 7, we have Subsequently, we have This completes the proof.

4. Stability in Distribution

For the convenience, we define the following notations:

Definition 12. Model (5) is globally attractive if , a.s., , and are two arbitrary solutions with initial conditions and , respectively.

Lemma 13. For any , there exists a constant which makes the solution of model (5) with any given initial value satisfy the property that

Proof. The proof is rather standard and hence is omitted.

From Lemma 13, there is a such that, for , . Note that is continuous; thus there is a constant such that , when . Denote ; then we have

Lemma 14. If , , , then model (5) will be asymptotically stable in distribution; i.e., when , there exists a unique probability measure such that the transition probability density of converges weakly to with any given initial value [45].

Proof. DenoteDefine as the factor of -th diagonal element of . Then applying Kirchhoff’s Matrix Tree Theorem [46, 47], we can see that .
Define Calculating the right differential yields From Theorem 2.3 in [48], Then we can obtain Namely, Subsequently, Note that ; then is integrable on .
In other words, . Moreover, from model (5), we have Thus, where . Therefore, are uniformly continuous. In other words, are continuously differentiable functions with respect to . By Lemma 13 and the conclusion of [45], one can observe that Suppose that is the transition probability density of the process and denotes the probability of event with the initial value . By Lemma 7 and Chebyshev’s inequality [49], we can see that the family of is tight. So we can get a compact subset such that for given .
Let be the probability measures on . For arbitrary two measures , we define the following Kantorovich metric: whereFor any and , we get From (121), there exists a such that, for , we have Obviously By the arbitrariness of , we have Thus, Therefore, is Cauchy in with metric .
There exists a unique such that . From (132), we can obtain Consequently,This completes the proof of Lemma 14.

5. Optimal Harvesting

We give the following extra notions to get the optimal harvesting policy: where , , and is the unit matrix.

Theorem 15. Suppose that , , and is positive definite. (i)If and when , we have ; then the optimal harvesting effort is and the maximum of ESY iswhere .(ii)When , there is or ; then the optimal harvesting policy does not exist.

Proof. Let . When (iv) of Theorem 11 holds, we find that, for every , and if the optimal harvesting effort exists, then it must belong to .
Firstly we prove (i). It is easy to see that , so is not empty. By (iv) of Theorem 11, for any , we have Applying to Lemma 14, model (18) has a unique invariant measure . According to Corollary 3.4.3 in [46], we can get that is strong mixing. At the same time, it is ergodic by Theorem 3.2.6 in [46]. Hence, it then follows from in [46] that Let represent the stationary probability density of model (18); we obtain Since the invariant measure of model (18) is unique, then, according to the one-to-one correspondence between and its corresponding invariant measure , we obtain That is to say, Assume that is the unique stagnation point of the following equation: It holds that . We can take use of the following Hessian matrix [42, 43]: is negatively defined, so is the unique extreme point of . In other words, if , i.e., and , then the optimal harvesting effort is and is the maximum value of ESY. We are now in the position to prove (ii). Suppose that the optimal harvesting effort exists. So ; i.e., . That is to say, if is the optimal harvesting effort, then must be the unique solution of (138). However, is also the solution of (138). Hence, , and . It contradicts with the condition.
This completes the proof of Theorem 15.

6. Numerical Simulations

In this section, we carry out extensive numerical simulations using MATLAB by choosing the following parameters to check the model. And the phase diagrams are given too. Some figures are as follows: . From Assumptions 4, 5, and 6, we can obtain the following results.

(A) Persistent and Extinction. If , then .(i)If , then species i, , go to extinction.(ii)If , then the predator goes to extinction and one prey is persistent in mean, and another goes to extinction.(iii)If , then the species 3 goes to extinction and species 1,2 are persistent in mean.(iv)If , then three species are persistent in mean.

In Figure 1, , ; it shows that all the populations are extinct.

In Figure 2, , . We can find that only the prey is persistent in mean and another prey and predator are extinct.

In Figure 3, , . We can find that only the two preys are persistent in mean and the predator is extinct.

In Figure 4, , , and the noise is small.

(B) Optimal Harvesting. Regarding the optimal harvesting effort, when , it is not difficult to estimate that is positive definite. Note that ; we can observe . Then we can find . Therefore, by Theorem 15, we can observe . Thus the optimal harvesting policy exists (see Figure 3).

In Figure 5, in order to check the conclusion, we choose another two types of harvesting data. It is obvious that the optimal harvesting policy leads to the maximum of expectation of sustainable yield.

7. Conclusion

This paper is about a stochastic three-species population model with time delays and Lévy jumps [49]. We also consider the optimal harvesting of preys [50]. To begin with, we establish the modified model. In Theorem 11, we obtain the sufficient criteria for extinction and persistence in mean of each species. Lévy jump is important to study. When , all species go to extinction. When , then the predator goes to extinction and is persistent in mean, goes to extinction. When , then goes to extinction and , are persistent in mean. When , then three species are persistent in mean. The main purpose of this paper is to study the optimal harvesting. After discussing the stability of distribution, we study the optimal harvesting and obtain the maximum yield of two preys.

From the numerical simulations, we list the following biological meanings.

We find the noise can cause the variation of species. When the noise is large, it in reality can suppress the increase of population, then it dies out.

Time delay and Lévy jump have important effects on the persistence in mean and the harvesting yield.

In traditional papers, scholars consider two species or three species without optimal harvesting. We consider a three-species model with Lévy jump and optimal harvesting. Time delay is ineluctable in the ecological environment and is necessary to consider delays. Recently, stochastic models with the telephone noise have been studied by many authors [51, 52]. In the future research, we hope to add more realistic conditions and study more interesting topics, for example, pulse process, Markov Chain, telephone noise, and partial differential system [5355].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the Research Fund for the Taishan Scholar Project of Shandong Province of China and the SDUST Research Fund (2014TDJH102).