Optimal harvesting for a stochastic competition system with stage structure and distributed delay

A stochastic competition system with harvesting and distributed delay is investigated, which is described by stochastic differential equations with distributed delay. The existence and uniqueness of a global positive solution are proved via Lyapunov functions, and an ergodic method is used to obtain that the system is asymptotically stable in distribution. By using the comparison theorem of stochastic differential equations and limit superior theory, sufficient conditions for persistence in mean and extinction of the stochastic competition system are established. We thereby obtain the optimal harvest strategy and maximum net economic revenue by the optimal harvesting theory of differential equations.


Introduction
In nature, relationships between species can be classified as either competition, predatorprey, or mutualism. Because of limited natural resources, competition among populations is widespread. Many scholars have researched competition models. Early studies mainly considered deterministic models [5,16]. Individual organisms experience a growth process, from infancy to adulthood, immaturity to maturity, and adulthood to old age, with viability varying by age. Young individuals have a weaker ability to cope with environmental disturbances, predators, and competitors' survival pressure, while the survival ability of adult individuals is strong, and they are able to conceive the next generation. The stage-structured model is popular among scholars, and the study of the stage-structured deterministic model, as a single-species model [7] or two-species competitive model [14], is comprehensive. Predatorprey models with stage structures have been discussed in the literature [4,17,18]. X. Y. Huang et al. presented the sufficient conditions of extinction for a two-species competitive stagestructured system with harvesting [6].
The effects of population competition are not immediate, hence, it is necessary to consider time delays in the governing equations [9,15,20]. We propose a competitive model with distributed delay and harvesting, where x i is the density of the ith species, i = 1, 2, 3, where x 1 , x 2 , respectively represent the juveniles and adults of one of two species. a 11 is the birth rate of juveniles and a 21 is the transformation rate from juveniles to adults. a 12 , a 22 denote inter-specific competitive coefficients of x 1 and x 2 . Considering x 1 is young and not competitive, we assume that only x 2 and x 3 are competitive. d 1 and d 2 are the loss rates of populations x 2 and x 3 in competition. r and k 3 are respectively the intrinsic growth rate and environmental capacity of species x 3 . The sum of the death and conversion rates of juveniles x 1 and the sum of the death rates of adults x 2 are expressed by s and β, respectively. q is the catchability coefficient of species x 3 . E denotes the effort used to harvest the population x 3 . All of the parameters are assumed to be positive constants. The kernel For the distributed delay, MacDonald [10] initially proposed that it is reasonable to use a Gamma distribution, as a kernel, where α i > 0, i = 1, 2 denote the rate of decay of effects of past memories, and n is called the order of the delay kernel f i (t). They are nonnegative integers. This article mainly considers the weak kernel case, i.e., f i = α i e −α i t for n = 0. The strong kernel case can be considered similarly. Let Then, by the linear chain technique [13], the system (1.1) is transformed to the following equivalent system: In addition, the population must be disturbed by realistic environmental noise, which is important in the study of bio-mathematical models [12,15,19], such as rainfall, wind, and drought. White noise is introduced to indicate the effects on the system disturbance. It is assumed that environmental disturbances will manifest themselves mainly as disturbances in population density x i (i = 1, 2, 3) of a system (1.2). Further, the following system of stochastic differential equations is obtained: where B i (t), i = 1, 2, are independent standard Brownian motions and σ 2 i , i = 1, 2, represent the intensity of the white noise. Because x 1 and x 2 live together, they are affected by the same noise.
The following assumption applies throughout this paper.
3) on t 0. Furthermore, the solution will remain in R 5 + with probability 1.
Proof. System (1.3) is locally Lipschitz continuous, so for any initial value s., where τ e is the explosion time [1].
We must show that τ e = ∞ a.s. Let m 0 > 0 be sufficiently large that the initial value x i (0) is in the interval 1 m 0 , m 0 . For each m > m 0 , define a stopping time, Obviously, τ m increases as m → ∞. Let τ ∞ = lim m→∞ τ m . Hence τ ∞ ≤ τ e a.s., which is enough to certify τ ∞ = ∞ a.s. In contrast, there is a pair of constants T > 0 and ε ∈ (0, 1), such that Hence an integer m 1 > m 0 exists, and for arbitrary m > m 1 , A Lyapunov function V : R 5 + → R + is defined as where a, b are positive constants to be determined later. The nonnegativity of this function can be seen because Let T > 0 be a random positive constant. For any 0 ≤ t ≤ τ m ∧ T, using Itô's formula, one obtains The following proof is similar to that of Bao and Yuan [2]. Apply inequality (2.3) to equation (2.1), and integrate from 0 to τ m ∧ T to obtain Taking the expectations, the above inequality becomes When m → ∞, it is easy to see that P(τ ∞ ≤ T) = 0. Owing to the arbitrariness of T, P(τ ∞ = ∞) = 1. The proof is completed.

Stability in distribution
3) with any given initial value. Then there exists a constant K 2 > 0, such that lim sup t→+∞ E|x(t)| ≤ K 2 .
Proof. The proof is similar to Theorem 3.1 in paper [2], and hence is omitted here.
Then one can further prove the following theorem.
Proof. Let x φ (t) and x ϕ (t) be two solutions of system (1.3), with initial values φ(θ) ∈ R 5 + and ϕ(t) ∈ R 5 + , respectively. Applying Itô's formula to Therefore, Because V(t) ≥ 0, according to the inequality above, That is, Moreover, it can be seen from the first Hence E x 1 (t) is uniformly continuous. Using the same method on the other equations of , and E u 2 (t) are uniformly continuous. According to [3], , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F t contains all P-null sets). Suppose p(t, φ, dy) is the transition probability density of the process x(t), and p(t, φ, A) is the probability of event x φ (t) ∈ A with initial value φ(θ) ∈ R 5 + . By Lemma 3.1 and Chebyshev's inequality, the family of transition probability p(t, φ, A) is tight. So, a compact subset K ∈ R 5 + can be obtained such that p(t, φ, K) ≥ 1 − * for any * > 0. Let P (R 5 + ) be probability measures on R 5 + . For any two measures P 1 , P 2 ∈ P, we define the metric For any g ∈ L and t, ι > 0, one obtains where U K = {x ∈ R 5 + : |x| ≤ K}, and U c K is a complementary set of U K . Since the family of p(t, φ, dy) is tight, for any given ι ≥ 0, there exists sufficiently large K such that p(ι, φ, U c K ) < * 4 . From (3.1), there exists T > 0 such that for t ≥ T, Consequently, it is easy to find that |Eg That is, There is a unique µ(·) ∈ P (R 5 + ) such that lim t→∞ d L p(t, 0, ·), µ(·) = 0. In addition, it follows from (3.1) that The proof is completed.

Optimal harvesting
For convenience, we introduce the following notation: . For x(t) ∈ R + , the following holds: (i) If there are positive constants T and δ 0 such that for any t ≥ T, where α, δ 1 , δ 2 are constants, then (ii) If there are positive constants T, δ, and δ 0 such that for any t ≥ T, then x * ≥ δ δ 0 a.s.
holds for all but finitely many n.
which, together with the positivity of Indeed, integration of the system (1.3) from 0 to t yields Thus Next, to obtain the optimal harvest strategy of system (1.3), we establish the following auxiliary systems: .
(iii) if a 11 k 2 < b 1 and b 3 > qE, then both x 1 and x 2 go to extinction a.s., and x 3 is persistent in mean a.s.
Proof. By the stochastic comparison theorem, we obtain So, it follows from (4.14), (4.19), and (4.21) that Applying Itô's formula to system (1.3) yields Integrate both sides of the above three equations from 0 to t, and divide by t to obtain Now, let us prove conclusion (i). We use Lemma 4.2 to obtain Then, for arbitrary 3 > 0, there exists T 2 > 0 such that Using the specific property of the limit superior in (4.25) gives By the assumption a 11 k 2 < b 1 , we can let 3 be sufficiently small that a 11 k 2 < b 1 − 3 , and by Lemma 4.1, lim t→∞ x 1 (t) = 0 and lim t→∞ x 1 (t) = 0. From Lemma (4.4), for the above 3 , there exists T 3 > 0 such that Letting dm(E) dE = 0, the optimal harvested efforts are and the maximum expectation of net economic revenue is

Numerical analysis
We use some hypothetical parameter values to verify Theorems 4.6 and 4.7. We choose k 1 = 50, k 2 = 50, k 3 = 100, and initial values x 1 (0) = 5, x 2 (0) = 5, x 3 (0) = 8. Assign different values to other parameters in Table 5.1, which satisfies Theorem 4.6 , to prove theoretical results.   Regarding the optimal harvesting effort, we still select the same parameters with the Fig. 5.4. By Theorem 4.7, we obtain E * = 1.788. Therefore, the optimal harvesting policy exists, we show it in Fig. 5.5. The maximum expectation of net economic revenue exists when E * = 1.788.

Conclusion
We investigated the dynamics of a stochastic stage-structured competitive system with distributed delay and harvesting. We took a weak kernel case as an example for convenience, and we similarly discuss the strong kernel case. Our objective was to study the optimal harvest strategy and the maximum net economic revenue. Some main results are as follows: (i) The existence and uniqueness of the positive solution of system (1.3) was proved, using a Lyapunov function to ensure the rationality of the system and provide support for later results.
(iii) The research of the optimal harvest and maximum expectation of net economic revenue of stochastic models has clear practical significance. Species extinction must be strictly prevented during fishing. First, sufficient conditions for persistence in mean and extinction were established. The optimal harvested efforts were E = 1 2pq pΓ 1 a 12 a 22 − r k 3 c , and the maximum expectation of net economic revenue was m(E ) = k 3 4pqr We only considered the effect of white noise and delay on the dynamics of the stagestructured competitive system. It is also interesting to consider the effect of telephone noise, toxins, and Markovian switching, and these will be topics of our further research.