Optimal Harvesting Policies for a Stochastic Food-Chain System with Markovian Switching

Optimal harvesting problem is an important and interesting topic from both biological and mathematical point of view. Since Clark’s works [1, 2], one of the most important areaoptimal harvesting problems have received a lot of attention and been studied widely. Among these studies, a large number of literatures were focused on deterministicmodels [3–9], with some on the stochastic versions [10–17], but only a few on the food-chain systems. Furthermore, it is well known that the theory of food chains illustrated the balance of nature and that no animal or plant can exist independently. Motivated by these arguments presented above, we are interested in the optimal harvesting problems on the following stochastic food-chain system:


Introduction
Optimal harvesting problem is an important and interesting topic from both biological and mathematical point of view.Since Clark's works [1,2], one of the most important areaoptimal harvesting problems have received a lot of attention and been studied widely.Among these studies, a large number of literatures were focused on deterministic models [3][4][5][6][7][8][9], with some on the stochastic versions [10][11][12][13][14][15][16][17], but only a few on the food-chain systems.Furthermore, it is well known that the theory of food chains illustrated the balance of nature and that no animal or plant can exist independently.Motivated by these arguments presented above, we are interested in the optimal harvesting problems on the following stochastic food-chain system: with the initial value ( 0 ,  0 ,  0 ).Where   (),  = 1, 2, 3, is a standard Brownian motion and  is the harvesting effort (control parameter), and (), (), () represent the population densities of three species (resource, consumer, and predator) at time , respectively.All parameters are positive constants and parametric functions are continuous and positive.  (⋅),  = 1, 2, 3, represent the intrinsic growth rate of species , , , respectively;  1 measures the strength of competition among individuals of species ;  2 is the maximum value of the per capita reduction rate of  due to ;  1 ,  2 , and  have similar meaning to  2 ;  1 (⋅) measures the extent to which the environment provides protection to species  and ;  2 (⋅) measures the extent to which the environment provides protection to species  and ; () be a right continuous Markov chain;   (⋅),  = 1, 2, 3, represents the intensity of the white noise.This system is the extension of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation which was discussed by Ji et al. [18], Song et al. [19], and Guo et al. [20], and there the factor of Markovian switching is not considered.
In the most literatures [1,2] the sustainable yield function is used as the harvesting function.Here the harvesting function associated with (1) is This type of harvesting function is also used by some other papers, such as Wang [21, chapter 4] and Zou and Wang [22] and defined as the time averaging yield function.The optimal harvesting problem considered in this paper is then stated as follows.Find a harvesting effort  * such that Based on the aforementioned discussion, obviously, the first and most important duty is to discuss the existence of lim  → ∞ (∫  0 ()/) and then the optimal harvesting problem.Therefore, the rest of the paper is organized as follows.
In Section 2, we show that system has a global positive solution.In Section 3, we obtain some long time behavior of the solution, especially the property of persistent in mean, which ensures the existence of the time averaging yield function and its explicit expression is given.In Section 4, the optimal harvesting policies are investigated.In Section 5, we illustrate our main results through several numerical examples.Last but not least, conclusions are drawn in Section 6.
On the other hand, for convenience, we give some notations and assumptions in the rest of this section.
Throughout this paper, unless otherwise specified, let (Ω, F, {F  } ≥0 , ) be a complete probability space with a filtration {F  } ≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F 0 contains all P-null sets).The standard Brownian motion   (),  = 1, 2, 3, is defined on this probability space.
The right continuous Markov chain () on this probability space taking values in a finite-state space  = {1, 2, . . ., } with the generator  = (  ) × is given by where  > 0.Here   is the transition rate from  to  and   ≥ 0 if  ̸ = , while We assume that the Markov chain (⋅) and the Brownian motion   (⋅) are independent of each other,  = 1, 2, 3.As a standing hypothesis we also assume in this paper that the Markov chain () is irreducible.This is very reasonable as it means that the system will switch from any regime to any other regime.This is equivalent to the condition that, for any In order to obtain some properties of the system, some assumptions are given in the following.These assumptions are conventional; they guarantee that the ecosystem is not collapsed as time lapses.
and  is positive and sufficiently small.
and  denotes a float constant in the rest of this paper, which expresses different constants in different positions.
The key method used in this paper is the comparison theorem for stochastic equations.This theorem for stochastic differential equations was developed by Ikeda and Watanabe [23] and has been used by many authors [24][25][26].

Positive and Global Solutions
As the state of the system ((), (), ()) is the population density of species in the system at time , it should be nonnegative.Moreover, in order for a stochastic differential equation to have a unique global (i.e., no explosion in a finite time) solution for any given initial data, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition [25].However, the coefficients of each equation in system obey neither the linear growth condition nor local Lipschitz continuous.In this section, we show existence and uniqueness of the positive solution.

The Long Time Behavior
Theorem 4 shows that the solution of the system (1) will remain in the positive cone  3 + .This nice property provides us with a great opportunity to discuss how the solution varies in  3  + in detail.In this section we will give some long time behavior of the solution, especially the property of persistent in mean, which ensures the existence of lim  → ∞ (∫  0 ()/).
The quadratic variation of 2 (()) ≤ , and by the strong law of large numbers for local martingales, we have Therefore, for all  > 0, ∃0 <  < ∞, we have On the other hand, from Lemma 5, we have By the arguments as above, when  >  ≥ , we can get Therefore, we obtain that is lim sup For the arbitrary of  > 0, we must have lim sup Hence, lim  → ∞ (ln y ()/) = 0 a.s., and the second conclusion can be proved similarly.
Based on the first conclusion in this theorem, the strong law of large numbers for local martingales, and the ergodic property of Markov chain, the second conclusion is proved.
Similarly to the proof of the second assertion of Theorem 7, the last two assertions can be proved.Definition 9 (see [5]).The system is said to be persistent in mean, if lim inf Theorem 10.If Assumptions 1 and 2 are satisfied, then the system is persistent in mean.

Mathematical Problems in Engineering
From the second assertion of Theorem 7, we have lim inf that is, lim inf  → ∞ (∫  0 ()/) > 0. This theorem is proved.

The Optimal Harvesting Policies
Based on the explicit expression of the time averaging yield function obtained in the last section, here we discuss the optimal harvesting problem mentioned in Section 1.
Theorem 11.If Assumptions 1 and 2 is satisfied, then the optimal harvesting effort is where and the optimal harvesting output is Proof.Based on Theorem 8, the optimization problem can be expressed as follows: From the definitions of  1 and Σ, we get  1 ≤ Σ.Therefore, the above optimization problem can be simplified as follows: Because the objective function is concave, and we can obtain the unique maximum point easily as here the  is obtained by letting ()/() = 0. Substituting it into the harvesting function, we obtain the optimal harvesting output This theorem is proved.
Remark 12. (i) That the feasible zone of optimization problem (46) is nonempty is guaranteed by Assumptions 1 and 2.
(ii) From the explicit expression of the optimal harvesting effort, we can easily investigate how the parameters influence on it, such that  * is decreasing in  2 , and this claim coincides with the fact that if the consumer's (()) consuming capacity is enhanced ( 2 augments), the harvesting effort must reduce ( * go down), or the resource (()) will be extinct and the whole ecosystem is crashed.

Numerical Results
We present numerical experiments in this section to show how the proposed model works in the constructive examples.The results enhance the readers to understand the theoretical conclusions from the practical applications.
For simplicity, assume that the random environments are modeled by a two-state Markov chain with state set  = {1, 2} and generator  = ( −7 7 5 5 ) .
Figures 1 and 2 show that the solutions of (1) are positive in the deterministic environment (without regime switching); that is, () ≡ 1 or 2. Figure 4 shows that the solutions of (1) are positive in the random environment (with regime swithcing); the random environment is described by Figure 3.They are all identical to Theorem 4. Figure 5 shows lim  → ∞ (ln ()/) = 0, lim  → ∞ (ln ()/) = 0, and lim  → ∞ (ln ()/) = 0 in the random environment described by Figure 3, and this is consistent with Theorems 7 and 8.

Conclusions
This paper studies an optimal harvesting problem for a foodchain system with markovian switching.Based on the properties, the food-chain system's solution is existing, unique, and positive; the system is persistent in mean, and the Mathematical Problems in Engineering rationality of the optimal harvesting problem is proved.Then the optimal harvesting policy is obtained.Nevertheless, there are rooms to continue work on this issue, such that more than one of control variables in the system are considered.The permanence and extinction of the system and the stability in distribution need to be investigated too.