The Optimal Harvesting of a Stochastic Gilpin-Ayala Model Under Regime Switching

In this paper, we consider a stochastic Gilpin-Ayala model under regime switching. Obtain the optimal harvesting effort and the maximum sustained yield by investigating the condition of average boundness of the system, and the ergodicity of the Markov chain. Also, through an example, we have proved our conclusion.


Introduction
In recently, many authors have discussed population systems subject to the white noise (see [1][2][3][4][5][6]10]). Also, the optimal harvesting in managing natural resources has received much attention. Because the growth of species in the natural world is inevitably affected by environmental noise, many scholars have considered the optimal harvesting of stochastic population systems. By solving the corresponding Fokker-Planck equation, Beddington and May(1977) established the optimal harvesting policy for a stochastic logistic model. Using the same method, Li and Wang (2010) obtained the optimal harvesting policy for a stochastic Gilpin-Ayala model. The optimal harvesting of the stochastic population model was also examined in Alvarez and Shepp(1998) A famous Gilpin-Ayala population model with harvesting is described by the ordinary differential equation (ODE) Where h is the harvesting effort and 1 θ > is a constant.
If we still use ( ) a t to denote the average growth rate, but incorporate white noise, and the intrinsic growth rate becomes As we known, there are various types of environmental noise. Let us now take a further step by considering another type of environmental noise, namely color noise, say telegraph noise (see e.g. [7,8]). In this context, telegraph noise can be described as a random switching between two or more environmental regimes, which differ in terms of factors such as nutrition or rainfall. The switching is memoryless and the waiting time for the next switch has an exponential distribution. We can hence model the regime switching by a finite-state Markov chain. Assume that there are n regimes and the system obeys When it is in regime 1, while it obeys another stochastic Gilpin-Ayala model in regime 2 and so on. Therefore, the system obeys . The switching between these n regimes is governed by a Markov chain ( ) r t on the state space S={1,2,...,n}. The population system under regime switching can therefore be described by the following stochastic model This system is operated as follows: Takeuchi et al. [7] investigated a 2-dimensional autonomous predator-prey Lotka-Volterra system with regime switching and revealed a very interesting and surprising result: If two equilibrium states of the subsystems are different, all positive trajectories of this system always exit from any compact set of 2 R + with probability 1; on the other hand, if the two equilibrium states coincide, then the trajectory either leaves any compact set of 2 R + or converges to the equilibrium state. In practice, two equilibrium states are usually different, in which case Takeuchi et al. [7] showed that the stochastic population system is neither permanent nor dissipative. This is an important result as it reveals the significant effect of environmental noise on the population system: both its subsystems develop periodically but switching between them makes them become neither permanent nor dissipative. Therefore, these factors motivate us to consider the Gilpin-Ayala population system subject to both white noise and color noise, described by (SDE) where for each , ) ) , ( ( i S a i b i ∈ and ( ) i α are all nonnegative constants and 1.
θ > Our aim is to reveal the optimal harvesting of the system (6) with the environmental noise affects.
δ > Here uv γ is the transition rate from u to v and It is well known that almost every sample path of ( ) · r is right continuous step function with a finite number of jumps in any finite subinterval of R + . As a standing hypothesis we assume in this paper that the Markov chain r(t) is irreducible. This is a very reasonable assumption, as it means that the system can switch from any regime to any other regime. This is equivalent to the condition that for any , u v S ∈ , one can find finite numbers 1 2 , ,... k i i i S ∈ such that

∑
For convenience and simplicity in the following discussion, define , then the system (9) can be written as Similarly to the Theorem 2.1 in [9], we have the following Lemma. Lemma 1. There exists a unique continuous solution N(t) to SDE (6) for any initial value N(0)=N0>0, which is global and represented by Since ( ) Y t and ( ) N t have the same monotone and extreme points in R + , then we can investigate the optimal harvesting of the system (10) instead of (6).
The solution of system (10) with initial value ( ) ( ) which is positive and global. We first give some definitions about the optimal harvesting of the system (10) with the environmental noise affects.
Definition. The harvesting effort h * is said to be optimal, if For system (10), we introduce the following basic assumptions: (H1) For each , For the system(10), we have the following results.

The Main Results
We firstly have the following lemma. Lemma 2. If assumption (H1) holds, for an arbitrary given positive constant p, the solution Y(t) of SDE (10) with any given positive initial value has the property that Proof By the generalized Itˆo formula, we have Integrating it from 0 to t and taking expectations of both sides, we obtain that If 0<p<1. we obtain Therefore, letting ( ) ( ( )).
Notice that if 0<p<1, the solution of equation By the definitions of z(t), we obtain the assertion (11).
Proof By Lemma 1, the solution Y(t) with positive initial value will remain in R + . We have We can also derive from this that But, by the well-known Burkholder-Davis-Gundy inequality and the H"older inequality, we derive that Applying the well-known Borel-Cantelli lemma, we obtain that for almost all ω ∈ Ω holds for all but finitely many k. Hence, there exists a 0 ( ) k ω , for almost all ω ∈ Ω , for which (23) holds whenever 0 k k > .
Consequently, for almost all ω ∈ Ω and By the ergodicity of the Markov chain r(t), as t → ∞ , On the optimal harvesting effort and the maximum sustainable yield, we have the following results.

Conclusions and example
In this paper, we investigate the optimal harvesting effort and the maximum sustainable yield of a stochastic Gilpin-Ayala model under regime switching, we get the optimal harvesting effort of the SDE (10) and estimate the value of the maximum sustainable yield. we get the value of the maximum sustainable yield of the subsystem of (10) without the SDE (10).
Making use of the results, we shall illustrate these conclusions through the following example.
Then the optimal harvesting effort of (34) is