The Optimal Harvesting of a Stochastic Gilpin-Ayala Model Under Regime Switching
Juan Hou, Yanqun Wang, Zhenguo Luo
College of Mathematics and Statistics, Hengyang Normal University, Hengyang, P. R. China
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To cite this article:
Juan Hou, Yanqun Wang, Zhenguo Luo. The Optimal Harvesting of a Stochastic Gilpin-Ayala Model under Regime Switching. Science Journal of Applied Mathematics and Statistics. Vol. 4, No. 6, 2016, pp. 276-283. doi: 10.11648/j.sjams.20160406.15
Received: September 30, 2016; Accepted: October 13, 2016; Published: November 7, 2016
Abstract: 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.
Keywords: Optimal Harvesting, Stochastic Differential Equation, Markov Chain
1. Introduction
In recently, many authors have discussed population systems subject to the white noise (see [1-6, 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), Braumann (2002), Lande et al.(1995), Liu and Bai(2014), Ludwig and Varah(1979), Song et al.(2011) and Zou and Wang (2014).
A famous Gilpin-Ayala population model with harvesting is described by the ordinary differential equation (ODE)
(1)
Where is the harvesting effort and is a constant. If we still use to denote the average growth rate, but incorporate white noise, and the intrinsic growth rate becomes
Where is white noise and represents the intensity of the noise. Then this environmentally perturbed system may be described by the Ito's equation
Where is the 1-dimensional standard Brownian motion with. 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
(2)
When it is in regime 1, while it obeys another stochastic Gilpin-Ayala model
(3)
in regime 2 and so on. Therefore, the system obeys
(4)
In regime . The switching between these n regimes is governed by a Markov chain on the state space S={1,2,...,n}. The population system under regime switching can therefore be described by the following stochastic model
(5)
This system is operated as follows: If, the system obeys Eq.(4) with until time when the Markov chain jumps tofrom ; the system will then obeys Eq.(4) with from until when the Markovian chain jumps to from . The system will continue to switch as long as the Markovian chain jumps. In other words, Eq.(5) can be regarded as Eqs. (4) switching from one to another according to the law of the Markov chain. The different Eqs.(4) are therefore referred to as the subsystem of Eq.(5).
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 with probability 1; on the other hand, if the two equilibrium states coincide, then the trajectory either leaves any compact set of 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)
(6)
where for each and are all nonnegative constants and Our aim is to reveal the optimal harvesting of the system (6) with the environmental noise affects.
2. Preliminaries
Throughout this paper, unless otherwise specified, let be a complete probability space with a filtration satisfying the usual conditions (i.e. it is right continuous and contains all P-null sets). Let be a scalar standard Brownian motion defined on this probability space. We also denote by the open interval, and denote by the interval. Let r(t) be a right-continuous Markov chain on the probability space, taking values in a finite state space S={1,2,...,n}, with the generator given by
(7)
where Here is the transition rate from u to v and if , while
we assume that the Markov chain is independent of the Brownian motion It is well known that almost every sample path of is right continuous step function with a finite number of jumps in any finite subinterval of . 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 , one can find finite numbers such that Note that always has an eigen value 0. The algebraic interpretation of irreducibility is that rank. Under this condition, the Markov chain has a unique stationary (probability) distribution which can be determined by solving the following linear equation
(8)
subject to
For convenience and simplicity in the following discussion, define
where is a constant vector.
Let be a solution of Eq.(6),by Itˆo’s formula
(9)
Let , then the system (9) can be written as
(10)
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 and have the same monotone and extreme points in, then we can investigate the optimal harvesting of the system (10) instead of (6).
The solution of system (10) with initial value is
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 effortis said to be optimal, if
where .
For system (10), we introduce the following basic assumptions:
(H1) For each ;
(H2) ;
(H3) For each,
For the system(10), we have the following results.
3. 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
(11)
where
(12)
Proof By the generalized Itˆo formula, we have
(13)
Integrating it from 0 to t and taking expectations of both sides, we obtain that
Then we have
(14)
If 0<p<1. we obtain
(15)
while if , we obtain
(16)
Therefore, letting we have
(17)
Notice that if 0<p<1, the solution of equation
obeys
similarly, if p≥1 the solution of equation
as is such that
Thus by the comparison argument we get
(18)
By the definitions of z(t), we obtain the assertion (11).
Lemma 3. Under (H1), the solution Y(t) of SDE (10) with any positive initial value has the property
(19)
Proof By Lemma 1, the solution Y(t) with positive initial value will remain in. We have
We can also derive from this that
From Lemma 2, we know that
(20)
But, by the well-known Burkholder-Davis-Gundy inequality and the H"older inequality, we derive that
(21)
Therefore
This, together with (20), yields
(22)
To prove assertion (19), we observe from (22) that there is a positive constant such that
Let be arbitrary. Then, by the well-known Chebyshev inequality, we have
Applying the well-known Borel-Cantelli lemma, we obtain that for almost all
(23)
holds for all but finitely many k. Hence, there exists a, for almost all, for which (23) holds whenever. Consequently, for almost alland,
Therefore
Letting we obtain the desired assertion (19). The proof is therefore complete.
Corollary 1. Under (H1), the solution of SDE (10) with any positive initial value has the property
(24)
Proof From Lemma 3, we have
Consequently, we will get some results about SDE(10).
Theorem 1. The solution of SDE (10) with any positive initial value has the property
(25)
Proof From Corollary 1, we have
Hence, we only need to prove the following conclusion
(26)
Let
The quadratic variation of this martingale is
By the strong law of large numbers for martingales, we therefore have
For any positive constant and small enough, there is a positive constant, such that
Therefore, for any t>s>T, there have
(27)
By (27), for any t>T, we have
There has a constant such that
That is There have
and
or
Therefore we obtain the assertion (26) and complete the proof.
Theorem 2. Suppose (H1), (H2) hold and the Markov chain r(t) is irreducible, then the solution Y(t) of SDE (10) with any positive initial value has the property
Proof From Itˆo equation,
(28)
Hence, we have
Obviously, , from Theorem 1, a.s for .
By the ergodicity of the Markov chain r(t), as,
that is exists and be equal to .
On the optimal harvesting effort and the maximum sustainable yield, we have the following results.
Theorem 3. Under (H1)，(H2) and (H3), the optimal harvesting effort of (10) is
(29)
and the maximum sustainable yield satisfied
(30)
Proof By the theorem 2, we have
(31)
and
(32)
It is easily to know that
Let
be the sustainable yield function, while F'(h)=0, we can get the unique extreme point. Noticed the h such that F'(h)=0 is independent on b(r(t)), therefore
and
this is the optimal harvesting effort, taking it into (31)(32) and we can get the (30) easily. The proof is complete.
Corollary 2. Assume for some , the subsystem of SDE(10) with Markov switching is
It has the optimal harvesting effort
and the maximum sustainable yield satisfied
4. 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.
Example. Consider a 3-dimensional stochastic differential equation with Markovian switching of the form
(33)
Where r(t) is a right-continuous Markov chain taking values in S = {1,2,3}, and
θ=2, r(t) and B(t) are independent. Here
We rewrite the (33) as (34)
(34)
We compute
Let the generator of the Markov chain r(t) be
By solving the linear equation (8) we obtain the unique stationary distribution
.
Then the optimal harvesting effort of (34) is
and the maximum sustainable yield separately, for i=1,2,3
We get the optimal harvesting effort of is , then the optimal harvesting effort of is.
Acknowledgments
This work was supported by the Science Foundation of Hengyang Normal University (15B18)(15B17); The Xinjiang Uygur Autonomous Region Natural Science Foundation (2015211B004); the Hunan Provincial Key Laboratory of Intelligent Information Processing and Application, Hengyang, 421002, China.
References