An optimal stopping problem in the stochastic Gilpin-Ayala population model
© Ai and Sun; licensee Springer 2012
Received: 7 September 2012
Accepted: 25 November 2012
Published: 10 December 2012
We present an explicit solution to an optimal stopping problem of the stochastic Gilpin-Ayala population model by applying the smooth pasting technique (Dixit in The Art of Smooth Pasting, 1993 and Dixit and Pindyck in Investment under Uncertainty, 1994). The optimal stopping rule is to find an optimal stopping time and an optimal stopping boundary of maximizing the expected discounted reward, which are given in this paper explicitly.
Optimal stopping problems of stochastic systems play an important role in the field of stochastic control theory. A special interest in such problems is attracted by many fields such as finance, biology models and so on.
The aim of the optimal stopping problems is to search for random times at which the stochastic processes should be stopped to make the expected values of the given reward functionals optimal. Lots of explicitly solvable stopping problems with exponentially discounted stopping problems are mainly those for one-dimensional diffusion processes. The optimal stopping times are the first time at which the underlying processes exit certain regions restricted by constant boundaries.
In this paper, the optimal stopping time for the stochastic Gilpin-Ayala model [1–4], whose solution is a diffusion process, is introduced, and the explicit expressions for the value functions and the boundaries in such optimal stopping problems are obtained. To our best knowledge, there have been few tries to research the optimal harvesting problems based on optimal stopping, and many scholars studied stochastic logistic models such as [5, 6]. There are only a few results about the corresponding stochastic Gilpin-Ayala model, which is our motivation.
where denotes the density of resource population at time t, is called the intrinsic growth rate and , K is the environmental carrying capacity. It is obvious that (1.1) becomes the classic logistic population model when .
where the constants r, b are mentioned in (1.1) and is one-dimensional Brownian motion .
The outline for this paper is as follows. Section 2 of this paper is concerned with the general problem of choosing an optimal stopping time for the stochastic Gilpin-Ayala population model. In Section 3, a closed-form candidate function for the value function is given. We verify the candidate for the expected reward is optimal and the optimal stopping boundary is expressed by the smooth pasting technique.
2 Formulation of the problem
for all , is one-dimensional Brownian motion (see ), and note that .
where the discounting exponent , is the profit at time τ and a represents a fixed fee and it is natural to assume that . The positive constant w represents the permanent assets. denotes the expectation with respect to the probability law of the process , starting at .
Note that it is trivial that the initial value . So we further assume that and the stopping time τ is bounded since .
- (2)smooth pasting condition(3.15)
In fact, is showed to be the unique solution of (3.15) by the following assumptions and Lemma 3.1.
We assume the following.
The following lemma provides an optimal stopping boundary.
Lemma 3.1 is the maximum value point of given by (3.12) with respect to , for fixed , .
where , , with some normalizing constant A for . Then by applying the Jensen inequality and considering the obvious fact that , we deduce , which gives the monotonicity of on (similar discussion can be found in ).
There exists a unique solution, which satisfies , of (3.16) on and note that on under Assumption 2.
- (2)The maximum value is given by(3.24)
under (3.9) and Assumption 1 on the interval . The proof is completed.
Now, let us give the following lemma for our main Theorem 3.3.
given by (2.4) for all , .
- (2)For , ,(3.25)
, , .
for , i.e., for and
for . This is easily done by routine calculation under Assumptions 1 and 2.
Let us give our main theorem.
If , then . So, we have by (3.31) and is optimal for .
- (b)Next, suppose . By Dynkin’s formula  and the fact that a.s. for , we have(3.32)
So, and is optimal, .
4 Conclusion and further studies/research
This paper describes the optimal harvesting problems of the stochastic Gilpin-Ayala population model as an optimal stopping problem, which is our first try. Meanwhile, we obtain the explicit optimal value function and optimal stopping time by using the smooth pasting technique. Finally, we prove the result. Furthermore, our work can lead a new way for the optimal harvesting problem in the real world. In further direction, the optimal harvesting problems for the stochastic predator-prey model and related stochastic models will be considered.
We are grateful to Prof. Wang Ke for a number of helpful suggestions for improving the article. The second author was supported by the Natural Science Foundation of the Education Department of Heilongjiang Province (Grant No. 12521116).
- Gilpin ME, Ayala FJ: Global models of growth and competition. Proc. Natl. Acad. Sci. USA 1973, 70: 3590–3593. 10.1073/pnas.70.12.3590View ArticleMATHGoogle Scholar
- Lian B, Hu S: Stochastic delay Gilpin-Ayala competition models. Stoch. Dyn. 2006, 6: 561–576. 10.1142/S0219493706001888MathSciNetView ArticleMATHGoogle Scholar
- Lian B, Hu S: Asymptotic behaviour of the stochastic Gilpin-Ayala competition models. J. Math. Anal. Appl. 2008, 339: 419–428. 10.1016/j.jmaa.2007.06.058MathSciNetView ArticleMATHGoogle Scholar
- Liu M, Wang K: Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system. Appl. Math. Lett. 2012, 25: 1980–1985. 10.1016/j.aml.2012.03.015MathSciNetView ArticleMATHGoogle Scholar
- Liu M, Wang K: Persistence and extinction in stochastic non-autonomous logistic systems. J. Math. Anal. Appl. 2011, 375: 443–457. 10.1016/j.jmaa.2010.09.058MathSciNetView ArticleMATHGoogle Scholar
- Liu M, Wang K: Asymptotic properties and simulations of a stochastic logistic model under regime switching. Math. Comput. Model. 2011, 54: 2139–2154. 10.1016/j.mcm.2011.05.023View ArticleMathSciNetMATHGoogle Scholar
- Abramovitz M, Stegun IA: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Wiley, New York; 1972. National Bureau of StandardsGoogle Scholar
- Clark CW: Mathematical Bioeconomics: the Optimal Management of Renewal Resources. Wiley, New York; 1976.MATHGoogle Scholar
- Clark CW: Mathematical Bioeconomics: the Optimal Management of Renewal Resources. 2nd edition. Wiley, New York; 1990.MATHGoogle Scholar
- Fan M, Wang K: Optimal harvesting policy for single population with periodic coefficients. Math. Biosci. 1998, 152: 165–177. 10.1016/S0025-5564(98)10024-XMathSciNetView ArticleMATHGoogle Scholar
- Zhang X, Shuai Z, Wang K: Optimal impulsive harvesting policy for single population. Nonlinear Anal., Real World Appl. 2003, 4: 639–651. 10.1016/S1468-1218(02)00084-6MathSciNetView ArticleMATHGoogle Scholar
- Alvarez LHR, Shepp LA: Optimal harvesting of stochastically fluctuating populations. Math. Biosci. 1998, 37: 155–177.MathSciNetMATHGoogle Scholar
- Alvarez LHR: Optimal harvesting under stochastic fluctuations and critical depensation. Math. Biosci. 1998, 152: 63–85. 10.1016/S0025-5564(98)10018-4MathSciNetView ArticleMATHGoogle Scholar
- Lungu EM, Øksendal B: Optimal harvesting from a population in a stochastic crowded environment. Math. Biosci. 1997, 145: 47–75. 10.1016/S0025-5564(97)00029-1MathSciNetView ArticleMATHGoogle Scholar
- Øksendal B: Stochastic Differential Equations. 6th edition. Springer, New York; 2005.MATHGoogle Scholar
- Wang K: Stochastic Biomathematics Models. Science Press, Beijing; 2010.Google Scholar
- Muller KE:Computing the confluent hypergeometric function, . Numer. Math. 2001, 90: 179–196. 10.1007/s002110100285MathSciNetView ArticleMATHGoogle Scholar
- Olver FWJ, Lozier DW, Boisvert RF, Clark CW: NIST Handbook of Mathematical Function. Cambridge University Press, Cambridge; 2010.MATHGoogle Scholar
- Dixit A: The Art of Smooth Pasting. Harwood Academic, Switzerland; 1993.MATHGoogle Scholar
- Dixit A, Pindyck R: Investment under Uncertainty. Princeton University Press, New Jersey; 1994.Google Scholar
- Gapeev PV, Markus R: An optimal stopping problem in a diffusion-type model with delay. Stat. Probab. Lett. 2006, 76: 601–608. 10.1016/j.spl.2005.09.006View ArticleMathSciNetMATHGoogle Scholar
- Halmos P: Measure Theory. Springer, Berlin; 1974.MATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.