Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 676871, 12 pages

doi:10.1155/2012/676871

Research Article

The Optimal Control and MLE of Parameters of a Stochastic Single-Species System

Huili Xiang and Zhijun Liu

Department of Mathematics, Hubei University for Nationalities, Hubei, Enshi 445000, China

Received 5 May 2012; Revised 4 September 2012; Accepted 20 September 2012

Academic Editor: Garyfalos Papaschinopoulos

Copyright © 2012 Huili Xiang and Zhijun Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper investigates the optimal control and MLE (maximum likelihood estimation) for a single-species system subject to random perturbation. With the help of the techniques of stochastic analysis and mathematical statistics, sufficient conditions for the optimal control threshold value, the optimal control moment, and the maximum likelihood estimation of parameters are established, respectively. An example is presented to illustrate the feasibility of our theoretical results.

1. Introduction

The Malthus model is usually expressed as

where , stands for the density of species at moment, and is the intrinsic growth rate. As everyone knows, model (1.1) has epoch-making significance in mathematics and ecology and later, many deterministic mathematical models have been widely studied (see [1–5]). In fact, a population system is inevitably affected by the environmental noise in the real world. As a consequence, it is reasonable to study a corresponding stochastic model. Notice that some recent results, especially on optimal control, for the following stochastic model

have been obtained (see [6–9]), where stands for the standard Brownian motion. However, for some pest populations, their generations are nonoverlapping (e.g., poplar and willow weevil, osier weevil and paranthrene tabaniformis) and the discrete models are more appropriate than the continuous ones. Compared with the continuous ones, the study on discrete mathematical models is more challenging. Inspired by [1–12], in this paper we will consider the following discrete model of system (1.2)

where , , and any two of them are independent. stands for the noise intensity. We will focus on the optimal control threshold value, the optimal control moment, and the maximum likelihood estimation of parameters. To the best of our knowledge, no work has been done for system (1.3).

The rest of this paper is organized as follows. In Section 2, some preliminaries are introduced. In Section 3, we give three results of this paper. As applications of our main results; an example is presented to illustrate the feasibility of our theoretical results in Section 4.

2. Preliminaries

In this section, we summarize several definitions, assumptions, and lemmas which are useful for the later sections.

Definition 2.1. Only when the quantity of pest population reaches one starts to control the pest population, and the real number is called to be a control threshold value.

Definition 2.2. Until the th generation, the total quantity of pest population first reaches the control threshold value, then one says that is the first reaching time.

Two main goals of this paper are to seek the optimal control threshold value and the optimal control moment from the point of view of the lowest control cost. Considering that the practical control to some pest population must be in the limited time range, we give the first assumption:

, where is the number of generation of pest population in a control period and is a positive integer.

Denote . Usually, at the beginning, the number of pest population is very small, so we give the second assumption:

The first reaching time .

Let the life period of pest population be , we should annihilate pest at moment from the point of view of the lowest control cost. We further give the third assumption.

The number of pest population will not reach the extent which can cause damage again after being annihilated.

By , we have

So we can give the expression of the total loss caused by pest and expending for annihilating pest, respectively. It is obvious that the loss caused by pest population comes from the quantity of population and damaging time. We need to the fourth assumption

  The generation of pest population is nonoverlapping.

On one hand, the loss caused by pest can be expressed as

where stands for the loss caused by unit number pest in one generation, is the mean function of . On the other hand, the expending for annihilating pest can be expressed as

where is defined by

that is,

where stands for the expending for annihilating pest once. Since is dependent on random variable and is dependent on random variable and threshold value , the total cost is a random variable, which can be expressed as

Thus, we need to search for such that is minimum and consequently, we can give the optimal control moment.

Next, we will give some lemmas which are very important to the proofs of three theorems in the following section.

Lemma 2.3. The solution of system (1.3) can be expressed as

Proof. By (1.3), we have

Thus, one has

By a simplification, we obtain

that is,

Lemma 2.4. If is the mean-value function of the solution of system (1.3), then one has

Proof. One has

Let . Since , we have

and the probability density function of random variable is

It follows from (2.13) and (2.15) that we have

Lemma 2.5. Let the life period of pest population be , let be the loss caused by unit number pest in one generation, and let be the expending for annihilating pests once time. The loss caused by pest can be expressed as

Proof. Consider

Lemma 2.6. Let , One has

where

Proof. By the definition of , we have

Then

By , we have

Furthermore,

By (2.24), we have

Moreover, one has

and then we obtain

Lemma 2.7. The mean-value function of the loss caused by pest population is

Proof. By the definition of mean value function, we have

then by Lemma 2.6, we obtain

Lemma 2.8. The following equality holds

Proof. By the definitions of and , we have

3. Main Results

In this section, we give three main results. We first give the optimal control threshold value.

Theorem 3.1. If the assumptions – are satisfied, then the optimal control threshold value of system (1.3) is the minimal nonnegative solution of the following equation about

where

Proof. By Lemmas 2.3–2.8, we obtain that the total loss can be expressed as

A calculation leads to

Denote is the minimal nonnegative solution of the above equation, it follows from (3.4) that and is the optimal control threshold value of system (1.3). The proof of Theorem 3.1 is complete.

In the following, we give the optimal control moment.

Theorem 3.2. If the assumptions – hold, then the optimal control moment of system (1.3) can be expressed as

where is defined in Theorem 3.1 and is the life period of the pest population.

Proof. By the definition of , we have . Furthermore, it follows from Lemma 2.6 that

The proof of Theorem 3.2 is complete.

Finally, we give the estimate of the maximum likelihood estimations of the parameters and of system (1.3).

Theorem 3.3. Let and be the maximum likelihood estimations of the parameters and , one has

where .

Proof. From system (1.3), we have

let , then we obtain

Since i.i.d , we have i.i.d . Let be the quantity of the th generation pest population, we can obtain corresponding values , then the likelihood function of parameters and is

Further, we have

From (3.11), we obtain the following likelihood equation

and the maximum likelihood estimations of and are

The proof of Theorem 3.3 is complete.

4. An Example

In this section, to illustrate the feasibility of our theoretical results, we will give the following example.

Example 4.1. Consider the following system

The choose the loss caused by the unit number pest , the expending for annihilating pest once , and initial value , . By Theorems 3.1 and 3.2, we can obtain the approximates of the optimal control threshold value and the optimal control moment .

Next, we give the MLE of the parameters and to compare the true value with estimation. In Table 1, for the given true value of parameters and , the number of the sample “size ” increases from to , the data of the columns -MLE and -MLE are obtained by the average of 10 MLEs from the data coming from system (1.3). The columns of AE shows the absolute error of MLE. Table 1 shows that, with the augment of the number of the sample, the absolute error of MLE of and will decrease, which implies that it is reasonable to estimate the parameters of system (1.3) by MLE.

Table 1: The average value and absolute error of MLE of parameters with different number of sample.

True
Size
Aver
AE

r-MLE
-MLE

500
0.3049901
0.1000706
0.0049901
0.0000706

1000
0.3048290
0.0999396
0.0048290
0.0000604

2000
0.3040423
0.1000054
0.0040423
0.0000054

Acknowledgments

The authors would like to thank anonymous reviewers for their helpful comments which improved the presentation of their work. The work is supported by the National Natural Science Foundation of China (no. 11261017), the Key Project of Chinese Ministry of Education (no. 210134, 212111) and the Innovation Term of Hubei University for Nationalities (no. MY2011T007).

References

R. K. Miller, “On Volterra's population equation,” SIAM Journal on Applied Mathematics, vol. 14, pp. 446–452, 1966.

G. Seifert, “On a delay-differential equation for single specie population variations,” Nonlinear Analysis A, vol. 11, no. 9, pp. 1051–1059, 1987.

Z. Liu, Y. Chen, Z. He, and J. Wu, “Permanence in a periodic delay logistic system subject to constant impulsive stocking,” Mathematical Methods in the Applied Sciences, vol. 33, no. 8, pp. 985–993, 2010.

X. Liu and L. Chen, “Global behaviors of a generalized periodic impulsive logistic system with nonlinear density dependence,” Communications in Nonlinear Science and Numerical Simulation, vol. 10, no. 3, pp. 329–340, 2005.

P. Howitt, “Steady endogenous gowth with population and R and D inputs gowth,” Journal of Political Economy, vol. 107, no. 4, pp. 715–730, 1999.

X. Mao, G. Marion, and E. Renshaw, “Environmental Brownian noise suppresses explosions in population dynamics,” Stochastic Processes and their Applications, vol. 97, no. 1, pp. 95–110, 2002.

X. Mao, S. Sabanis, and E. Renshaw, “Asymptotic behaviour of the stochastic Lotka-Volterra model,” Journal of Mathematical Analysis and Applications, vol. 287, no. 1, pp. 141–156, 2003.

D.-Q. Jiang and B.-X. Zhang, “Existence, uniqueness, and global attractivity of positive solutions and MLE of the parameters to the logistic equation with random perturbation,” Science in China A, vol. 50, no. 7, pp. 977–986, 2007.

S. Y. Li , “The optimal threshold and moment for preventing the random walking insect population,” Or Transactions, vol. 7, no. 1, pp. 91–96, 2003.

B. Lian and S. Hu, “Asymptotic behaviour of the stochastic Gilpin-Ayala competition models,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 419–428, 2008.

L. M. Ricciardi, Stochastic Population Theory: Birth and Death Processes, Springer, Berlin, Germany, 1986.

Y. Li and H. Gao, “Existence, uniqueness and global asymptotic stability of positive solutions of a predator-prey system with Holling II functional response with random perturbation,” Nonlinear Analysis A, vol. 68, no. 6, pp. 1694–1705, 2008.