Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment

doi:10.1016/j.jtbi.2010.03.008

Journal of Theoretical Biology

Volume 264, Issue 3, 7 June 2010, Pages 934-944

https://doi.org/10.1016/j.jtbi.2010.03.008 Get rights and content

Abstract

A new single-species model disturbed by both white noise and colored noise in a polluted environment is developed and analyzed. Sufficient criteria for extinction, stochastic nonpersistence in the mean, stochastic weak persistence in the mean, stochastic strong persistence in the mean and stochastic permanence of the species are established. The threshold between stochastic weak persistence in the mean and extinction is obtained. The results show that both white and colored environmental noises have sufficient effect to the survival results.

Introduction

In the world today, with the rapid development of modern industry and agriculture, a great quantity of toxicants and contaminants enter into ecosystems one by one. Environmental pollution is one of the most important socio-ecological problems. The presence of a variety of toxicants in the environment is a threat to the survival of exposed populations, including mankind (see, e.g. Nelson, 1970, Jensen and Marshall, 1982, Shukla et al., 1989). This motivates scholars to investigate the effects of toxins on the population and the assessment of the risks taken by the species. Those studies are becoming more and more important.

Recently, many works have been done to study the effects of toxicant on populations utilizing mathematical models. In 1980s, Hallam et al., 1983a, Hallam et al., 1983b, Hallam and Deluna, 1984 proposed some deterministic models to deal with effects of pollutants on various ecosystems for the first time. In particular, under the assumption that the growth rate of the population depends linearly upon the uptake concentration of the toxicant but the corresponding carrying capacity does not depend upon the concentration of toxicant present in the environment, Hallam et al. (1983b) considered the effect of toxicant present in the environment on a single-specie population and proposed a basic model. Hallam and Ma (1986) proposed the conception of persistence in the mean, and a new method which was used to determine the thresholds between persistence and extinction of nonautonomous models had been given. Ma et al. (1990) studied the survival of a single logistic species in a polluted environment by taking into account the organism's uptake of toxicant from the environment and egestion of toxicant into the environment. Thomas et al. (1996) proposed and analyzed a mathematical model to understand the effect of pollutant on a single-species population, and then got some sufficient conditions to ensure the survival of the population. More investigation and improvement of such deterministic models can be found in Hallam and Ma, 1986, Hallam and Ma, 1987, Luna and Hallam (1987), Ma et al., 1989, Ma et al., 1997, Freedman and Shukla (1991), Shukla and Dubey (1996), Buonomo et al. (1999), Pan et al. (2000), Bai and Wang (2006), He and Wang, 2007, He and Wang, 2009, Jiao et al. (2009), and Agarwal and Devi (2009).

Those important and useful works on deterministic models provide a great insight into the effect of the pollution, but in the real world, the natural growth of populations are inevitably affected by random fluctuations. Generally, there are two main random noises. One is white noise, another is colored noise. There are a few results dealing with behavior of model disturbed by white noise in polluted environment. Gard (1992) proposed a stochastic single-species model in a polluted environment and analyzed it by assuming that the concentration of toxicant in the organism was a constant. Liu and Wang (2009) studied a stochastic single-specie growth model in a polluted environment and gave a threshold between local extinction and stochastic weak persistence in the mean. However, colored noise was not taken into account in either Gard (1992) and Liu and Wang (2009) (see Remark 1 in Section 3). Indeed, so far as we know there is few results on dynamics of the polluted population disturbed by colored noise, and little is known of the impacts of colored noise on the survival of species living in a polluted environments. So in this paper, we propose and study a model which accounts for both colored and white noise in a polluted environment. The important contributions of this paper are therefore clear.

The rest of the paper is arranged as follows. In Section 2, we are going to develop a new stochastic single-specie model under regime switching in a polluted environment. In Section 3, we will carry out the survival analysis for our model and obtain some sufficient conditions for extinction, stochastic nonpersistence in the mean, stochastic weak persistence in the mean, stochastic strong persistence in the mean and stochastic permanence of the species. The threshold between stochastic weak persistence in the mean and extinction is obtained. In Section 4, we shall introduce some examples and figures to illustrate the various theorems obtained before. Section 5 gives the conclusions and future directions of research.

Section snippets

Modelling environmental switching with colored noise

From now on, unless otherwise specified, we shall always work on a given complete probability space $(Ω, F, P)$ with a filtration ${F_{t}}_{t \geq 0}$ satisfying the usual conditions. Let B(t) stand for a given standard Brownian motion defined on the probability space.

Our study relies on the assumption that the environment is of complete spatial homogeneity and there is no migration. The individual organisms in the population are assumed to be nongrowing. Let x(t) be the population size at time t; C₀(t) stands

The survival analysis for (SM)

In this section, we shall carry out the survival analysis for model (SM). For this end, we list the following notations and definitions which are commonly used.

Notations and useful definitions:

〈 f (t) 〉 = \frac{1}{t} \int_{0}^{t} f (s) ds

f_{*} = \underset{t \to + \infty}{\lim \inf} f (t)

f^{*} = \underset{t \to + \infty}{\lim \sup} f (t)

R_{+} = [0, + \infty)

Extinction:

\lim_{t \to + \infty} x (t) = 0

a.s.

Stochastic nonpersistence in the mean:

〈 x 〉^{*} = 0

a.s.

Stochastic weak persistence in the mean:

〈 x 〉^{*} > 0

a.s.

Stochastic strong persistence in the mean:

〈 x 〉_{*} > 0

a.s.

Stochastic permanence: there are constants

β > 0, M > 0

such

Examples and numerical simulations

In this section we will give two examples to show that both white noise and colored noise have sufficient effect on the persistence or extinction of the species. In applications, the generator Q=(q_ij) of the Markov chain $γ (t)$ is predicable, however, the stationary distribution $(π_{1}, \dots, π_{m})$ of this Markov chain can be obtained through simple calculation (for the sake of convenience, we prepare those approaches in Appendix C. The reader is referred to Ross, 2006 for more details). So in the

Conclusions and future directions

Environmental pollution is one of the most important socio-ecological problems in the world today. And in the real world, the natural growth of population is inevitably affected by some random disturbance and those disturbance should not be neglected in many cases. For example, when the size of population is small and when the mean and variance of perturbations are large, only considering the deterministic model is not enough to understand the real world. Stochastic population dynamics with or

Acknowledgments

The authors thank the referees for their very important and helpful comments and suggestions. We also thank Q.Y. Li and X.L. Zou for valuable program files of the figures. This research was supported by The National Natural Science Foundation of PR China (nos. 10701020 and 10971022).

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Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment

Abstract

Introduction

Section snippets

Modelling environmental switching with colored noise

The survival analysis for (SM)

Examples and numerical simulations

Conclusions and future directions

Acknowledgments

Nonlinear Anal. Real World Appl.

Math. Bios.

J. Comput. Appl. Math.

Bull. Math. Biol.

Ecol. Model.

J. Theor. Biol.

Appl. Math. Model.

Nonlinear Anal. Real World Appl.

Environ. Pollut.

Nonlinear Anal. Real World Appl.

J. Math. Anal. Appl.

J. Comput. Appl. Math.

Ecol. Model.

J. Math. Anal. Appl.

J. Math. Anal. Appl.

Stochastic Process. Appl.

Ecol. Model.

J. Math. Anal. Appl.

Math. Biosci.

J. Math. Anal. Appl.

A time-delay model for the effect of toxicant in a single species growth with stage-structure

Nonlinear Anal. Real World Appl.

Stochastic Differential Equations