A Stochastic Two Species Competition Model: Nonequilibrium Fluctuation and Stability

G P Samanta

doi:10.1155/2011/489386

Abstract

The object of this paper is to study the stability behaviours of the deterministic and stochastic versions of a two-species symmetric competition model. The logistic parameters of the competitive species are perturbed by colored noises or Ornstein-Uhlenbeck processes due to random environment. The Fokker-Planck equation has been used to obtain probability density functions. Here, we have also discussed the relationship between stability behaviours of this model in a deterministic environment and the corresponding model in a stochastic environment.

Highlights

In recent years, scientists have become increasingly aware of the fact that most natural phenomena do not follow strictly deterministic laws but rather oscillate randomly about some average behaviour
The influence of the environment has to be taken into account. This is often done by arbitrarily augmenting the deterministic equations with stochastic terms or taking the environmental parameters as time dependent and rapidly varying
We have studied the relationship between stability behaviours of a twospecies symmetric competition model in which environmental parameters are prescribed constant deterministic environment and the corresponding model in which these parameters have an element of random fluctuation stochastic environment, where it is assumed that the fluctuations in the environment manifest themselves as random fluctuations in the logistic parameters about some mean value, as they usually do in nature 8, 9

Summary

Introduction

Scientists have become increasingly aware of the fact that most natural phenomena do not follow strictly deterministic laws but rather oscillate randomly about some average behaviour. This is especially true in the ecological models where environmental influences should be taken as stochastic. The influence of the environment has to be taken into account This is often done by arbitrarily augmenting the deterministic equations with stochastic terms or taking the environmental parameters as time dependent and rapidly varying. We have studied the relationship between stability behaviours of a twospecies symmetric competition model in which environmental parameters are prescribed constant deterministic environment and the corresponding model in which these parameters have an element of random fluctuation stochastic environment , where it is assumed that the fluctuations in the environment manifest themselves as random fluctuations in the logistic parameters about some mean value, as they usually do in nature 8, 9

Two-Species Symmetric Competition

Stationary Probability Density

Concluding Remarks