Sufficient and necessary conditions on the existence of stationary distribution and extinction for stochastic generalized logistic system

Advances in Difference Equations

Lei Liu¹  Yi Shen² 

[1] College of Science, Hohai University, Nanjing, China;School of Automation, Huazhong University of Science and Technology, Wuhan, China

关键词: stochastic generalized logistic system;    extinction;    stationary Fokker-Planck equation;    stationary distribution;    Itô’s formula;   

DOI  :  10.1186/s13662-014-0345-y

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

In this paper, we consider the existence of stationary distribution and extinction for a stochastic generalized logistic system. Sufficient and necessary conditions for the existence of a stationary distribution and extinction are obtained. (a) The system has a unique stationary distribution if and only if the noise intensity is less than twice the intrinsic growth rate. The probability density function has been solved by the stationary Fokker-Planck equation. (b) The system will become extinct when and only when the noise intensity is no less than twice the intrinsic growth rate, and the exponential extinction rate is estimated precisely by two parameters of the systems. A new perspective is provided to explain the recurrence phenomenon in practice. Nontrivial examples are provided to illustrate our results.

【 授权许可】

CC BY