# American Institute of Mathematical Sciences

September  2015, 20(7): 2069-2088. doi: 10.3934/dcdsb.2015.20.2069

## Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion

 1 Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China 2 School of Science, Beijing University of Civil Engineering and Architecture, Beijing, 100044, China, China

Received  May 2014 Revised  February 2015 Published  July 2015

Using stochastic differential equations with Lévy jumps, this paper studies the effect of environmental stochasticity and random catastrophes on the permanence of Lotka-Volterra facultative systems. Under certain simple assumptions, we establish the sufficient conditions for weak permanence in the mean and extinction of the non-autonomous system, respectively. In particular, a necessary and sufficient condition for permanence and extinction of autonomous system with jump-diffusion are obtained. We generalize some former results under weaker assumptions. Finally, we discuss the biological implications of the main results.
Citation: Dan Li, Jing'an Cui, Yan Zhang. Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2069-2088. doi: 10.3934/dcdsb.2015.20.2069
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