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Conditions for permanence and ergodicity of certain stochastic predator–prey models

Published online by Cambridge University Press:  24 March 2016

Nguyen Huu Du ,
Dang Hai Nguyen  and
G. George Yin
Nguyen Huu Du
Affiliation:
Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam. Email address: dunh@vnu.edu.vn
Dang Hai Nguyen
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. Email address: dangnh.maths@gmail.com
G. George Yin*
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA.
*
**** Email address: gyin@math.wayne.edu
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Abstract

In this paper we derive sufficient conditions for the permanence and ergodicity of a stochastic predator–prey model with a Beddington–DeAngelis functional response. The conditions obtained are in fact very close to the necessary conditions. Both nondegenerate and degenerate diffusions are considered. One of the distinctive features of our results is that they enable the characterization of the support of a unique invariant probability measure. It proves the convergence in total variation norm of the transition probability to the invariant measure. Comparisons to the existing literature and matters related to other stochastic predator–prey models are also given.

Keywords

Ergodicityextinctionpermanencepredator–preyBeddington–DeAngelis functional responsestationary distribution
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 1 , March 2016 , pp. 187 - 202
Copyright
Copyright © Applied Probability Trust 2016 

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