Stochastic bifurcations, a necessary and sufficient condition for a stochastic Beddington–DeAngelis predator–prey model

A probabilistic predator-prey model is constructed using linked discrete-time branching-type processes. A necessary and sufficient condition for positive probability of survival of both populations is given.

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The Galton-Watson predator-prey process

Journal of Applied Probability ◽

10.2307/3214736 ◽

1991 ◽

Vol 28 (1) ◽

pp. 9-16 ◽

Cited By ~ 1

Author(s):

John Coffey ◽

Wolfgang J. Bühler

Keyword(s):

Discrete Time ◽

Sufficient Condition ◽

Positive Probability ◽

Necessary And Sufficient Condition ◽

Predator Prey ◽

Probability Of Survival ◽

Predator Prey Model ◽

Prey Model ◽

Necessary And Sufficient

A probabilistic predator-prey model is constructed using linked discrete-time branching-type processes. A necessary and sufficient condition for positive probability of survival of both populations is given.

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Positive Solutions for a General Gause-Type Predator-Prey Model with Monotonic Functional Response

Abstract and Applied Analysis ◽

10.1155/2011/547060 ◽

2011 ◽

Vol 2011 ◽

pp. 1-16 ◽

Cited By ~ 2

Author(s):

Guohong Zhang ◽

Xiaoli Wang

Keyword(s):

Functional Response ◽

Positive Solutions ◽

Sufficient Conditions ◽

Index Theory ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Fixed Point Index Theory ◽

Local And Global Bifurcations ◽

Necessary And Sufficient

We study a general Gause-type predator-prey model with monotonic functional response under Dirichlet boundary condition. Necessary and sufficient conditions for the existence and nonexistence of positive solutions for this system are obtained by means of the fixed point index theory. In addition, the local and global bifurcations from a semitrivial state are also investigated on the basis of bifurcation theory. The results indicate diffusion, and functional response does help to create stationary pattern.

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Multiple Periodic Solutions of Generalized Gause-Type Predator-Prey Systems with Nonmonotonic Numerical Responses and Impulse

ISRN Mathematical Analysis ◽

10.1155/2014/347201 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Zhenguo Luo ◽

Liping Luo ◽

Yunhui Zeng

Keyword(s):

Periodic Solutions ◽

Coincidence Degree ◽

Degree Theory ◽

Continuation Theorem ◽

Coincidence Degree Theory ◽

Sufficient Condition ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Multiple Periodic Solutions

We consider an impulsive periodic generalized Gause-type predator-prey model with nonmonotonic numerical responses. Using the continuation theorem of coincidence degree theory, we present an easily verifiable sufficient condition on the existence of multiple periodic solutions. As corollaries, some applications are listed. In particular, our results extend and improve some known criteria.

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