The stochastic interactions between predator and prey under Markovian switching: competitive interaction between multiple prey

Volume 10, Issue 11, pp 5622--5645 http://dx.doi.org/10.22436/jnsa.010.11.03

Publication Date: November 10, 2017 Submission Date: April 15, 2017

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Authors

Yanqing Li - College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, P. R. China. Long Zhang - College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, P. R. China.

Abstract

In this paper, a class of predator-prey model with prey competition is proposed, in which the interactions of predation between predator and prey are randomised and subsequently evaluated under Markovian switching. By constructing appropriate Lyapunov functions and applying various analytical methods, sufficient conditions for the existence of unique global positive solution, stochastic permanence and mean extinction are established. In the permanence case, we also estimate the superior and inferior limits of the sample path in a time-averaged Markov decision. We conclude that the interactions between predator and two prey, two competitive prey themselves and the dynamical properties of switching subsystems are not only dependent on subsystem coefficients but also on the transition probability of the Markov chain (switching from one state to another). Specific examples and numerical simulations are provided to demonstrate our theoretical results.

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Keywords

• Random selection
• competition between prey
• Markovian switching
• stochastic permanence
• extinct in mean

MSC

•  92D25
•  34D20
•  34D10

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