Stochastic delay foraging arena predator–prey system with Markov switching

Abstract

In this paper, we introduce white noise, telegraph noise and time delay to the two-dimensional foraging arena population system describing the prey and predator abundance. The aim is to find out how the interactions between white noise, telegraph noise and time delay affect the dynamics of the population system. Firstly, the existence of a global positive solution is verified. Then the long-time properties including the stochastically ultimate boundedness, extinction and some other asymptotic pathwise estimation of this population system are studied. Finally, the main results are illustrated by two examples.

Keywords:

• Stochastic predator–prey model
• Brownian motion
• Markov chain
• time delay
• ultimate boundedness

MATHEMATICS SUBJECT CLASSIFICATION:

• 37A50
• 76M35
• 92D25

Acknowledgments

The authors would like to thank the reviewers and the editors for their very professional comments and suggestions. The first two authors would like to thank the University of Strathclyde for awarding them the PhD studentships. The first author would also like to thank China Scholarship Council and the MASTS pooling initiative (The Marine Alliance for Science and Technology for Scotland) for their financial support. MASTS is funded by the Scottish Funding Council (grant reference HR09011). The third author would like to thank the Royal Society (WM160014, Royal Society Wolfson Research Merit Award) and the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship) for their financial support.

Disclosure statement

No potential conflict of interest was reported by the authors.