Chapter 2. Long-Time Behavior of Stochastic Models of Population Dynamics with Jumps

$39.50

Oleksandr Borysenko1 and Olga Borysenko2
1Taras Shevchenko National University of Kyiv, Ukraine
2National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine

Part of the book: Stochastic Processes: Fundamentals and Emerging Applications

Abstract

The results of the study of the non-autonomous stochastic models of population dynamics under the action of environmental noises such as white noise, a centered Poisson noise, and a non-centered Poisson noise are presented. We consider the non- autonomous logistic model with two types of stochastic disturbances, the stochastic two-species mutualism model, the non- autonomous stochastic predator-prey model with a modified version of Leslie-Gower term and Holling-type II functional response and the non-autonomous stochastic predator-prey model with Beddington-DeAngelis functional response. Theorems on existence and uniqueness of a positive global solution to the corresponding stochastic differential equations are presented. Sufficient conditions of stochastic ultimate boundedness, stochastic permanence, non-persistence in the mean, almost sure weak persistence, weak and strong persistence in the mean and extinction of the population in the considered stochastic models are proposed.

Keywords: logistic model, two-species mutualism model, stochastic predator-prey model,
global solution, stochastic ultimate boundedness, stochastic permanence, non-persistence
in the mean, weak persistence in the mean, strong persistence in the mean, extinction


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