Considering various factors are stochastic rather than deterministic in the
evolution of populations growth, in this paper, we propose a single predator
multiple prey stochastic model with seasonal variation. By using the method
of solving an explicit solution, the existence of global positive solution
of this model are obtained. The method is more convenient than Lyapunov
analysis method for some population models. Moreover, the stochastically
ultimate boundedness are considered by using the comparison theorem of
stochastic differential equation. Further, some sufficient conditions for the
extinction and strong persistence in the mean of populations are discussed,
respectively. In addition, by constructing some suitable Lyapunov functions,
we show that this model admits at least one periodic solution. Finally,
numerical simulations clearly illustrate the main theoretical results and
the effects of white noise and seasonal variation for the persistence and
extinction of populations.
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