Stationary Distribution and Periodic Solution of Stochastic Toxin-Producing Phytoplankton–Zooplankton Systems

Chunjin Wei,Yingjie Fu

doi:10.1155/2020/4627571

Abstract

In this paper, we investigate the dynamics of autonomous and nonautonomous stochastic toxin-producing phytoplankton–zooplankton system. For the autonomous system, we establish the sufficient conditions for the existence of the globally positive solution as well as the solution of population extinction and persistence in the mean. Furthermore, by constructing some suitable Lyapunov functions, we also prove that there exists a single stationary distribution which is ergodic, what is more important is that Lyapunov function does not depend on existence and stability of equilibrium. For the nonautonomous periodic system, we prove that there exists at least one nontrivial positive periodic solution according to the theory of Khasminskii. Finally, some numerical simulations are introduced to illustrate our theoretical results. The results show that weaker white noise and/or toxicity will strengthen the stability of system, while stronger white noise and/or toxicity will result in the extinction of one or two populations.

Highlights

As well known, mathematical models describing the plankton dynamics have played an important role in understanding the various mechanisms involved in toxin-producing phytoplankton. ere are many scienti c works have been carried out to investigate the e ects of toxin-producing phytoplankton on plankton ecosystems [1,2,3,4,5,6,7]
− −g, where ( ) and ( ) denote the density of toxin-producing plankton (TPP) population and the zooplankton population at time, respectively, subject to the nonnegative initial condition (0) = 0 ≥ 0 and (0) = 0 ≥ 0. and represent the intrinsic growth rate and the environmental carrying capacity of TPP population, respectively. is the rate of predation of zooplankton on TPP population, is the ratio of biomass consumed by zooplankton for its growth, and denotes the mortality rate of zooplankton due to nature death, denotes the rate of toxin liberation by TPP population. represents the predational response function and g describes the distribution of toxic substances
We intend to study toxin-producing phytoplankton–zooplankton model with environmental uctuations, and we extend this model into a nonautonomous stochastic model by taking into account seasonal variation

Summary

Introduction

Mathematical models describing the plankton dynamics have played an important role in understanding the various mechanisms involved in toxin-producing phytoplankton. ere are many scienti c works have been carried out to investigate the e ects of toxin-producing phytoplankton on plankton ecosystems [1,2,3,4,5,6,7]. Ere are many scienti c works have been carried out to investigate the e ects of toxin-producing phytoplankton on plankton ecosystems [1,2,3,4,5,6,7]. E obtained result indicates that there is a threshold limit of toxin liberation by the phytoplankton species below which the system does not have any excitable nature and above which the system shows excitability These important and useful works on deterministic phytoplankton–zooplankton model provide a great insight into. To the best of our knowledge, there is little work on the existence of stochastic periodic solution for nonautonomous toxin-producing phytoplankton zooplankton model. We intend to study toxin-producing phytoplankton–zooplankton model with environmental uctuations, and we extend this model into a nonautonomous stochastic model by taking into account seasonal variation .

Existence and Uniqueness of the Global Positive Solution

Extinction and Persistence in Mean

Stationary Distribution and Ergodicity

The Existence of Periodic Solution of Nonautonomous System

Numerical Simulation