Abstract
Taking white noise into account, a stochastic nonautonomous logistic model is proposed and investigated. Sufficient conditions for extinction, nonpersistence in the mean, weak persistence, stochastic permanence, and global asymptotic stability are established. Moreover, the threshold between weak persistence and extinction is obtained. Finally, we introduce some numerical simulink graphics to illustrate our main results.
Highlights
IntroductionThe classical nonautonomous logistic equation can be expressed as follows: dx t dt x t a t −b t x t
The classical nonautonomous logistic equation can be expressed as follows: dx t dt x t a t −b t x t .1.1 for t ≥ 0 with initial value x 0 x0 > 0, x t is the population size at time t, a t denotes the rate of growth, and a t /b t stands for the carrying capacity at time t
Population dynamics is inevitably affected by environmental noise which is an important component in an ecosystem see e.g., 6–9
Summary
The classical nonautonomous logistic equation can be expressed as follows: dx t dt x t a t −b t x t. The deterministic models assume that parameters in the systems are all deterministic irrespective environmental fluctuations. By the wellknown central limit theorem, the error term follows a normal distribution and is sometimes dependent on how much the the current population sizes differ from the equilibrium state see, e.g., 11–13. Owing to the model 1.4 describes a population dynamics, it is necessary to investigate the survival of the logistic population which involves extinction, persistence, and global asymptotical stability see, e.g., 14–16. It follows from the above definitions that stochastic permanence implies stochastic weak persistence, extinction means stochastic nonpersistence in the mean. The last section gives the conclusions and future directions of the research
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