Abstract
Abstract A stochastic mutualism model is proposed and investigated in this paper. We show that there is a unique solution to the model for any positive initial value. Moreover, we show that the solution is stochastically bounded, uniformly continuous and globally attractive. Under some conditions, we conclude that the stochastic model is stochastically permanent and persistent in mean. Finally, we introduce some figures to illustrate our main results.
Highlights
Population systems have long been an important theme in mathematical biology due to their universal existence and importance
Since stochastic differential equation ( ) describes population dynamics, it is necessary for the solution of the system to be positive and not to explode to infinity in a finite time
2.4 Global attractivity Here we show that the solution of ( ) is globally attractive
Summary
Population systems have long been an important theme in mathematical biology due to their universal existence and importance. The above model is not so good in describing the mutualism of two species (see [ ]). As a matter of fact, population systems are often subject to environmental noise, i.e., due to environmental fluctuations, parameters involved in population models are not absolute constants, and they may fluctuate around some average values Based on these factors, more and more people began to be concerned about stochastic population systems (see [ – ]). Since stochastic differential equation ( ) describes population dynamics, it is necessary for the solution of the system to be positive and not to explode to infinity in a finite time.
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