Mean-square dissipativity of numerical methods for a class of resource-competition models with fractional Brownian motion

Abstract

ABSTRACTThe concept of dissipativity in dynamical systems generalizes the idea of a Lyapunov stability. In this paper the dissipativity is used to designate that the system possesses a bounded absorbing set. Specifically, this paper studies mean-square dissipativity of two numerical methods for a class of resource-competition models with fractional Brownian motion (fBm). Some conditions under which the underlying systems are mean-square dissipative are established. It is shown that the mean-square dissipativity is preserved by the split-step θ-method (SSθ) under some constraints, while the split-step backward Euler method (SSBE) could inherit mean-square dissipativity without any restriction on stepsize. The results of this study indicate that the split-step backward Euler method (SSBE) outperforms the split-step θ-method (SSθ) in terms of mean-square dissipativity. Finally, an example is given to illustrate the effectiveness of the results.