The impact of memory effect on time-delay logistic systems driven by a class of non-Gaussian noise

This paper investigates the effects of different memory effects on the evolution of species population densities. A stochastic logistic model driven by non-Gaussian noise, with discrete and distributed time-delay, is considered. Taking memory intensity as a parameter, we investigated the evolution of population dynamics. In order to reduce the stochastic time delay equation to the average Itoˆ equation during its derivation, we adopted the stochastic averaging method and the generalized central manifold theory. It is used to capture the one-dimensional slow variables of a system near the critical state of the transition between two steady states during the evolution of the system. With the help of the one-dimensional Itoˆ equation governed by this slow variable, we perform dynamic analysis of the population progression. The results show a change in the population system from a single steady-state to a bi-periodic oscillation to a single steady-state again as memory intensity increased from weak to strong. Finally, numerical simulation verifies the rationality of the theoretical results. This general approach used in our research can also be extended to other fields for the study of similar problems, such as the analysis of the dynamics in network congestion control models. •The effect of memory effects on population dynamics is provided.•Analyzed the dynamic evolution of a single species population density under the coupling effect of distributed delay and generalized harmonic noise generated by the memory kernel.•We consider a logistic system with distributed delay and propose an efficient method of transforming it into a logistic system with constant delay for discussion.•The generalized harmonic noise considered in the system is the internal noise generated by the distributed delay kernel function.•The logistic model driven by harmonic noise proposed in the paper gives a more realistic response to the evolution of population density.