Permanence and extinction in non-autonomous logistic system with random perturbation and feedback control

Hongxiao Hu,Ling Zhu

doi:10.1186/s13662-016-0904-5

Abstract

In this paper, we study a stochastic non-autonomous logistic system with feedback control. Sufficient conditions for stochastic asymptotically bounded, extinction, non-persistence in the mean, weak persistence, and persistence in the mean are established. The critical number between weak persistence and extinction is obtained. A very important fact is found in our results, that is, the feedback control is harmless to the permanence of species under the randomized environment.

Highlights

The classical non-autonomous logistic equation can be expressed as follows: x(t) = x(t) r(t) – a(t)x(t), ( . )where x(t) denotes the population size at time t, r(t) is the intrinsic growth rate and r(t)/a(t) is the carrying capacity at time t
Sometimes we should search for certain schemes to ensure the system still have the same dynamic property as system ( . ) under the same conditions
In [ ], Gopalsamy and Weng motivated by control theory and studied the global asymptotic stability of positive equilibrium of a regulated logistic growth with a delay in the state feedback of the control model

Summary

Introduction

Where x(t) denotes the population size at time t, r(t) is the intrinsic growth rate and r(t)/a(t) is the carrying capacity at time t It has been studied extensively and many important results on the global dynamics of solutions have been found (see [ – ] and references therein). In many works (see [ – ]), the authors obtained the result that the feedback controls are harmless to the permanence for the deterministic systems. Motivated by the above analysis, we will study the following non-autonomous randomized logistic system with feedback control: dx(t) = x(t)(r(t) – a(t)x(t) – c(t)u(t)) dt + σ (t)x(t) dBt, du(t) = (–e(t)u(t) + f (t)x(t)) dt, where r(t) is a continuous bounded function on [ , +∞) and a(t), c(t), σ (t), e(t), and f (t) are nonnegative continuous bounded function on [ , +∞). ). We will find that, in our results, the feedback control is harmless to the permanence of species with stochastic perturbation

Preliminaries

Future directions