Dynamics of a single species under periodic habitat fluctuations and Allee effect

Abstract

The dynamics of a single species that goes extinct when rare is described by a nonlinear differential equation N ′ = r N ( 1 − N K ) ( N K − A K ) , where a parameter A ( 0 < A < K ) is associated with the Allee effect, r is the intrinsic growth rate and K is the carrying capacity of the environment. The purpose of this paper is to study the existence of periodic solutions and their stability properties assuming that r , A and K are continuous T -periodic functions. Using rather elementary techniques, we completely describe population dynamics analyzing influences of both strong ( A > 0 ) and weak ( A < 0 ) Allee effects. Thus, we answer questions regarding the location of positive periodic solutions and their stability complementing the research in a recent paper by Padhi et al. [Seshadev Padhi, P.D.N. Srinivasu, G. Kiran Kumar, Periodic solutions for an equation governing dynamics of a renewable resource subjected to Allee effects, Nonlinear Anal. RWA 11 (2010) 2610–2618]. Bounds for periodic solutions and estimates for backward blow-up times are also established. Furthermore, we demonstrate advantages of our approach on a simple example to which the results in the cited paper fail to apply.