Dynamics of a single species under periodic habitat fluctuations and Allee effect - Rizaner, Fatma Bayramoglu; Rogovchenko, Svitlana P. (Pergamon-Elsevier Science Ltd, 2012)
The dynamics of a single species that goes extinct when rare is described by a nonlinear differential equation N' = rN(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. (C) 2011 Elsevier Ltd. All rights reserved.
Keywords: Nonlinear differential equation; Allee effect; Periodic solutions; Stability; Blow up; Existence; Positive solutions