Allee effects on population dynamics in continuous (overlapping) case H. Merdan * , O. Duman, O ¨ . Akın, C. C ¸ elik TOBB University of Economics and Technology, Faculty of Arts and Sciences, Department of Mathematics, So ¨g ˘u ¨ to ¨ zu ¨ 06530, Ankara, Turkey Accepted 20 June 2007 Abstract This paper presents the stability analysis of equilibrium points of a continuous population dynamics with delay under the Allee effect which occurs at low population density. The mathematical results and numerical simulations show the stabilizing role of the Allee effects on the stability of the equilibrium point of this population dynamics. Ó 2007 Elsevier Ltd. All rights reserved. 1. Introduction It is well known that population dynamical models are the prototypes of nonlinear systems. These models may show complex dynamics including periodic behavior, limit cycles and chaos [4–10,12–17]. If we want to represent the sexual reproduction, we have to consider genetical relations on the models. It is also known that sexual reproduction can sta- bilize the population dynamics generally; by leading to the coexistence of different genotypes and decreasing (in fact diminishing) fluctuations in the density [5]. The first step for the sexual reproduction is the opposite of multiplication because of the necessity of the fusion gametes. It is well recognized that individuals of many species can benefit from the presence of conspecific. This concept is broadly referred as Allee effect [1]. The Allee effect can be regarded not only as a suite of problems associated with rarity, but also as the basis of animal sociality [14]. Allee effects play an important role in dictating animal mating systems (see, for instance, [1,4–6,8,10,11,14,15,17]). The stability analysis is a very important research topic in many areas including population dynamics (see, for instance, [2–4,7–10,13,15–17]). In this work, we concentrate on the Allee effect that leads to a different and dynamically important consequence of sexual reproduction. The purpose of this paper is to study the general non-linear, delay dif- ferential equation of the following form: dN dt ¼ kNf ðN ðt  T ÞÞ ¼: F ðN ðtÞ; N ðt  T ÞÞ ð1Þ 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.062 * Corresponding author. E-mail addresses: merdan@etu.edu.tr (H. Merdan), oduman@etu.edu.tr (O. Duman), omerakin@etu.edu.tr (O ¨ . Akın), canan.celik@etu.edu.tr (C. C ¸ elik). Chaos, Solitons and Fractals 39 (2009) 1994–2001 www.elsevier.com/locate/chaos