Research Article Received 15 November 2011 Published online 19 April 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/mma.2509 MOS subject classification: 92D25; 34C23; 34C60; 34D23; 37G15 Limit cycles in a Gause-type predator–prey model with sigmoid functional response and weak Allee effect on prey Eduardo González-Olivares * † and Alejandro Rojas-Palma Communicated by M. A Lachowicz The goal of this work is to examine the global behavior of a Gause-type predator–prey model in which two aspects have been taken into account: (i) the functional response is Holling type III; and (ii) the prey growth is affected by a weak Allee effect. Here, it is proved that the origin of the system is a saddle point and the existence of two limit cycles surround a stable positive equilibrium point: the innermost unstable and the outermost stable, just like with the strong Allee effect. Then, for determined parameter constraints, the trajectories can have different !  limit sets. The coexistence of a stable limit cycle and a stable positive equilibrium point is an important fact for ecologists to be aware of the kind of bistability shown here. So, these models are undoubtedly rather sensitive to disturbances and require careful management in applied contexts of conservation and fisheries. Copyright © 2012 John Wiley & Sons, Ltd. Keywords: Allee effect; sigmoid functional response; predator–prey models; limit cycle; bifurcation 1. Introduction A continuous-time predator–prey model of Gause type is analyzed in this work, modifying the Yodzis model [1], in which we assume that: (1) The prey growth is affected by a weak Allee effect [2, 3], and (2) The functional response or consumption rate is Holling type III, or sigmoid, or S-shaped [1, 4–7]. It is well known that Gause-type predator–prey models [8, 9] are described by the following autonomous bidimensional differential equations system: X : ( dx dt D xg .x/  h .x/ y dy dt D ..x/  c/ y, (1) where x D x.t/ and y D y.t/ indicate the population size for t > 0, respectively; functions g.x/, h.x/, and .x/ represent the prey growth rate, the functional response or consumption rate function of the predator, and the numerical response, respectively. They have appropriate properties to satisfy initial-value problems [10, 11]. For the Gause-type predation model (1), properties establishing stability, periodic orbit, and bifurcations [5,6] have been shown. For instance, global stability of the unique positive equilibrium point is proved in [7], the uniqueness of limit cycles is demonstrated in [11–13], and conditions for the nonexistence of limit cycles are given in [5, 11]. Necessary and sufficient conditions for the uniqueness of limit cycles of the system (1) have been given in [6], for the general sigmoid functional response described by h.x/ D x n x n Ca n , for a > 0, n > 1, and n 2 N. However, the determination of the quantity of limit cycles that can be originated throughout the bifurcation of a center focus [14] is a problem that has not been generally studied. This problem is related to the well-known Hilbert 16th problem for polynomial systems Grupo de Ecología Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Chile *Correspondence to: Eduardo González-Olivares, Grupo de Ecología Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Chile. † E-mail: ejgonzal@ucv.cl Copyright © 2012 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2012, 35 963–975 963