On a predator-prey system of Holling type

Abstract

Abstract. We consider the predator-prey system with a fairly general functional response of Holling type and give a necessary and sufficient condition under which this system has exactly one stable limit cycle. Our result extends previous results and is an answer to a conjecture which was recently presented by Sugie, Miyamoto and Morino.

Global stability of the predator-prey model with a sigmoid functional response

“…If E * is locally stable, then it is globally stable. The above properties have also been confirmed to be true for large range of n for the general Holling type functional response (3) [4,12,19,20].…”

On the Dynamics of a General Predator-Prey System

“…We discuss the uniqueness of limit cycles of the general system (1). Our criterion improves and partially generalizes those of (Hasik, 2000;Kuang and Freedman, 1988;Sugie et al, 1997). We first start with the most famous uniqueness result which was the motivation for several authors and criteria.…”

“…Then the system (3) has exactly one limit cycle which is globally asymptotically stable. In view of Theorem 4.1 and those of (Sugie et al, 1997;Hasik, 2000), we give the following uniqueness theorem for the limit cycles of our general system (1).…”

“…The functions g(x) and p(x) denote the functional response of the prey and predator, respectively and satisfy the assumptions p(0) = 0 and g(0)>0, for all x>0. There have been considerable interests in the dynamics of the predatorprey models of some special cases of (1) by several authors (Attili, 2001;Attili and Mallak, 2006;Xiao and Ruan, 2001;Hesaaraki and Moghadas, 2001; Kar and Matsuda, 2007;Moghadas and Corbett, 2008;Moghadas and Alexander, 2005;Sugie et al, 1997;Ruan and Xiao, 2001). Hasik (2000); Sesay et al (2010); Moghadas and Alexander (2006); Saha and Bandyopadhyay (2005) and Tao et al (2011) the authors considered Eq.…”

Stability and bifurcation in an age-structured model with stocking rate and time delays

“…Mathematical model, that describes this interaction, could help to understand its dynamic processes and to make practical predictions. Such model could be various type of differential equations: ordinary differential equations [11,23,34], delay differential equations [3,24,31,37], reaction-diffusion equations [4,7,32,36], first order partial differential equations with age-structure [6,13,26,30,35] and so on. In this paper, we will consider the following model, which is formulated from the one in [26] by incorporating the fertility rate τ 1 into the model,…”