Global stability and sliding bifurcations of a non-smooth Gause predator-prey system

A non-smooth Gause predator–prey model with a constant refuge is proposed and analyzed. Firstly, the existence and stability of regular, virtual, pseudo-equilibria and tangent points are addressed. Then the relations between the existence of a regular equilibrium and a pseudo-equilibrium are studied, and the results indicate that the two types of equilibria cannot coexist. The sufficient and necessary conditions for the global stability of limit cycle, sliding touching cycle, canard cycle, focus point and pseudo-equilibrium are provided by using qualitative analysis techniques of non-smooth Filippov dynamic systems. Furthermore , sliding bifurcations related to boundary node (focus) and touching bifurcations were investigated by employing theoretical and numerical techniques. Finally, we compare our results with previous studies on a non-smooth Gause predator–prey model without involving a carrying capacity for the prey population, and some biological implications are discussed. In order to search for evidence of stable population limit cycles in a microcosm experiment, Gause [1,2] carried out three experiments after Lotka and Volterra formulated their theoretical equation on predator–prey population dynamics. In none of these experiments were the population dynamics consistent with the Lotka–Volterra neutrally stable limit cycles [3]. The experiments, conducted in either a homogeneous environment or in heterogeneous environments with a refuge for prey, provided strong evidence that population dynamics oscillated periodically and were independent of the numbers in the initial populations. Gause's experiments confirmed that a certain threshold concentration of prey cannot be destroyed by predators. This was because the prey could avoid the predator via a habitat shift if they moved to the refuge when their density was low, which indicated that the prey individuals were effectively protected in a refuge when at low concentrations [4]. The prey population could reappear and become accessible to predators once again when their density increased and exceeded the threshold density [3]. In such a case, the prey and predator can coexist and oscillate periodically. In order to take into account the observed experimental results theoretically, Gause et al. [2] extended the classical Lotka–Volterra model by using a piecewise saturating function to replace the linear consumption rate [5]. Denote HðZÞ ¼ x À R C with Z ¼ ðx; yÞ T 2 R 2 þ where R C describes the critical prey population threshold, and then the parameter can be defined as follows 0096-3003/$-see front matter Ó 2013 Elsevier Inc. All rights reserved.