Bifurcation and Stability in a Delayed Predator–Prey Model with Mixed Functional Responses

Abstract

The model analyzed in this paper is based on the model set forth by Aziz Alaoui et al. [Aziz Alaoui & Daher Okiye, 2003;Nindjin et al., 2006] with time delay, which describes the competition between the predator and prey. This model incorporates a modified version of the Leslie-Gower functional response as well as that of Beddington-DeAngelis. In this paper, we consider the model with one delay consisting of a unique nontrivial equilibrium E * and three others which are trivial. Their dynamics are studied in terms of local and global stabilities and of the description of Hopf bifurcation at E * . At the third trivial equilibrium, the existence of the Hopf bifurcation is proven as the delay (taken as a parameter of bifurcation) that crosses some critical values.

A survey on constructing Lyapunov functions for reaction-diffusion systems with delay and their application in biology

“…This approach is based essentially on finding the original Lyapunov function for the corresponding reactiondiffusion system and generalizing it to delay cases by adding some spatio-temporal integral terms. This method has been applied to several delayed reaction-diffusion biological models, for example, see [14][15][16][17][18][19][20][21][22][23][24].…”