Unsustainable zip-bifurcation in a predator–prey model involving discrete delay

Abstract

A third-order system of ordinary differential equations, modelling two predators competing for a single prey species, is analysed in this paper. A delay term modelling the delayed logistic growth of the prey is included. Fixed points of the system are identified, and a linearized stability analysis is carried out. For some parameter regime, there exists a continuum of equilibria and these equilibria may undergo a zip bifurcation. The main results presented herein are that this zip bifurcation is 'unsustainable' for certain ranges of values of the time-delay parameter. Finally, spatial diffusion is incorporated in the delay differential equation model, and it is shown that the zip bifurcation remains unsustainable.

Zip and velcro bifurcations in competition models in ecology and economics

“…For additional references and examples in more general contexts and the corresponding analysis, see Section 7.4 of Farkas' book [14]. More recent work by Ferreira and Rao deals with zipbifurcation in a predator-prey model with diffusion [17], and in systems involving discrete delay [18,19] and crossdiffusion [20]. Zip bifurcations are also discussed by Escobar-Callejas et al [21], and by Echeverri et al [22].…”

A model of competing species that exhibits zip bifurcation

“…Farkas has proven (see [8], [9], [10]) that the phenomenon called zip bifurcation is general, in the sense that it is present in all tri-dimensional models that satisfy certain theoretical conditions. From Farkas' research, the zip bifurcation phenomenon has been studied from several points of view: particularization of Farkas' general system for models including numerical experiments [6]; 3-dimensional competition models with generalized Holling type III functional response for the predators [19]; generalization to four-dimensional systems coming from economy and politology [2] and from population dynamics [11]; generalization to n−dimensional ordinary differential systems with and without diffusion [12], [13], [14]; tri-dimensional predator-prey model with a delay term for the growth of the prey [15]; and non-smooth systems that exhibit a corresponding non-smooth zip bifurcation [1].…”

“…It is important to remark that the most usual functions to build equations that model competing populations species exhibiting zip bifurcation are of two kinds (see [1], [2], [6]- [15], [19]): The logistic function: G(S) = γS 1 − S K for the per-capita growth rate of the prey, where S is the prey density and γ > 0 denotes the maximum growth rate, assuming that the carrying capacity of the habitat K > 0, K ≥ S ≥ 0 and the Holling-family functions: P (S) = m S n a n +S n for the functional response of the predators, where n ≥ 1 is an arbitrary integer that defines the type of Holling function, m > 0 is the supremum of P , and a > 0 is the half-saturation constant. The logistic function describes the situation when the increasing rate is slow at the beginning, accelerates in the middle and slows up at the end.…”