Zip bifurcation in a competitive system with diffusion

Abstract

We study the existence of a global attractor in a reaction-diffusion system which describes the interaction among n + 1 species, amongst which n species of predators compete for a single prey. Also, we prove the persistence of the zip bifurcation phenomenon for the reaction-diffusion system, which was introduced by Farkas [5] for a three dimensional ODE prey-predator system.

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

“…The following result may be found in [13]. Hence, the polynomial P k (ν) is stable independently of the diffusion matrix D = diag(δ 0 , δ 1 , δ 2 ).…”

“…In [10,13] it was concluded that in a delay-free system the sustainability of the zip bifurcation is achieved in the presence of diffusion. The above result is surprising since it shows that the zip bifurcation phenomenon cannot be sustained in the presence of a discrete delay in system (3.1).…”

“…Positivity and global existence of the solutions for (4.1) and (4.2), when τ = 0, have been studied in [13]. Here, K represents the carrying capacity of the environment that depends on the state x.…”

“…Note that in the case of no delay the corresponding PDE model with different diffusion coefficients was first studied in [13], and it is described by the system…”

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].…”