Coexistence states in a cross-diffusion system of a predator–prey model with predator satiation term

Abstract

In this paper, we study the existence and non-existence of coexistence states for a cross-diffusion system arising from a prey–predator model with a predator satiation term. We use mainly bifurcation methods and a priori bounds to obtain our results. This leads us to study the coexistence region and compare our results with the classical linear diffusion predator–prey model. Our results suggest that when there is no abundance of prey, the predator needs to be a good hunter to survive.

Unilateral global bifurcation for a class of quasilinear elliptic systems and applications

“…We also refer to [6] where the authors developed a bifurcation result to analyze a predator-prey system which is a particular case of (1). Actually, here we extend this result to the general system (1).…”

“…with some boundary conditions, specially motivated by the well-known model proposed by Shigesada, Kawasaki and Teramoto [36]. Moreover, several techniques have been used to deal with (6), for instance fixed point index [32,33,34,27,17], sub-supersolution methods [30] and bifurcation theory [13,11,20,29,21,41,19] and references therein.…”

“…to deal with (6), for instance fixed point index [32,33,34,27,17], sub-supersolution methods [30] and bifurcation theory [13,11,20,29,21,41,19] and references therein.…”

“…It should be noted that a standard approach to deal with (6) is to apply the change of variables U = ϕ(u, v) and V = ψ(u, v), transforming (6) into a semilinear system, decoupled in the diffusion, where one can apply the techniques which worked for such systems, for instance, the bifurcation theorems of [23]. However, (1) can not be written as (6) because there does not exist an immediate change variable that transforms (1) into a semilinear system.…”

Radially symmetric solutions for a Keller-Segel system with flux limitation and nonlinear diffusion

“…There exists an extensive mathematical literature studying such systems, see for instance the surveys Horstmann [28], [29], Bellomo et al [4], Hillen and Painter [27] and references therein for more details concerning previous results of the time dependent problem. Elliptic systems of chemotaxis have also captured the attention of the mathematical community, see for instance Boccardo and Orsina [11], Cintra, Morales-Rodrigo and Suárez [14] among others. Logistic terms have been included in the system to limit the growth of the biological species, among a broad literature on parabolic-elliptic chemotaxis elliptic systems, see for instance Tello and Winkler [41], Galakhov, Tello and Salieva [21] for bounded domains and Salako and Shen [38], [39] in the whole N -dimensional space.…”