Bifurcation structure of coexistence states for a prey–predator model with large population flux by attractive transition

This paper is concerned with a prey–predator model with population flux by attractive transition. Our previous paper (Oeda and Kuto, 2018, Nonlinear Anal. RWA, 44, 589–615) obtained a bifurcation branch (connected set) of coexistence steady states which connects two semitrivial solutions. In Oeda and Kuto (2018, Nonlinear Anal. RWA, 44, 589–615), we also showed that any positive steady-state approaches a positive solution of either of two limiting systems, and moreover, one of the limiting systems is an equal diffusive competition model. This paper obtains the bifurcation structure of positive solutions to the other limiting system. Moreover, this paper implies that the global bifurcation branch of coexistence states consists of two parts, one of which is a simple curve running in a tubular domain near the set of positive solutions to the equal diffusive competition model, the other of which is a connected set characterized by positive solutions to the other limiting system.

Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms

<p style='text-indent:20px;'>In this paper, we present results about existence and non-existence of coexistence states for a reaction-diffusion predator-prey model with the two species living in a bounded region with inhospitable boundary and Holling type II functional response. The predator is a specialist and presents self-diffusion and cross-diffusion behavior. We show the existence of coexistence states by combining global bifurcation theory with the method of sub- and supersolutions.</p>

Nonlocal singular elliptic system arising from the amoeba–bacteria population dynamics

In this paper, we prove the existence of coexistence states for a nonlocal singular elliptic system that arises from the interaction between amoeba and bacteria populations. Our study is based on fixed point arguments using a version of the Bolzano’s theorem, for which we will first analyze a local system by bifurcation theory. Moreover, we study the behavior of the coexistence region obtained and we interpret our results according to the growth rate of both species.

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

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.