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In this work, we study a spatially heterogeneous predator–prey model where the interaction is governed by a Holling II functional response. The existence of coexistence states of the two species by using the bifurcation theory is discussed. As a result, it is shown that the degeneracy for the prey and the predator has distinctly different effects on the coexistence states of the two species when the intrinsic growth rate of the prey is above a certain critical value. We also study the asymptotic behavior of positive stationary solutions as the intrinsic growth rate of the predator tends to a certain critical value. [ABSTRACT FROM AUTHOR]

This paper is concerned with positive solutions of a predator–prey model with cross-diffusion. By virtue of the Leray–Schauder degree theory and the bifurcation theory, some sufficient conditions for the existence of positive solutions of the system are established. In particular, we derive the multiplicity results when some parameters are suitably large. Moreover, the existence and stability of nonconstant solutions are studied by the techniques of space decomposition and the implicit function theorem. Finally, the uniqueness of positive solution is studied when the spatial dimension is one. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with a -dimensional competition model between plasmid-bearing and plasmid-free organisms in the unstirred chemostat, which incorporates the Beddington–DeAngelis functional response and inhibitor. By the application of degree theory in cones, bifurcation theory and perturbation technique, we obtain the uniqueness, multiplicity and stability of the coexistence solutions of the steady-state system when the effects of the inhibitor is large enough. [ABSTRACT FROM AUTHOR]

We study a reaction-diffusion system of the unstirred chemostat model with toxins in the N-dimensional case. Firstly, some sufficient conditions for the existence of positive steady-state solutions are obtained by means of the fixed point index theory. Secondly, the local structure of positive steady-state solutions and their stability are discussed by the bifurcation theory. Lastly, the effect of the toxins is investigated by the perturbation technique. The results show that when the parameter β, which measures the effect of toxins, is large enough, this model has no coexistence solution pHrovided that the maximal growth rate b of the species v is greater than σ ˆ 1 1 − k, and all positive solutions of this model are controlled by a limit system provided that b belongs to the interval (σ 1 1 − k, σ ˆ 1 1 − k). [ABSTRACT FROM AUTHOR]