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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: A diffusive predator–prey model with predator competition is considered under Dirichlet boundary conditions. Some existence and non-existence results are firstly obtained. Then by investigating the bifurcation of positive solutions, the multiplicity of positive solutions is established for suitably large . Furthermore, by meticulously analyzing the asymptotic behaviors of positive solutions when goes to , we find that there is at most a positive solution for any when is sufficiently large. At last, some numerical simulations are presented to supplement the analytic results in one dimension. [Copyright &y& Elsevier]

Abstract: This paper deals with an unstirred chemostat model with the Beddington–DeAngelis functional response. First, a sufficient condition to the existence of positive steady state solutions is established. Second, the effect of the parameter in the Beddington–DeAngelis functional response which models mutual interference between species is considered. The result shows that if is sufficiently large, the solution of this model is determined by a limiting equation. The main tool used here includes the fixed point index theory, the perturbation technique and the bifurcation theory. [Copyright &y& Elsevier]

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]

Abstract: In this paper, we consider positive solutions of a predator–prey model with Beddington–DeAngelis functional response. First, we establish the existence of positive solutions and present some sufficient and necessary conditions mainly using the standard fixed point index theory in cone. Second, the multiplicity, uniqueness and stability of positive solutions are obtained by means of bifurcation theory, perturbation theory of eigenvalues and the fixed point index theory. The parameters and play very important roles in deciding the number of positive solutions, which are showed explicitly in this paper. [Copyright &y& Elsevier]

Abstract We establish a reaction–diffusion ecological–mathematical model with predator–prey dependent functional response and the Neumann boundary conditions. First, we study the bifurcation solution emitting from the unique positive constant solution by considering the local bifurcation. Finally, we mainly analyze the coexistence of the prey and predator with the help of the fixed point index theory, where we use a different approach in calculating one of the fixed point indices. [ABSTRACT FROM AUTHOR]

In this paper, we consider a reaction–diffusion system known as the Brusselator model with homogenous Neumann boundary conditions. The model is a system of two differential equations which describes a type of complex chemical or biochemical reaction system. We mainly study the coexistence of non-constant positive solutions to the model. [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]