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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 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]

The traveling wave solutions of a reaction–diffusion model with general functional responses are considered in this paper. A necessary and sufficient condition is established for the existence of traveling wave solutions. Moreover, the minimum speed, asymptotic behavior and uniqueness of traveling wave solutions are also obtained in relation to the interaction parameters. [ABSTRACT FROM AUTHOR]

This paper deals with the unstirred chemostat model with crowding effects. The introduction of crowding effects makes the conservation law invalid, and the equations cannot be combined to eliminate one of the variables. Consequently, the usual reduction of the system to a competitive system of one order lower is lost. Thus the system with predation and competition is non-monotone, and the single population model cannot be reduced to a scalar system. First, the uniqueness and asymptotic behaviors of the semi-trivial solutions are established. Second, the existence and structure of coexistence solutions are given by the degree theory and bifurcation theory. It turns out that the positive bifurcation branch connects one semi-trivial solution branch with another. Finally, the stability and asymptotic behaviors of coexistence solutions are discussed in some cases. It is shown that crowding effects are sufficiently effective in the occurrence of coexisting, and overcrowding of a species has an inhibiting effect on itself. [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]