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Abstract: In this paper, we consider an ecosystem of phytoplankton–nutrient which comes from pelagic zone of the ocean, by using remote sensing. We are concerned with the pattern formation of the model. We first give the asymptotic behavior of positive solutions of parabolic dynamical system. Then, we discuss the stability of constant positive solution and the boundedness of stationary solutions. Finally, the non-existence and existence results of the pattern are investigated by considering some parameters, the result shows that under certain range of these parameters, the pattern of the model occurs. [Copyright &y& Elsevier]

Abstract: We consider a diffusive predator–prey model with Beddington–DeAngelis functional response under homogeneous Dirichlet boundary conditions. The effect of large k which represents the extent of mutual interference between predators is extensively studied. By making use of the fixed point index theory, we obtain a complete understanding of the existence, uniqueness and stability of positive steady-states when k is sufficiently large. Moreover, we present some numerical simulations that supplement the analytic results in one dimension. [Copyright &y& Elsevier]

Abstract: In this paper the existence and non-existence of traveling wave solutions are established for a bio-reactor model with stage-structure. The method to prove the existence of wave solutions is to construct an invariant cone of initial functions defined in a large but bounded domain, to apply a fixed point theorem on this cone and then extend to the unbounded spatial domain by a limiting argument. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the existence and stability of periodic traveling curved fronts for reaction–diffusion equations with time-periodic bistable nonlinearity in two-dimensional space. By constructing supersolution and subsolution, we prove the existence of periodic traveling wave fronts. Furthermore, we show that the front is globally stable. [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 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]

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]

Abstract: This paper is concerned with an autocatalysis model subject to no-flux boundary conditions. The existence of Hopf bifurcation are firstly obtained. Then by the center manifold theory and the normal form method, the direction of Hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions are established. On the other hand, the steady-state bifurcations from simple and double eigenvalues are intensively studied. The techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalues. Finally, some numerical simulations are shown to verify the analytical results. [Copyright &y& Elsevier]