Existence for positive steady states of an eco-epidemiological model

An eco-epidemiological model with an epidemic in the predator and with a Holling type Ⅱ function is considered.A system with diffusion under the homogeneous Neumann boundary condition is studied.The existence for a positive solution of the corresponding steady state problem is mainly discussed.First,a prior estimates（positive upper and lower bounds） of the positive steady states of the reaction-diffusion system is given by the maximum principle and the Harnack inequation.Then,the non-existence of non-constant positive steady states by using the energy method is given.Finally,the existence of non-constant positive steady states is obtained by using the topological degree.

Positive Steady States of a Competitor-Competitor-Mutualist Model

In this paper we deal with the positive steady states of a Competitor-Competitor-Mutualist modelwith diffusion and homogeneous Dirichlet boundary conditions.We first give the necessary conditions,and thenestablish the sufficient conditions for the existence of positive steady states.

Non-constant Positive Steady States of the Epidemic Model with Non-monotonic Incidence Rate

In this paper, the positive steady states of the epidemic model with non-monotonic incidence rate are considered. Firstly, it is proved that the unique positive constant steady state is stable for the ODE system and the PDE system. Secondly, a priori estimate of positive steady states is given, and the non-existence of non-constant positive steady states is established by using Poincare inequality and Young inequality. Finally,the existence and bifurcation of non-constant positive steady states are studied by using the degree theory and the global bifurcation theorem.

Stationary Patterns of a Ratio-dependent Prey-predator Model with Cross-diffusion

This paper is concerned with a ratio-dependent predator-prey system with diffusion and cross- diffusion in a bounded domain with no flux boundary condition. We show that under certain hypotheses, the cross-diffusion can create non-constant positive steady states even though the corresponding model without cross-diffusion fails.

Global dynamics of a nonlocal population model with age structure in a bounded domain:A non-monotone case

We study the global dynamics of a nonlocal population model with age structure in a bounded domain. We mainly concern with the case where the birth rate decreases as the mature population size become large. The analysis is rather subtle and it is inadequate to apply the powerful theory of monotone dynamical systems. By using the method of super-sub solutions, combined with the careful analysis of the kernel function in the nonlocal term, we prove nonexistence, existence and uniqueness of positive steady states of the model.Moreover, due to the mature individuals do not diffuse, the solution semiflow to the model is not compact. To overcome the difficulty of non-compactness in describing the global asymptotic stability of the unique positive steady state, we first establish an appropriate comparison principle. With the help of the comparison principle,we can employ the theory of dissipative systems to obtain the global asymptotic stability of the unique positive steady state. The main results are illustrated with the nonlocal Nicholson's blowflies equation and the nonlocal Mackey-Glass equation.

Turing patterns in a reaction-diffusion epidemic model

In this paper, we investigate the spatiotemporal dynamics of a reactio^diffusion epi- demic model with zero-flux boundary conditions. The value of our study lies in two aspects： mathematically, by using maximum principle and the linearized stability theory, a priori estimates of the steady state system and the local asymptotic stability of positive constant solution are given. By using the implicit function theorem, the exis- tence and nonexistence of nonconstant positive steady states are shown. Applying the bifurcation theory, the global bifurcation structure of nonconstant positive steady states is established. Epidemiologically, through numerical simulations, under the conditions of the existence of nonconstant positive steady states, we find that the smaller the space, the easier the pattern formation; the bigger the diffusion, the easier the pattern formation. These results are beneficial to disease control, that is, we must do our best to control the diffusion of the infectious to avoid disease outbreak.