On multiplicity and stability of positive solutions of a diffusive prey-predator model

In the paper, we consider the following diffusive prey-predator model:

     

Positive solutions of a prey-predator model with predator saturation and competition

In this paper, we study the existence, multiplicity, bifurcation and stability of positive solutions to a prey-predator model with predator saturation and competition

     

A simplified stationary energy-transport model with temperature-dependent conductivity

A simplified stationary energy-transport model in semiconductors is studied while the conductivity depends on both electron density and temperature. The existence and uniqueness of classical solutions to the model is proved for small variations of electron density and temperature.     

The stationary solution of a one-dimensional bipolar quantum hydrodynamic model

In this paper, we consider the existence and uniqueness of stationary solution to the bipolar quantum hydrodynamic model in one dimensional space with general non-constant doping profile. The existence of the stationary solution is proved by Leray-Schauder fixed-point theorem and a crucial truncation technique is used to derive the positive upper and lower bounds of the stationary solution. The uniqueness of the stationary solution is shown by a delicate energy estimate.     

Positive solutions of a diffusive prey–predator model in a heterogeneous environment

In this paper, we investigate a diffusive prey–predator model in a spatially degenerate heterogeneous environment. We are concerned with the positive solutions of the model, and obtain some results for the existence and non-existence of positive solutions. Moreover, the multiplicity, stability and asymptotical behaviors of positive solutions with respect to the parameters are also studied.     

Entire solutions for a discrete diffusive equation

We study entire solutions of a discrete diffusive equation with bistable nonlinearity. It is well known that there are three different wavefronts connecting any two of those three equilibria, say, 0,a,1. We construct three different types of entire solutions. The first one is a solution which behaves as two opposite wavefronts (connecting 0 and 1) of the same positive speed approaching each other from both sides of the real line. The second one is a solution which behaves as two different wavefronts (connecting a and one of {0,1}) approaching each other from both sides of the real line and converging to the wavefront connecting 0 and 1. The third one is a solution which behaves as a wavefront connecting a and 0 and a wavefront connecting 0 and 1 approaching each other from both sides of the real line.     

Qualitative analysis on a diffusive prey-predator model with ratio-dependent functional response

In this paper, we investigate a prey-predator model with diffusion and ratio-dependent functional response subject to the homogeneous Neumann boundary condition. Our main focuses are on the global behavior of the reaction-diffusion system and its corresponding steady-state problem. We first apply various Lyapunov functions to discuss the global stability of the unique positive constant steady-state. Then, for the steady-state system, we establish some a priori upper and lower estimates for positive steady-states, and derive several results for non-existence of positive non-constant steadystates if the diffusion rates are large or small. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10801090, 10726016, 10771032) and the Scientific Innovation Team Project of Hubei Provincial Department of Education (Grant No. T200809)     

Positive solutions for a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-Ⅱ functional response

We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response.The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional response.Here,a positive solution corresponds to a coexistence state of the model.Firstly,we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter plane.In addition,we present the uniqueness of positive solutions in one dimension case.Secondly,we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system,and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane.     

带混合边界条件的Holling-Ⅱ型捕食模型正稳态解的存在性

研究了一个带混合边界条件的Holling-Ⅱ型捕食模型．通过构造一个微分紧算子K,借助锥内不动点指数理论、拓扑度理论、椭圆方程估计理论和极值原理证得了算子K存在正的不动点,此结论表明捕食模型存在正解,即物种共存状态．     

Asymptotic periodicity of a food-limited diffusive population model with time-delay

In this paper, a general reaction-diffusion food-limited population model with time-delay is proposed. Accordingly, the existence and uniqueness of the periodic solutions for the boundary value problem and the asymptotic periodicity of the initial-boundary value problem are considered. Finally, the effect of the time-delay on the asymptotic behavior of the solutions is given.