一类带收获率的捕食-食饵模型的共存解

讨论了齐次Neumann边界条件下食饵有外界常收获率的捕食-食饵模型的共存态首先分析了正常数解的稳定性以及非常数正平衡解不存在的条件.其次,基于对平衡解的先验估计,利用拓扑度理论,给出了此平衡态系统非常数正解的存在性.     

一具有外源和内部感染的捕食-食饵模型的定性分析

研究了一类具有外源和内部感染的捕食-食饵模型在Neumann边界条件下的共存态问题.首先利用特征值理论证明了该模型正常数平衡解的渐近稳定性;然后,给出了正平衡解的先验估计;最后利用度理论研究了非常数正平衡解的存在性.     

一类捕食-食饵-互惠模型非常数正平衡解的存在性及分歧

讨论了一类捕食-食饵-互惠反应扩散系统的非常数正平衡解.首先分析了常数正平衡解的稳定性,其次;利用最大值原理和Harnack不等式给出了正解的失验估计.在此基础上,利用积分性质进一步讨论了非常数正解的不存在性,相应地证明了当扩散系数d_2 d_3大于特定正常数且扩散系数d_1有界时此模型没有非常数正解.同时利用度理论证明了当模型的线性化算子的正特征值的代数重数是奇数且扩散系数d_3不小于给定正常数时此模型至少存在一个非常数正解,最后研究了非常数正解的分歧.     

一个具有扩散和比例依赖响应函数捕食模型的定性分析

本文考虑了一个具有扩散项和比例依赖响应函数的捕食模型. 该模型带有齐次Neumann边界条件. 本文主要关心该反应扩散系 统解的大时间行为及其对应的平衡态问题. 首先通过构造各种Lyapunov函数, 讨论 唯一的正常数平衡解的全局稳定性. 然后, 对于平衡态问题, 建立了正平衡解上下界 的先验估计, 并且导出了当物种的扩散系数很大或者很小时非常数正平衡解的一些不存在性结果.     

一类捕食-被捕食模型的正平衡解存在性

针对自然界中捕食者染病的现象,建立了捕食者染病的捕食-被捕食模型,研究了捕食者为躲避疾病进行扩散,并且具有HollingⅡ功能性反应函数和齐次Neumann边界条件的问题,利用Harnack不等式和最大值原理给出反应扩散问题的正平衡解的先验估计,并利用拓扑度理论证明该问题的非常数正平衡解的存在性.讨论了对应平衡态问题的非常数正平衡解存在性。     

一类具有内部存储的非均匀恒化器模型的共存解

本文研究一类具有内部存储的非均匀恒化器模型正平衡解的存在性.由于模型中比率项的奇性,通常的线性化方法、分歧理论等均不适用.为克服比率项的奇性,首先建立模型正平衡解细致的先验估计,该估计表明模型的正平衡解含于一个特殊的锥内.然后借助于一类非线性特征值问题的主特征值及锥上的不动点指标理论给出了模型正平衡解存在的充分条件.     

一类具有交叉扩散的捕食模型非常数正解的存在性

研究了一类具有扩散和交叉扩散项的Holling-Tanner捕食-食饵模型.首先利用最大值原理和Harnack不等式给出正解的先验估计,进一步利用度理论得到非常数正解的存在性与不存在性,从而给出非常数正解存在的充分条件.     

一个趋化性模型非常数平衡解的存在性

对带两个趋化性参数的趋化性模型平衡解的存在性问题进行研究.在参数满足特定的条件下,应用局部分岔理论得到非常数平衡解的局部分岔结构,从而证明了该趋化性模型存在无穷多个非常数正平衡解.     

带有交错扩散的Leslie-Gower型三种群系统的稳态模式

讨论了带有Neumann边界条件的一类Leslie-Gower型三种群系统,在一定的条件之下,虽然系统对应的扩散(没有交错扩散)系统的唯一正平衡解是稳定的,系统中的交错扩散可导致Turing不稳定性的产生.特别地,建立了该系统非常数共存解的存在性.结果表明,交错扩散可引起系统中出现非常数正稳态解(稳态模式).     

一类半线性反应扩散方程组的非平凡平衡解

利用正锥上的度理论,结合精细的先验估计技巧,讨论了一类强非线性弱耦合的反应扩散方程组,得到了其非平凡平衡解的存在性以及解的结构．     

A note on positive non-constant steady-state solutions to a reaction-diffusion model

In the present paper, we investigate a reaction-diffusion system with feedback effect subject to the homogeneous Neumann boundary condition and study the positive steady-state solutions. We establish a priori estimates for positive steady-state solutions and derive some results for non-existence of positive non-constant steady-state solutions. Our analysis complements the existing results on this model.     

NON-CONSTANT POSITIVE STEADY-STATES OF A PREDATOR-PREY-MUTUALIST MODEL

In this paper, the authors deal with the non-constant positive steady-states of a predator-prey-mutualist model with homogeneous Neumann boundary condition. They first give a priori estimates (positive upper and lower bounds) of positive steady-states, and then study the non-existence, the global existence and bifurcation of non-constant positive steady-states as some parameters are varied. Finally the asymptotic behavior of such solutions as d3→∞ is discussed.     

A qualitative study on general Gause-type predator–prey models with non-monotonic functional response

In this paper, we study a diffusive predator–prey model with general growth rates and non-monotonic functional response under homogeneous Neumann boundary condition. A local existence of periodic solutions and the asymptotic behavior of spatially inhomogeneous solutions are investigated. Moreover, we show the existence and non-existence of non-constant positive steady-state solutions. Especially, to show the existence of non-constant positive steady-states, the fixed point index theory is used without estimating the lower bounds of positive solutions. More precisely, calculating the indexes at the trivial, semi-trivial and positive constant solutions, some sufficient conditions for the existence of non-constant positive steady-state solutions are studied. This is in contrast to the works in previous papers. Furthermore, on obtaining these results, we can observe that the monotonicity of a prey isocline at the positive constant solution plays an important role.     

A note on a diffusive predator-prey model and its steady-state system

In this note, a diffusive predator-prey model subject to the homogeneous Neumann bound- ary condition is investigated and some qualitative analysis of solutions to this reaction-diffusion system and its corresponding steady-state problem is presented. In particular, by use of a Lyapunov function, the global stability of the constant positive steady state is discussed. For the associated steady state problem, a priori estimates for positive steady states are derived and some non-existence results for non-constant positive steady states are also established when one of the diffusion rates is large enough. Consequently, our results extend and complement the existing ones on this model.     

一类带扩散的捕食模型非常数正解的存在性

本文考虑了一类带扩散的捕食模型的平衡态问题.首先给出了正解的先验估计,进而,分别借助于能量方法和拓扑度理论得出了因参数的变化而引起的非常数正解的不存在性和存在性结果.     

Positive steady-state solutions of the Noyes–Field model for Belousov–Zhabotinskii reaction

In the paper, we investigate the Noyes–Field model for Belousov–Zhabotinskii reaction and study positive steady-state solutions of this model with the homogeneous Neumann boundary condition. We obtain the existence and non-existence of non-constant positive steady-state solutions.     

A qualitative study on general Gause-type predator-prey models with constant diffusion rates

In this paper, we study the qualitative behavior of non-constant positive solutions on a general Gause-type predator-prey model with constant diffusion rates under homogeneous Neumann boundary condition. We show the existence and non-existence of non-constant positive steady-state solutions by the effects of the induced diffusion rates. In addition, we investigate the asymptotic behavior of spatially inhomogeneous solutions, local existence of periodic solutions, and diffusion-driven instability in some eigenmode.     

On pattern formation in the Gray-Scott model

In the paper, we investigate an elliptic system well-known as the Gray-Scott model and present some further results for positive solutions of this model. More precisely, we give the refined a priori estimates of positive solutions, and improve some previous results for the non-existence and existence of positive non-constant solutions as the parameters are varied, which imply some certain conditions where the pattern formation occurs or not.     

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)     

Non-constant positive steady states of the Sel'kov model

This paper deals with the reaction-diffusion system known as the Sel'kov model with the homogeneous Neumann boundary condition. This model has been applied to various problems in chemistry and biology. We first give a priori estimates (positive upper and lower bounds) of positive steady states, and then study the non-existence, bifurcation and global existence of non-constant positive steady states as the parameters λ and θ are varied.