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.     

捕食者食饵均染病的入侵反应扩散捕食系统中扩散的作用

研究了捕食者食饵均染病的入侵反应扩散捕食系统.利用特征值方法和构造Lyapunov函数,获得了入侵扩散对正常数平衡解的影响, 当入侵扩散系数充分大时, 导致平衡态失稳.进一步, 利用拓扑度方法, 证明了在一定条件下入侵扩散系数很大, 自扩散充分小时, 有非常数正平衡解存在.     

一类糖酵解模型正平衡解的存在性分析

研究生化反应中具有代表性的一类糖酵解模型.运用先验估计讨论非常数正平衡解的不存在性,得到非常数正平衡解存在的必要条件.在常数平衡解Turing不稳定的基础上,利用度理论方法和解的先验估计,进一步给出非常数正平衡解存在的充分条件.     

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.     

具有空间扩散和年龄结构竞争模型的正平衡态

讨论了具有空间扩散和年龄结构的竞争模型在Neumann边界条件下正常数解的稳定性,并用两种不同的方法给出了非常数正解不存在的条件,即在此条件下不会发生斑图现象.     

一类带有交错扩散的捕食模型非常数正解的进一步分析

本文讨论一个带有交错扩散的捕食模型的齐次Neumann问题.首先,利用Harnack等式以及椭圆方程正则理论讨论了当扩散系数至少一个取极限时非常数正解的渐近性,再利用渐近性质以及奇异扰动方法讨论了当扩散系数取极限的情况下非常数正解的存在性.     

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.     

带有交叉扩散的捕食模型的非常数正稳态解的存在性

本文研究了下列带有交叉扩散的捕食模型的稳态问题的非常数正解的存在性,证明了当d4>1/m1v-u时存在(g1,d2,d3)使得稳态问题存在非常数正解;而当d4≤1/m1v-u或者d1≥m1v-u/u1或者a(m1b,a2(b))时稳态问题不存在非常数正解．     

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。     

Analysis of a prey-predator model with disease in prey

In this paper, a system of reaction-diffusion equations arising in eco-epidemiological systems is investigated. The equations model a situation in which a predator species and a prey species inhabit the same bounded region and the predator only eats the prey with transmissible diseases. A number of existence and non-existence results about the non-constant steady states of a reaction diffusion system are given. It is proved that if the diffusion coefficient of the predator is treated as bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steady-state solution under some conditions.     

Analysis of a nutrient-phytoplankton model in the presence of viral infection

In this paper, a system of reaction-diffusion equations arising in a nutrient-phytoplankton populations is investigated. The equations model a situation in which phytoplankton population is divided into two groups, namely susceptible phytoplankton and infected phytoplankton. A number of existence and non-existence results about the non-constant steady states of a reaction diffusion system are given. If the diffusion coefficient of the infected phytoplankton is treated as bifurcation parameter, non-constant positive steady-state solutions may bifurcate from the constant steady-state solution under some conditions.     

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.     

Non-constant positive steady state of one resource and two consumers model with diffusion

In this paper, a predator-prey reaction-diffusion system with one resource and two consumers is considered. Assume that one consumer species exhibits Holling II functional response while the other consumer species exhibits Beddington-DeAngelis functional response, and they compete for the common resource. First, it is proved that the unique positive constant steady state is stable for the ODE system and the reaction-diffusion system. Second, a prior estimates of positive steady state is given. Finally, the non-existence of non-constant positive steady state, the existence and bifurcation of non-constant positive steady state are studied.     

A generalized Degn–Harrison reaction–diffusion system: Asymptotic stability and non-existence results

In this paper we study the Degn–Harrison system with a generalized reaction term. Once proved the global existence and boundedness of a unique solution, we address the asymptotic behavior of the system. The conditions for the global asymptotic stability of the steady state solution are derived using the appropriate techniques based on the eigen-analysis, the Poincaré–Bendixson theorem and the direct Lyapunov method. Numerical simulations are also shown to corroborate the asymptotic stability predictions.Moreover, we determine the constraints on the size of the reactor and the diffusion coefficient such that the system does not admit non-constant positive steady state solutions.     

Stationary pattern of a diffusive prey–predator model with trophic intersections of three levels

The paper is concerned with a diffusive prey–predator model subject to the homogeneous Neumann boundary condition, which models the trophic intersections of three levels. We will prove that under certain assumptions, even though the unique positive constant steady state is globally asymptotically stable for the dynamics with diffusion, the non-constant positive steady state can exist due to the emergence of cross-diffusion. We demonstrate that the cross-diffusion can create stationary pattern. Moreover, we treat the cross-diffusion parameter as a bifurcation parameter and discuss the existence of non-constant positive solutions to the system with cross-diffusion.     

Spatiotemporal pattern formation and multiple bifurcations in a diffusive bimolecular model

In this paper, we have investigated a homogeneous reaction–diffusion bimolecular model with autocatalysis and saturation law subject to Neumann boundary conditions. We mainly consider Hopf bifurcations and steady state bifurcations which bifurcate from the unique constant positive equilibrium solution of the system. Our results suggest the existence of spatially non-homogeneous periodic orbits and non-constant positive steady state solutions, which implies the possibility of rich spatiotemporal patterns in this diffusive biomolecular system. Numerical examples are presented to support our theoretical analysis.     

Uniform boundedness and stability of global solutions in a strongly coupled three-species cooperating model

Using the energy estimate and Gagliardo–Nirenberg-type inequalities, the existence and uniform boundedness of global solutions for a strongly coupled reaction–diffusion system are proved. This system is the Shigesada–Kawasaki–Teramoto three-species cooperating model with self- and cross-population pressure. Meanwhile, some criteria on the global asymptotic stability of the positive equilibrium point for the model are also given by Lyapunov function. As a by-product, we proved that only constant steady states exist if the diffusion coefficients are large enough.     

Diffusive logistic equation with constant yield harvesting and negative density dependent emigration on the boundary

The structure of positive steady state solutions of a diffusive logistic population model with constant yield harvesting and negative density dependent emigration on the boundary is examined. In particular, a class of nonlinear boundary conditions that depends both on the population density and the diffusion coefficient is used to model the effects of negative density dependent emigration on the boundary. Our existence results are established via the well-known sub-super solution method.