Qualitative analysis for a ratio-dependent predator-prey model with disease and diffusion

This paper is purported to study a reaction diffusion system arising from a ratio-dependent predator-prey model with disease. We study the dynamical behavior of the predator-prey system. The conditions for the permanent and existence of steady states and their stability are established. We can obtain the bounds for positive steady state of the corresponding elliptic system. The non-existence results of non-constant positive solutions are derived.     

Qualitative analysis for a ratio-dependent predator–prey model with stage structure and diffusion

This paper is purported to study a reaction diffusion system arising from a ratio-dependent predator–prey model with stage structure. The global asymptotic stability of the unique positive constant equilibrium is obtained under suitable conditions; existence and non-existence results of non-trivial solution are derived.     

Qualitative analysis of a reaction–diffusion system

In this paper, we study the qualitative analysis of a reaction–diffusion system. The local stability of the trivial steady state, as well as the blow-up behavior of the solution are obtained.     

Qualitative analysis of autotroph-herbivore system with nutrient diffusion

We consider the mathematical model of a resource-based autotroph-herbivore system where diffusion of nutrient within the soil is taken into account. We have derived the conditions for the existence of steady-state of the system and studied the stability and instability of the spatially homogeneous case. Then we have introduced the diffusion term to the system and studied the qualitative behaviour of the spatially inhomogeneous case. Finally we have discussed the salient features of the analytical results giving also, the ecological interpretations.     

Qualitative analysis for a mathematical model of aids

In this paper a mathematical model of AIDS is investigated. The conditions of the existence of equilibria and local stability of equilibria are given. The existences of transcritical bifurcation and Hopf bifurcation are also considered, in particular, the conditions for the existence of Hopf bifurcation can be given in terms of the coefficients of the characteristic equation. The method extends the application of the Hopf bifurcation theorem to higher differential equations which occur in biological models, chemical models, and epidemiological models etc.This project is supported by the National Science Foundation Tian Yuan Terms and LNM Institute of Mechanics Academy of Science.This project is supported by the National and Yunnan Province Natural Science Foundation of China.     

Qualitative analysis for a cross-diffusion model of forest

The existence of a bounded global attractor for a cross-diffusion model of forest with homogeneous Dirichlet boundary condition is proved under some condition on the parameters Project supported by the National Natural Science Foundation of China (Grant No. 19671005).     

Asymptotic analysis of a model for substitutional-interstitial diffusion

A substitutional-interstitial model for impurity diffusion in semi-conductors is discussed. In particular we consider a surface-source problem and obtain asymptotic solutions in the limit of the surface concentration of impurity being much greater than the equilibrium vacancy concentration. In the absence of vacancy generation, a double error function impurity curve is obtained. These double profiles reproduce some of the qualitative features of diffusion in many III–V semiconductor systems. We also discuss how vacancy generation modifies the analysis and show that in the limit of high vacancy generation, the problem becomes one of linear diffusion with the diffusion curves then being single error function complements.     

Qualitative analysis of hepatitis B virus infection model with impulsive vaccination and time delay

According to the epidemic state, propagation mode and transformation between HBV infection states,the Hepatitis B Virus (HBV) infection with impulsive vaccination and time delay are modelled and analyzed. The control methods of impulsive vaccination and active therapy are adopted. By using comparative theorem of impulsive differential equation, the sufficient conditions that Hepatitis B Virus will be eliminated eventually or be persistent are derived.     

Qualitative analysis of a delayed non-Fickian model

This article focusses on the mathematical analysis of a delayed integro-differential model in which flux does not obey the classical Fick's law. The well-posedness of the integro-differential model in the Hadamard's sense is established. The dependence on the delay parameter of the total amount of desorpted/sorpted mass is studied. Numerical results that show the effectiveness of the model are included.     

Qualitative analysis of a ratio-dependent Holling-Tanner model

In this paper, we consider a Holling-Tanner system with ratio-dependence. First, we establish the sufficient conditions for the global stability of positive equilibrium by constructing Lyapunov function. Second, through a simple change of variables, we transform the ratio-dependent Holling-Tanner model into a better studied Liénard equation. As a result, the uniqueness of limit cycle can be solved.     

Qualitative analysis of a mathematical model for tissue inflammation dynamics

The non-linear differential eguation $$\dot X = \frac{{rX}}{{I + X}} - \frac{{XY}}{{X + Y}},\dot Y = \alpha (I + \theta X - Y)$$ is studied. It is the main aim of this paper to show the existence of bifurcation of saddle connection type; and to show the creation of limit cycles under certain conditions of the parameters, together with their biological significance.     

Qualitative analysis of a chemostat model with inhibitory exponential substrate uptake

In this paper, we consider a simple chemostat model involving a single species feeding on redundant substrate with a constant yield term. Many experiments indicate that very high substrate concentrations actually inhibit growth. Instead of assuming the prevalent Monod kinetics for growth rate of cells, we use a non-monotonic functional response function to describe the inhibitory effect. A detailed qualitative analysis about the local and global stability of its equilibria (including all critical cases) is carried out. Numerical simulations are performed to show that the dynamical properties depend intimately upon the parameters.     

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)     

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.     

An impulsive model for Wolbachia infection control of mosquito-borne diseases with general birth and death rate functions

Mosquito-borne diseases are global health problems, which mainly affect low-income populations in tropics and subtropics. In order to prevent the transmission of mosquito-borne diseases, the intracellular symbiotic bacteria named as Wolbachia is becoming a promising candidate to interrupt the virus transmission. In this paper, an impulsive mosquito population model with general birth and death rate functions is established to study the cytoplasmic incompatibility (CI) effect caused by mating of Wolbachia-infected males and uninfected females. The dynamics of the spread of Wolbachia in mosquito population are studied, and the strategies of mosquito extinction or replacing Wolbachia-uninfected mosquitoes with Wolbachia-infected mosquitoes are analyzed. Moreover, the results are applied to models with specific birth and death rate functions. It is shown that strategies may be different due to different birth and death rate functions, the type of Wolbachia strains and the initial number of Wolbachia-infected mosquitoes. Furthermore, numerical simulations are conducted to illustrate our conclusions.     

具斑块迁移的病毒模型的稳定性分析

研究了一类未感染细胞斑在两斑块间迁移的病毒模型.获得了无病平衡点全局稳定性以及地方病平衡点稳定性的充分条件.研究结果表明阈值R <1且感染细胞生成病毒的速度相对小而病毒细胞死亡率足够大时,病毒趋于灭绝;而阈值R> 1且感染细胞生成病毒的速度相对大而病毒细胞死亡率足够小时,病毒持续流行.     

Qualitative analysis of a coupled reaction-diffusion model in biology with time delays

Certain biochemical reaction can be modeled by a coupled system of time-delayed ordinary differential equations and linear parabolic partial differential equations. In a three-compartment model these equations are coupled through the boundary conditions. The aim of this paper is to give a qualitive analysis of this unusual coupled system. The analysis includes the existence and uniqueness of a global solution, explicit upper and lower bounds of the solution, and global stability of a steady-state solution. The global stability result is with respect to any nonnegative initial perturbation and is independent of the time delays in the process of reaction. Special attention is given to the Goodwin model for biochemical control of genes by a negative feedback mechanism with time delay and diffusion.