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.     

Positive steady-state solutions for predator–prey systems with prey-taxis and Dirichlet conditions

This paper is concerned with positive steady-state solutions of a class of cross-diffusion systems which arise in the study of the predator–prey systems with prey-taxis. Under homogeneous Dirichlet boundary conditions, we use the theory of fixed point index in positive cones to establish the existence of positive steady-state solutions. By analyzing two related eigenvalues, we further obtain the coexistence region with respect to the growth rates of two species and characterize the differences of coexistence region if different predator–prey interactions are adopted. Additionally, we investigate the limiting behavior of positive steady-state solutions as some parameter tends to infinity. Our results not only generalize the previously known one, but also present some new conclusions.     

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.     

Positive solutions for ratio-dependent predator-prey interaction systems

In this paper, we study the dynamics of predator-prey interaction systems between two species with ratio-dependent functional responses. First we provide sufficient and necessary conditions for positive steady-state solutions, and then we investigate the relationships between positive equilibria and positive solutions of the system over a large domain. Furthermore, we deal with the uniqueness and the stability of positive steady-states solutions with some assumptions. In addition, we discuss the extinction and the persistence results of time-dependent positive solutions to the system.     

A CONDITION OF THE EXISTENCE OF STABLE POSITIVE STEADY-STATE SOLUTIONS FOR A ONE PREDATOR TWO PREY SYSTEM

One predator two prey system is a research topic which has both the theoretical and practical values.This paper provides a natural condition of the existence of stable pcsitive steady-state solutions for the one predator two prey system.Under this conditon we study the existence of the positive steady-state solutions at vicinity of the triple eigenvalue by implicit function theorem,discuss the positive stable solution problem bifureated from the semi-trivial solutions containing two positive components with the help of bifurcation and perturbation methods.     

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.     

Global asymptotic behavior in a Lotka–Volterra competition system with spatio-temporal delays

This paper is concerned with a Lotka–Volterra competition system with spatio-temporal delays. By using the linearization method, we show the local asymptotic behavior of the nonnegative steady-state solutions. Especially, the global asymptotic stability of the positive steady-state solution is investigated by the method of upper and lower solutions. The result of global asymptotic stability implies that the system has no nonconstant positive steady-state solution.     

捕食—诱饵系统平衡态的分歧

本文利用分歧方法和一些适当的技巧,获得捕食—诱饵系统在 Diri-chlet 边界条件下有关正平衡态的结果,并给出一个关于解结构的猜想.     

Some nonexistence results for nonconstant stationary solutions to the Gray–Scott model in a bounded domain

In the present paper, we are concerned with a reaction–diffusion system well-known as the Gray–Scott model in a bounded domain and study the corresponding steady-state problem. We establish some results for the nonexistence of nonconstant positive stationary solutions.     

Global asymptotic stability of positive equilibrium of three-species Lotka–Volterra mutualism models with diffusion and delay effects

In the mutualism system with three species if the effects of dispersion and time delays are both taken into consideration, then the densities of the cooperating species are governed by a coupled system of reaction–diffusion equations with time delays. The aim of this paper is to investigate the asymptotic behavior of the time-dependent solution in relation to a positive uniform solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition, including the existence and uniqueness of a positive steady-state solution. A simple and easily verifiable condition is given to ensure the global asymptotic stability of the positive steady-state solution. This result leads to the permanence of the mutualism system, the instability of the trivial and all forms of semitrivial solutions, and the nonexistence of nonuniform steady-state solution. The condition for the global asymptotic stability is independent of diffusion and time-delays as well as the net birth rate of species, and the conclusions for the reaction–diffusion system are directly applicable to the corresponding ordinary differential system and 2-species cooperating reaction–diffusion systems. Our approach to the problem is based on inequality skill and the method of upper and lower solutions for a more general reaction–diffusion system. Finally, the numerical simulation is given to illustrate our results.     

Asymptotic stability of reaction-diffusion systems in chemical reactor and combustion theory

Some coupled reaction-diffusion systems arising from chemical diffusion processes and combustion theory are analyzed. This analysis includes the existence and uniqueness of positive time-dependent solutions, upper and lower bounds of the solution, asymptotic behavior and invariant sets, and the stability of steady-state solutions, including an estimate of the stability region. Explicit conditions for the asymptotic behavior and the stability of a steady-state solution are given. These conditions establish some interrelationship among the physical parameters of the diffusion medium, the reaction mechanism, the initial function and the type of boundary condition. Under the same set of physical parameters and reaction function, a comparison between the Neumann type and Dirichlet or third type boundary condition exhibits quite different asymptotic behavior of the solution. For the general nonhomogeneous system, multiple steady-state solutions may exist and only local stability results are obtained. However, for certain models it is possible to obtain global stability of a steady-state solution by either increasing the diffusion coefficients or decreasing the size of the diffusion medium. This fact is demonstrated by a one-dimensional tubular reactor model commonly discussed in the literature.     

带有第三边值的捕食模型的正稳态解的存在性

本文研究了一个捕食者带有第三类边界条件、被捕食者带有Neumann边界条件的捕食模型．获得了捕食模型正稳态解的存在性和非存在性结果．并且,证明了它的正稳态解的局部稳定性和唯一性．     

具有扩散的广义Brusselator系统的Hopf分歧

在齐次Neumann边界条件下,考虑广义Brusselator系统．首先讨论常微分系统Hopf分歧的存在性,得到渐近稳定的周期解．其次讨论具有扩散的偏微分系统,在扩散系数满足一定的条件下,得到超临界的Hopf分歧,并利用规范形理论和中心流形定理给出空间齐次周期解的渐近稳定性．最后,借助Matlab软件进行数值模拟,证明了定理的结论．同时,正平衡态解和空间非齐次周期解的描绘补充了理论分析结果．     

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.     

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.     

The method of multiple scales for forced oscillators with some real-power nonlinearities in the stiffness and damping force

The steady-state response of forced damped nonlinear oscillators is considered, the restoring force of which has a non-negative real power-form nonlinear term and the linear term of which can be negative, zero or positive. The damping term is also assumed in a power form, thus covering polynomial and non-polynomial damping. The method of multiple scales with a new expansion parameter is presented in order to cover the cases when the nonlinearity is not necessarily small. Amplitude-frequency equations and approximate solutions for the steady-state response at the frequency of excitation are obtained and compared with numerical results, showing good agreement.     

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.     

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.     

A diffusive predator-prey model in heterogeneous environment

In this paper, we demonstrate some special behavior of steady-state solutions to a predator-prey model due to the introduction of spatial heterogeneity. We show that positive steady-state solutions with certain prescribed spatial patterns can be obtained when the spatial environment is designed suitably. Moreover, we observe some essential differences of the behavior of our model from that of the classical Lotka-Volterra model that seem to arise only in the heterogeneous case.