一类具有扩散项的流行性传染病模型中的行波解

讨论了一类具有扩散项的流行性传染病模型中的行波解的存在性.首先,将对该模型所对应的反应扩散系统的行波解的讨论转化为对二阶常微分系统的上下解的讨论;然后,通过上下解方法建立了这个具有扩散项的传染病模型中行波解的存在性条件,并进一步讨论了扩散因素对行波解的波速的影响,得到被感染人群的流动对病毒的传播有一定的影响.     

Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations

For abstract functional differential equations and reaction-diffusion equations with delay, an exponential ordering is introduced which takes into account the spatial diffusion. The induced monotonicity of the solution semiflows is established and applied to describe the threshold dynamics (extinction or persistence/convergence to positive equilibria) for a nonlocal and delayed reaction-diffusion population model.     

On the influence of diffusion on the competition between bacteria and innate immune system of invertebrates

The influence of diffusion on the competition between bacteria and innate immune system of invertebrate animals is described by means of a system of two nonlinear reaction-diffusion equations, with constant coefficients. The existence of absorbing sets, corresponding to the coexistence of the two competing populations, is proven. The linear and non-linear stability properties of the solutions are analysed as well as the stability switches due to the diffusion.     

Perron Theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations

In this paper we revisit the existence of traveling waves for delayed reaction-diffusion equations by the monotone iteration method. We show that Perron Theorem on existence of bounded solution provides a rigorous and constructive framework to find traveling wave solutions of reaction-diffusion systems with time delay. The method is tried out on two classical examples with delay: the predator-prey and Belousov-Zhabotinskii models.     

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.     

一类带有扩散与时滞的流行性传染病模型的行波解

讨论了一类带有扩散与时滞的流行性传染病模型的行波解的存在性.首先,将系统的行波解的存在性问题转化为一个二阶常微分方程组的单调解的存在性问题;应用单调方法和不动点方法,进一步地将问题转化为方程组的上下解的构造问题;应用所建立的引理与定理,通过构造适合的上下解,证明了系统单调行波解的存在性.     

Traveling Waves in Degenerate Diffusion Equations

In this survey, we present a literature review on the study of traveling waves in degenerate diffusion equations by illustrating the interesting and singular wave behavior caused by degeneracy. The main results on wave existence and stability are presented for the typical degenerate equations, including porous medium equations, flux limited diffusion equations, delayed degenerate diffusion equations, and other strong degenerate diffusion equations.     

Analysis of reaction-diffusion systems by the method of linear determining equations

Exact solutions to two-component systems of reaction-diffusion equations are sought by the method of linear determining equations (LDEs) generalizing the methods of the classical group analysis of differential equations. LDEs are constructed for a system of two second-order evolutionary equations. The results of solving the LDEs are presented for two-component systems of reaction-diffusion equations with polynomial nonlinearities in the diffusion coefficients. Examples of constructing noninvariant solutions are presented for the reaction-diffusion systems that possess invariant manifolds.     

Traveling waves for nonlocal and non-monotone delayed reaction-diffusion equations

We study the existence of traveling wave solutions for a nonlocal and non-monotone delayed reaction-diffusion equation. Based on the construction of two associated auxiliary reaction diffusion equations with monotonicity and by using the traveling wavefronts of the auxiliary equations, the existence of the positive traveling wave solutions for c 〉 c. is obtained. Also, the exponential asymptotic behavior in the negative infinity was established. Moreover, we apply our results to some reactiondiffusion equations with spatio-temporal delay to obtain the existence of traveling waves. These results cover, complement and/or improve some existing ones in the literature.     

Approximating the traveling wavefront for a nonlocal delayed reaction-diffusion equation

In this paper a boundary layer method is combined with an asymptotic expansion method to approximate the traveling wave solution of a nonlocal delayed reaction-diffusion model. In particular, assuming that the diffusion coefficients of the mature and immature populations are small, the wave solution is approximated in three steps. First, the model is reduced by considering the Dirac delta function as the kernel function of the integral term. Second, a boundary layer method is employed to approximate the wave solution of the reduced model. Third, using this result and the generalized Watson’s lemma, the wave solution of the general model is approximated. By considering various birth functions, the approximate wave solutions are numerically compared with the exact wave solutions.     

The effect of boundary conditions on the mesoscopic lattice Boltzmann method: Case study of a reaction-diffusion based model for Min-protein oscillation

Min-protein oscillation in Escherichia coli has an essential role in controlling the accurate placement of the cell division septum at the middle-cell zone of the bacteria. This biochemical process has been successfully described by a set of reaction-diffusion equations at the macroscopic level. The lattice Boltzmann method (LBM) has been used to simulate Min-protein oscillation and proved to recover the correct macroscopic equations. In this present work, we studied the effects of LBM boundary conditions (BC) on Min-protein oscillation. The impact of diffusion and reaction dynamics on BCs was also investigated. It was found that the mirror-image BC is a suitable boundary treatment for this Min-protein model. The physical significance of the results is extensively discussed.     

Improved Duality Estimates and Applications to Reaction-Diffusion Equations

We present a refined duality estimate for parabolic equations. This estimate entails new results for systems of reaction-diffusion equations, including smoothness and exponential convergence towards equilibrium for equations with quadratic right-hand sides in two dimensions. For general systems in any space dimension, we obtain smooth solutions of reaction-diffusion systems coming out of reversible chemistry under an assumption that the diffusion coefficients are sufficiently close one to another.     

Qualitative analysis for a Wolbachia infection model with diffusion

We consider a reaction-diffusion model which describes the spatial Wolbachia spread dynamics for a mixed population of infected and uninfected mosquitoes. By using linearization method, comparison principle and Leray-Schauder degree theory, we investigate the influence of diffusion on the Wolbachia infection dynamics. After identifying the system parameter regions in which diffusion alters the local stability of constant steady-states, we find sufficient conditions under which the system possesses inhomogeneous steady-states. Surprisingly, our mathematical analysis, with the help of numerical simulations, indicates that diffusion is able to lower the threshold value of the infection frequency over which Wolbachia can invade the whole population.     

NUMERICAL STUDY OF THE TWO-DIMENSIONAL SPRUCE BUDWORM REACTION-DIFFUSION EQUATION WITH DENSITY DEPENDENT DIFFUSION

The spruce budworm model is one of the interesting single species reaction-diffusion problems describing insect dispersal behavior. In this paper, we investigate a two-dimensional model with linear diffusion dependence and a convective wind. This system has been successfully solved using an operator splitting method for various domains and initial conditions. The numerical results show that populations can grow and diffuse in such a way as to produce steady state outbreak populations or steady state inhomogeneous spatial patterns in which they aggregate with low population densities.     

Shadow systems and attractors in reaction-diffusion equations

For a pair of reaction diffusion equations with one diffusion coefficient very large, there is associated a reaction diffusion equation coupled with an ordinary differential equation (the shadow system) with nonlocal effects which has the property that it contains all of the essential dynamics of the original equations. Key words: Attractors, shadow systems, reaction-diffusion equations     

一类反应扩散方程的孤立周期波和局部临界周期分支

研究了一类含有五次非线性反应项和常数扩散项的反应扩散方程的小振幅孤立周期波解,以及它的行波方程局部临界周期分支问题.运用行波变换将反应扩散方程转换为对应的行波系统,应用奇点量方法和计算机代数软件MATHEMATICA计算出该系统的前8个奇点量,得到该系统奇点的两个中心条件,并证明行波系统原点处可分支出8个极限环,对应的非线性反应扩散方程存在8个小振幅孤立周期波解;通过周期常数的计算,得到了行波系统原点的细中心阶数,并证明该系统最多有3个局部临界周期分支,且能达到3个局部临界周期分支;通过分析行波系统的临界周期分支,得到该反应扩散方程有3个临界周期波长.     

Dynamics of a delayed Lotka–Volterra competition model with directed dispersal

We consider a diffusive Lotka–Volterra competition system with stage structure, where the intrinsic growth rates and the carrying capacities of the species are assumed spatially heterogeneous. Here, we also assume each of the competing populations chooses its dispersal strategy as the tendency to have a distribution proportional to a certain positive prescribed function. We give the effects of dispersal strategy, delay, the intrinsic growth rates and the competition parameters on the global dynamics of the delayed reaction diffusion model. Our result shows that competitive exclusion occurs when one of the diffusion strategies is proportional to the carrying capacity, while the other is not; while both populations can coexist if the competition favors the latter species. Finally, we point out that the method is also applied to the global dynamics of other kinds of delayed competition models.     

Exponential decay toward equilibrium via entropy methods for reaction-diffusion equations

In this work, we show how the entropy method enables to get in an elementary way (and without linearization) estimates of exponential decay towards equilibrium for solutions of reaction-diffusion equations corresponding to a reversible reaction. Explicit rates of convergence combining the dissipative effects of diffusion and reaction are given.     

A locally adaptive time stepping algorithm for the solution to reaction diffusion equations on branched structures

In this paper, we present a numerical method for solving reaction-diffusion equations on one dimensional branched structures. Through the use of a simple domain decomposition scheme, the many branches are decoupled so that the equations can be solved as a system of smaller problems that are tri-diagonal. This technique allows for locally adaptive time stepping, in which the time step used in each branch is determined by local activity. Though the method is presented in the specific context of electrical activity in neural systems, it is sufficiently general that it can be applied to other classes of reaction-diffusion problems and higher dimensions. Information in neurons, which can be effectively modeled as one-dimensional branched structures, is carried in the form of electrical impulses called action potentials. The model equations, based on the Hodgkin-Huxley cable equations, are a set of reaction equations coupled to a single diffusion process. Locally adaptive time stepping schemes are well suited to neural simulations due to the spatial localization of activity. The algorithm significantly reduces the computational cost compared to existing methods, especially for large scale simulations.     

On Time-space Fractional Reaction-diffusion Equations with Nonlocal Initial Conditions

This paper investigates the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. Based on the operator semigroup theory, we transform the time-space fractional reaction-diffusion equation into an abstract evolution equation. The existence and uniqueness of mild solution to the reaction-diffusion equation are obtained by solving the abstract evolution equation. Finally, we verify the Mittag-Leffler-Ulam stabilities of the nonlinear time-space fractional reaction-diffusion equations with nonlocal initial conditions. The results in this paper improve and extend some related conclusions to this topic.