Point condensations of a model for fungal development

We study the stationary solutions for a reaction-diffusion system of activator-inhibitor type which arises as a model for fungal development. Under the condition that the activator diffuses slowly and the inhibitor diffuses very quickly we rigorously construct solutions which show single peak pattern near the boundary or in the interior in the activator component and have nearly constant values in the other. We also establish the linear stability and instability of such solutions.     

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.     

比率依赖型捕食者-食饵系统行波解的存在性

本文讨论一类比率依赖型捕食者 -食饵系统的反应扩散方程组 .首先 ,我们证明了时间周期定常解的存在性和稳定性 .其次 ,我们给出了扩散引起正常数平衡解失稳的条件 .最后 ,我们证明了比率依赖型捕食者 -食饵系统行波解的存在性和渐近性 .     

Interface dynamics of the Fisher equation with degenerate diffusion

We consider a degenerate parabolic reaction-diffusion equation with a monostable nonlinearity arising in population dynamics. In some suitable scaling limit, we prove the generation and propagation of an interface with constant normal velocity in the case that the initial condition has a convex compact support.     

The nonsymmetric case of the Keller-Segel model in chemotaxis: some recent results

Looking at the nonsymmetric case of a reaction-diffusion model known as the Keller-Segel model, we summarize known facts concerning (global in time) existence and prove new blowup results for solutions of this system of two strongly coupled parabolic partial differential equations. We show in Section 4, Theorem 4, that if the solution blows up under a condition on the initial data, blowup takes place at the boundary of a smooth domain . Using variational techniques we prove in Section 5 the existence of nontrivial stationary solutions in a special case of the system. Received April 2000     

Turing instability and travelling waves in diffusive plankton models with delayed nutrient recycling

In this paper we propose a reaction-diffusion system with twodistributed delays to stimulate the growth of plankton communitiesin the lakes/oceans in which the plankton feeds on a limitingnutrient supplied at a constant rate. The limiting nutrientis partially recycled after the death of the organisms and adistributed delay is used to model nutrient recycling. The seconddelay is involved in the growth response of the plankton tonutrient uptake. We first show that there are oscillations (Hopfbifurcations) in the delay model induced by the second delay.Then we study Turing (diffusion-driven) instability of the reaction-diffusionsystem with delay. Finally, it is shown that if the delay modelhas a stable periodic solution, then the corresponding reaction-diffusionmodel with delay has a family of travelling waves.     

Asymptotic behavior of solutions for a class of predator-prey reaction-diffusion systems with time delays

The aim of this paper is to investigate the asymptotic behavior of solutions for a class of three-species predator-prey reaction-diffusion systems with time delays under homogeneous Neumann boundary condition. Some simple and easily verifiable conditions are given to the rate constants of the reaction functions to ensure the convergence of the time-dependent solution to a constant steady-state solution. The conditions for the convergence are independent of diffusion coefficients and time delays, and the conclusions are directly applicable to the corresponding parabolic-ordinary differential system and to the corresponding system without time delays.     

Pattern formation in the Brusselator system

In the paper, we deal with a reaction-diffusion system well known as the Brusselator model and some improved results for the steady states of this model are presented. We first give an a priori estimates (positive upper and lower bounds) of positive steady states. Then, we obtain the non-existence and existence of positive non-constant steady states as the parameters λ, θ and b are varied, which means some certain conditions under which the pattern formation occurs or not.     

Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions

We study the blow-up behavior for a semilinear reaction-diffusion system coupled in both equations and boundary conditions. The main purpose is to understand how the reaction terms and the absorption terms affect the blow-up properties. We obtain a necessary and sufficient condition for blow-up, derive the upper bound and lower bound for the blow-up rate, and find the blow-up set under certain assumptions.     

Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations

For a balanced bistable reaction-diffusion equation, an axisymmetric traveling front has been well known. This paper proves that an axially asymmetric traveling front with any positive speed does exist in a balanced bistable reaction-diffusion equation. Our method is as follows. We use a pyramidal traveling front for an unbalanced reaction-diffusion equation whose cross section has a major axis and a minor axis. Preserving the ratio of the major axis and a minor axis to be a constant and taking the balanced limit, we obtain a traveling front in a balanced bistable reaction-diffusion equation. This traveling front is monotone decreasing with respect to the traveling axis, and its cross section is a compact set with a major axis and a minor axis when the constant ratio is not 1.     

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.     

Diffusive logistic equation with constant yield harvesting, I: Steady States

We consider a reaction-diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth. We prove the existence, uniqueness and stability of the maximal steady state solutions under certain conditions, and we also classify all steady state solutions under more restricted conditions. Exact global bifurcation diagrams are obtained in the latter case. Our method is a combination of comparison arguments and bifurcation theory.

     

Generalized quasilinearization method for reaction-diffusion equations under nonlinear and nonlocal flux conditions

In this paper we consider an initial boundary value problem for a reaction-diffusion equation under nonlinear and nonlocal Robin type boundary condition. Assuming the existence of an ordered pair of upper and lower solutions we establish a generalized quasilinearization method for the problem under consideration whose characteristic feature consists in the construction of monotone sequences converging to the unique solution within the interval of upper and lower solutions, and whose convergence rate is quadratic. Thus this method provides an efficient iteration technique that produces not only improved approximations due to the monotonicity of its iterates, but yields also a measure of the convergence rate.     

Spatiotemporal complexity of a diffusive planktonic system with prey-taxis and toxic effects

In this paper, we propose a three-species reaction-diffusion planktonic system with prey-taxis and toxic effects, in which the zooplankton can recognize the nontoxic and toxin-producing phytoplankton and can make proper response. We first establish the existence and stability of the unique positive constant equilibrium solution by utilizing the linear stability theory for partial differential equations. Then we obtain the existence and properties of nonconstant positive solutions by detailed steady state bifurcation analysis. In addition, we obtain that change of taxis rate will result in the appearance of time-periodic solutions. Finally, we conduct some numerical simulations and give the conclusions.     

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)     

Entropy functionals for finding requirements in hierarchical reaction-diffusion models for inflammations

Based on an established model for liver infections, we open the discussion on the used reaction terms in the reaction-diffusion system. The mechanisms behind the chronification of liver infections are widely unknown, therefore we discuss a variety of reaction functions. By using theorems about existence, uniqueness, and nonnegativity, we identify properties of reaction functions which are indispensable to modelling liver infections. We introduce an entropy functional for reaction-diffusion models of this type, which allows predictions of the longtime behavior of the solutions. As a result, we find more conditions on the reaction functions to derive a model covering different inflammation courses. Finally, we discuss the models in the frame of a hierarchical model family.     

Oscillations spatio-temporelles engendrees par un automate cellulaire

We study a planar cellular automaton which is a simple model of a reaction-diffusion mechanism in excitable media; we are especially interested in the spatio-temporal organization which it generates. Under suitable assumptions, the sequence of the states of the plan is ultimately either stationary or periodic. In the latter case, we prove that there exists only one admissible period which is independent of the initial conditions and that a spatial organization of the plan appears, consisting of parallel equidistant target-wave-fronts growing with constant speed.     

Stage-structured Harvest Models

We formulate a stage-structured population model where the population is divided to two classes, the juveniles and the adults. Then, we include harvest in the model and assume that the harvesting is only on adults. The cases where the harvesting rate is constant, proportional to the amount of adults, or of Holling-II type are studied. While the model dynamics are relatively simple when the harvesting rate is proportional, the model system with a constant or a Holling-II type harvesting rate can have multiple positive equilibria. We explore the existence of all possible equilibria and investigate their stability. We also give numerical examples to confirm our findings.     

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.