Coexistence, permanence and resilience in biological reaction-diffusion systems

Reaction–diffusion systems are widely used to describespatio-temporal phenomena in a variety of scientific fields,including population ecology. In this paper, I demonstrate thatexisting results for coexistence and permanence of general Lotka–Volterrasystems with absorbing boundaries can be applied in a complementarymanner to address a variety of boundary conditions, includingthe insulating problem. Furthermore, the condition is applicableeven to systems containing positive feedback mechanisms in thedynamics. A single (vector) inequality, the first iterate condition,is derived which serves as a sufficient condition for coexistence,permanence and resilience. Additionally, I demonstrate thatthis inequality condition is but the first in a series of conditionsthat can be used to describe the behaviour of such systems.Finally, I provide a comparison between the iterate conditionsand an alternative test for solution resiliency.     

Pattern formation in a 2D simple chemical system with general orders of autocatalysis and decay

In this paper, we investigate pattern formation in a coupledsystem of reaction–diffusion equations in two spatialdimensions. These equations arise as a model of isothermal chemicalautocatalysis with termination in which the orders of autocatalysisand termination, m and n, respectively, are such that 1 <n < m. We build on the preliminary work by Leach & Wei(2003, Physica D, 180, 185–209) for this coupled systemin one spatial dimension, by presenting rigorous stability analysisand detailed numerical simulations for the coupled system intwo spatial dimensions. We demonstrate that spotty patternsare observed over a wide parameter range.     

Absorbing boundary conditions for reaction-diffusion equations

We treat here of the question of absorbing boundary conditionsfor nonlinear diffusion equations. We use the conditions designedfor the linear equation, we prove them to be well posed forthe nonlinear problem, and through numerical experiments thatthey are well suited for reaction–diffusion equations.     

The effects of a complexing agent on travelling waves in autocatalytic systems with applied electric fields

The effects of electric fields on the reaction fronts that arisein a system governed by an autocatalytic reaction and a complexationreaction between the autocatalyst and a complexing agent areconsidered. The complexation reaction is assumed to be fastrelative to the autocatalytic reaction and the equations forthis limit are derived. The corresponding travelling waves arediscussed, the case of quadratic autocatalysis being treatedin detail. The existence of minimum speed waves is examined,being dependent on the ratio of diffusion coefficients D, theconcentration S₀ and equilibrium constant K of the complexationreaction as well as the electric field strength E. It is seenthat, for some parameter values, minimum speed waves have negativeautocatalayst concentrations, and waves which have the lowestspeed consistent with non-negative concentrations are also obtained.Numerical integrations of the initial-value problem are performedfor representative parameter values. These show the developmentof the appropriate travelling wave (when it exists) as the largetime behaviour of the system, and, in cases where no travellingwave exists, the numerical integrations show the electrophoreticseparation of substrate and autocatalyst.     

A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction-diffusion problems

A coupled system of two singularly perturbed linear reaction–diffusiontwo-point boundary value problems is examined. The leading termof each equation is multiplied by a small positive parameter,but these parameters may have different magnitudes. The solutionsto the system have boundary layers that overlap and interact.The structure of these layers is analysed, and this leads tothe construction of a piecewise-uniform mesh that is a variantof the usual Shishkin mesh. On this mesh central differencingis proved to be almost first-order accurate, uniformly in bothsmall parameters. Supporting numerical results are presentedfor a test problem.     

Monotone wave-fronts in a structured population model with distributed maturation delay

^** Email: s.gourley{at}surrey.ac.uk We analyse a stage-structured reaction–diffusion modelfor a single species on an infinite 1D domain. Recognising thatnot all individuals may take the same amount of time to mature,the maturation delay is incorporated via a probability distributionfunction, leading to a distributed delay system. The systemis non-local in space, because individuals may have moved whileimmature. A detailed investigation of travelling front solutionsconnecting the extinction state with the positive equilibriumis carried out, focussing attention on the minimum speed andthe qualitative form of the profile, which appears always tobe monotone. A rigorous proof of existence is provided for aspecial, but realistic, choice of the probability distributionfunction representing the maturation delay. Numerical simulationsof the initial value problem are also presented.     

Traveling wavefronts of a delayed lattice reaction-diffusion model

We investigate a system of delayed lattice differential system which is a model of pioneer-climax species distributed on one dimensional discrete space. We show that there exists a constant $c^*&gt;0$, such that the model has traveling wave solutions connecting a boundary equilibrium to a co-existence equilibrium for $c\geq c^*$. We also argue that $c^*$ is the minimal wave speed and the delay is harmless. The Schauder's fixed point theorem combining with upper-lower solution technique is used for showing the existence of wave solution.     

Global stability of traveling wavefronts for nonlocal reaction-diffusion equations with time delay

This paper is concerned with the stability of traveling wavefronts for a population dynamics model with time delay. Combining the weighted energy method and the comparison principle, the global exponential stability of noncritical traveling wavefronts(waves with speeds c c_*, where c = c~* is the minimal speed) is established, when the initial perturbations around the wavefront decays to zero exponentially in space as x →-∞, but it can be allowed arbitrary large in other locations, which improves the results in [9, 18, 21].     

Maximum norm a posteriori error estimates for a 1D singularly perturbed semilinear reaction-diffusion problem

^{A singularly perturbed semilinear two-point boundary-value problemis discretized on arbitrary non-uniform meshes. We present second-ordermaximum norm a posteriori error estimates that hold true uniformlyin the small parameter. Their application to monitor-functionequidistribution and a posteriori mesh refinement are discussed.Numerical results are presented that support our theoreticalestimates.}     

On existence of solutions for some hyperbolic-parabolic-type chemotaxis systems

In this paper, we discuss the local and global existence ofweak solutions for some hyperbolic–parabolic systems modellingchemotaxis.     

Resilience in reaction-diffusion systems

Reaction-diffusion systems with zero-flux Neumann boundariesare widely used to model various kinds of interaction in, forexample, the scientific fields of ecology, biology, chemistry,medicine and industry. The physical systems within these fieldsare often known to be (conditionally or unconditionally) resilientwith respect to shocks, disturbances or catastrophies in theimmediate environment. In order to be good mathematical modelsof such situations the reaction-diffusion systems must havethe same resilient or asymptotic behaviour as that of the physicalsituation. Three fundamentally different kinds of reaction termsare usually distinguished according to the entry signs of thereaction Jacobian: mutualism, mixed (predator-prey) interactionand competition. The asymptotic stability (in the Poincarésense) of mutualistic systems has already been studied extensively,but the results cannot be generalized (globally) to the othertwo fundamental types, which are not order-preserving. A partial(local) generalization is, however given here for these twotypes, involving simple Jacobian inequalities and knowledge(often prompted by the underlying physical situation) of invariantsets in solution space. The return time of resilient systemsand the approach rate of asymptotically stable solutions arealso estimated.     

Travelling wavefronts of reaction-diffusion equations in cylindrical domains

Abstract

Function spaces with asymptotics is a usual tool in the analysis on manifolds with singularities. The asymptotics are singular ingredients of the kernels of pseudodifferential operators in the calculus. They correspond to potentials supported by the singularities of the manifold, and in this form asymptotics can be treated already on smooth configurations. This article is aimed at describing refined asymtotics in the Dirichlet problem in a ball. The beauty of explicit formulas actually highlights the structure of asymptotic expansions in the calculi on singular varieties.     

Persistence of travelling wavefronts in a generalized Burgers-Huxley equation with long-range diffusion

In this paper, we study the persistence of travelling wavefronts in a generalized Burgers-Huxley equation with long-range diffusion. When the influence of long-range diffusion effect is sufficiently small, we prove the persistence of these waves by using geometric singular perturbation theory. When the influence becomes large, the behavior of these waves can only be investigate numerically. In this case, we find that the solutions lose monotonicity by using Matlab program bvp4c. Some previous results are extended.     

Persistence of wavefronts in delayed nonlocal reaction-diffusion equations

We develop a perturbation argument based on existing results on asymptotic autonomous systems and the Fredholm alternative theory that yields the persistence of traveling wavefronts for reaction-diffusion equations with nonlocal and delayed nonlinearities, when the time lag is relatively small. This persistence result holds when the nonlinearity of the corresponding ordinary reaction-diffusion system is either monostable or bistable. We then illustrate this general result using five different models from population biology, epidemiology and bio-reactors.     

On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems

In this paper we consider reaction-diffusion systems in which the conditions imposed on the nonlinearity provide global existence of solutions of the Cauchy problem, but not uniqueness. We prove first that for the set of all weak solutions the Kneser property holds, that is, that the set of values attained by the solutions at every moment of time is compact and connected. Further, we prove the existence and connectedness of a global attractor in both the autonomous and nonautonomous cases. The obtained results are applied to several models of physical (or chemical) interest: a model of fractional-order chemical autocatalysis with decay, the Fitz-Hugh-Nagumo equation and the Ginzburg-Landau equation.     

Fast-diffusion limit with large noise for systems of stochastic reaction-diffusion equations

We consider a class of a stochastic reaction-diffusion equations with additive noise. In the limit of fast diffusion, one can approximate solutions of the stochastic reaction–diffusion equations by the solution of a suitable system of ordinary differential equation only describing the reactions, but due to nonlinear interaction of large diffusion and fluctuations in the limit new effective reaction terms appear. We focus on systems with polynomial nonlinearities and illustrate the result by applying it to a predator-prey system and a cubic auto-catalytic reaction between two chemicals.     

On the derivation of fractional diffusion equation with an absorbent term and a linear external force

Recently, the generalized fractional reaction–diffusion equation subject to an external linear force field has been proposed to describe the transport processes in disordered systems. The solution of this generalized model can be formally expressed in closed form through the Fox function. For the sack of completeness, we dedicate this work to construct a neatly derivation of the generalized fractional reaction–diffusion equation. Remarkably, such derivation could in general offer some novel and inspiring inspection to the phenomena of anomalous transport. For instance, there is a strong evidence that the fractional calculus offers some physical insight into the origin of fractional dynamics for a systems which exhibit multiple trapping.     

Functional Inequalities for Particle Systems on Polish Spaces

Various Poincaré–Sobolev type inequalities are studied for a reaction–diffusion model of particle systems on Polish spaces. The systems we consider consist of finite particles which are killed or produced at certain rates, while particles in the system move on the Polish space interacting with one another (i.e. diffusion). Thus, the corresponding Dirichlet form, which we call reaction–diffusion Dirichlet form, consists of two parts: the diffusion part induced by certain Markov processes on the product spaces Eⁿ (n≥1) which determine the motion of particles, and the reaction part induced by a Q-process on ℤ₊ and a sequence of reference probability measures, where the Q-process determines the variation of the number of particles and the reference measures describe the locations of newly produced particles. We prove that the validity of Poincaré and weak Poincaré inequalities are essentially due to the pure reaction part, i.e. either of these inequalities holds if and only if it holds for the pure reaction Dirichlet form, or equivalently, for the corresponding Q-process. But under a mild condition, stronger inequalities rely on both parts: the reaction–diffusion Dirichlet form satisfies a super Poincaré inequality (e.g., the log-Sobolev inequality) if and only if so do both the corresponding Q-process and the diffusion part. Explicit estimates of constants in the inequalities are derived. Finally, some specific examples are presented to illustrate the main results. Mathematics Subject Classifications (2000) 4FD0F, 60H10. Feng-Yu Wang: Supported in part by the DFG through the Forschergruppe “Spectral Analysis, Asymptotic Distributions and Stochastic Dynamics”, the BiBoS Research Centre, NNSFC(10121101), and RFDP(20040027009).     

TRAVELLING WAVEFRONTS IN DELAYED COOPERATIVE AND DIFFUSIVE SYSTEMS WITH NON-LOCAL EFFECTS

In this paper, traveling wavefront solutions are established for two cooperative systems with time delay and non-local effects. The results are an extension of the existing results for delayed logistic scale equations and diffusive Nicholson equations with non-local effects to systems. The approach used is the upper-lower solution technique and Schauder fixed point Theorem developed by Ma（J Differential Equations,2001,171：294-314. ）.