On the formation of acceleration and reaction-diffusion wavefronts in autocatalytic-type reaction-diffusion systems

^** Email: d.j.needham{at}reading.ac.uk We consider generalization of the theory for the evolution ofreaction–diffusion and accelerating wavefronts in KPP-typesystems as developed in Needham (2004, Proc. R. Soc. Lond. A,460, 1921–1934) (DN). These generalizations allow forthe removal of a number of technical restrictions imposed inthe paper of DN.     

The evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay

In this paper, we study a reaction–diffusion system for an isothermal chemical reaction scheme governed by a quadratic autocatalytic step A+B→2B$A+B\to 2B$ and a decay step B→C$B\to C$, where A, B, and C are the reactant, the autocatalyst, and the inner product, respectively. Previous numerical studies and experimental evidences demonstrate that if the autocatalyst is introduced locally into this autocatalytic reaction system where the reactant A initially distributes uniformly in the whole space, then a pair of waves will be generated and will propagate outwards from the initial reaction zone. One crucial feature of this phenomenon is that for the strong decay case, the formation of waves is independent of the amount of the autocatalyst B introduced into the system. It is this phenomenon of KPP-type which we would like to address in this paper. To study the propagation of reactant and autocatalyst analytically, we first use the tail behavior of waves to construct a pair of generalized super-/sub-solutions for the approximate system of the autocatalytic reaction system. Note that the autocatalytic reaction system does not enjoy comparison principle. Together with a family of truncated problems, we can establish the existence of a family of traveling waves with the minimal speed. Second, we use this pair of generalized super-/sub-solutions to show that the propagation of waves is fully determined by the rate of decay of the initial data at infinity in the sense of Aronson–Weinberger formulation, which in turn confirms the aforementioned numerical and experimental results.     

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.     

Reaction-diffusion waves in an isothermal chemical system with general orders of autocatalysis and spatial dimension

The possibility of initiating reaction-diffusion waves in an autocatalytic system represented schematically byAB, ratekab ^p (p >- 1, witha, b being the concentrations ofA andB respectively) is considered through the local input ofB, measured by the parameter ₀, into an otherwise uniform expanse ofA. It is shown that for 1 <-p < 1 + (2/N) (whereN is the space dimension) waves develop no matter how small the value of ₀, while forp > 1 + (2/N) there is some threshold value of ₀ below which waves are not formed, with diffusion playing the dominant role throughout. A lower bound for this threshold value is found. The permanent-form travelling wave equations are then discussed and the behaviour of the solution asp 1 is considered in detail. It is shown that a three-region structure develops with the asymptotic wave speedv being singular (of the formv 2–2.3381 (p- 1)^2/3) asp 1.     

A weak solution approach to a reaction–diffusion system modeling pattern formation on seashells

We investigate a reaction–diffusion system proposed by H. Meinhardt as a model for pattern formation on seashells. We give a new proof for the existence of a local weak solution for general initial conditions and parameters upon using an iterative approach. Furthermore, the solution is shown to exist globally for suitable initial data. The behavior of the solution in time and space is illustrated through numerical simulations. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

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.     

Stability and pattern formation in a coupled arbitrary order of autocatalysis system

Spatiotemporal structures arising in two identical cells, each governed by arbitrary order autocatalator kinetics and coupled via the diffusive interchange of a reactant, are discussed. The stability of two homogeneous steady states is obtained with the use of linear stability analysis. By studying the linearized equations, it is found that two steady states, in the uncoupled and coupled system respectively, may give rise to the possibility of bifurcations to spatially nonuniform pattern forms. Further information about Turing bifurcation solutions close to these bifurcation points are obtained by weakly nonlinear theory. It is seen that the coupling leads to bifurcations not present in the uncoupled system which give rise to locally stable nonuniform pattern forms. Finally the stability of the equilibrium points of the amplitude equation is discussed by weakly nonlinear theory, with the bifurcation branches about small coupled system with 0<α?1$0<\alpha ?1$ and large coupling for α?1$\alpha ?1$.     

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.     

Pattern formation in a general two-cell Brusselator model

In this article, we are concerned with the following general coupled two-cell Brusselator-type system:

     

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.}     

Multi-agent system model with mixed coupling topologies for pattern formation and formation splitting

In this article, a new type of multi-agent system model with mixed coupling topologies is proposed for realizing pattern formations with specific geometric shapes and formation splitting. The interactions among individual agents are assumed to be universally repulsive and selectively attractive. By designing the form of attractive coupling matrix, one can obtain a variety of formations with specific shapes in the system through self-assembly of agents. Both symmetric coupling case and asymmetric coupling case are considered. Analysis and simulation results show symmetric ones result in convergent dynamics to steady-state formations, whereas, for asymmetric case, the system exhibits complex dynamic behaviours, including collective rotation and chaotic motion. By breaking the graph defined by attractive couplings into disjoint subgraphs, one can make the formation of agents to split into small sizes. The results are relevant for the design of coordination and cooperative control for multi-agent systems.     

Sharp error estimates for discretizations of the 1D convection-diffusion equation with Dirac initial data

This paper derives sharp estimates of the error arising fromexplicit and implicit approximations of the constant-coefficient1D convection–diffusion equation with Dirac initial data.The error analysis is based on Fourier analysis and asymptoticapproximation of the integrals resulting from the inverse Fouriertransform. This research is motivated by applications in computationalfinance and the desire to prove convergence of approximationsto adjoint partial differential equations.     

Strong stabilization of the system of linear elasticity by a Dirichlet boundary feedback

In this paper, by using the Nagy–Foias–Foguel theoryof decomposition of continuous semigroups of contractions, we prove that the system of linear elasticity is strongly stabilizable by a Dirichlet boundary feedback. We also give a concise proofof a theorem of Dafermos about the stability of thermoelasticity.     

Pattern formation of a diffusive predator-prey model with herd behavior and nonlocal prey competition

In this paper, we study the influence of the nonlocal interspecific competition of the prey population on the dynamics of the diffusive predator-prey model with prey social behavior. Using the linear stability analysis, the conditions for the positive constant steady state at which undergoes Hopf bifurcation, T-H bifurcation (Turing-Hopf bifurcation) are investigated. The Turing patterns occur in the presence of the nonlocal competition and cannot be found in the original system. For determining the dynamical behavior near T-H bifurcation point, the normal form of the T-H bifurcation has been used. Some graphical representations are provided to illustrate the theoretical results.     

Pattern formation in reaction-diffusion neural networks with leakage delay

Due to the heterogeneity of the electromagnetic field in neural networks, the diffusion phenomenon of electrons exists inevitably. In this paper, we investigate pattern formation in a reaction-diffusion neural network with leakage delay. The existence of Hopf bifurcation, as well as the necessary and sufficient conditions for Turing instability, are studied by analyzing the corresponding characteristic equation. Based on the multiple-scale analysis, amplitude equations of the model are derived, which determine the selection and competition of Turing patterns. Numerical simulations are carried out to show the possible patterns and how these patterns evolve. In some cases, the stability performance of Turing patterns is weakened by leakage delay and synaptic transmission delay.     

A Lyapunov functional for a triangular reaction–diffusion system with nonlinearities of exponential growth

The aim of this work is to study the global existence of solutions to a triangular system of reaction–diffusion equations, which describes epidemiological or chemical situations. On the basis of the construction of a suitable Lyapunov functional, we show that for any initial data, classical global solutions exist even when the nonlinearities are of exponential growth. Copyright © 2012 John Wiley & Sons, Ltd.     

Aggregation of microglia in 2D with string gradient weighted moving finite elements

Alzheimer's disease (AD) is a severe neurodegenerative disorder characterised by cognitive impairment and dementia. In the AD‐affected brain, microglia cells are up‐regulated and accumulate at senile plaques, the most prominent pathological feature of AD. In order to further study and predict the movement of activated microglia, we utilised their chemotactic properties. Specifically, we formulated the string gradient weighted moving finite element method for a system of partial differential equations in two dimensions, which includes nonlinear diffusion of a different variable found in chemotaxis models. The method was applied successfully to solve highly nonlinear chemorepulsion–chemorepellent models in two dimensions, and the results were compared with one‐dimensional results found previously in the literature. We conclude that the string gradient weighted moving finite element method is easily applied to chemotaxis models, in particular movement and aggregation of microglia, resulting in the ability to study the models extended in two dimensions efficiently. Our study highlights the feasibility and power of mathematical modelling to advance our understanding of pathophysiological processes in neurodegenerative diseases, including AD. Copyright © 2012 John Wiley & Sons, Ltd.