Download or read book Propagation Phenomena in a Bistable Reaction Diffusion System written by John Rinzel and published by . This book was released on 1981 with total page 63 pages. Available in PDF, EPUB and Kindle. Book excerpt: Consideration is given to a system of reaction diffusion equations which have qualitative significance for several applications including nerve conduction and distributed chemical/biochemical systems. These equations are of the FitzHugh-Nagumo type and contain three parameters. For certain ranges of the parameters the system exhibits two stable singular points. A singular perturbation construction is given to illustrate that there may exist both pulse type and transition type traveling waves. A complete, rigorous, description of which of these waves exist for a given set of parameters and how the velocities of the waves vary with the parameters is given for the case when the system has a piecewise linear nonlinearity. Numerical results of solutions to these equations are also presented. These calculations illustrate how waves are generated from initial data, how they interact during collisions, and how they are influenced by local disturbances and boundary conditions.
Download or read book Propagation Phenomena of Integro difference Equations and Bistable Reaction diffusion Equations in Periodic Habitats written by Weiwei Ding and published by . This book was released on 2014 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation is concerned with propagation phenomena of some evolution equations in periodic habitats. The main results consist of the following two parts. In the first part, we investigate the spatial spreading phenomena of some spatially periodic integro-difference equations. Firstly, we establish a general theory on the existence of spreading speeds for noncompact evolution systems, under the hypothesis that the linearized systems have principal eigenvalues. Secondly, we introduce the notion of uniform irreducibility for finite Radon measures on the circle. It is shown that, any generalized convolution operator generated by such a measure admits a principal eigenvalue. Finally, applying the above general theories, we prove the existence of spreading speeds for some integro-difference equations with uniformly irreducible dispersal kernels. In the second part, we study the front propagation phenomena of spatially periodic reaction-diffusion equations with bistable nonlinearities. Firstly, we focus on the propagation solutions in the class of pulsating fronts. It is proved that, under various assumptions on the reaction terms, pulsating fronts exist when the spatial period is small or large. We also characterize the sign of the front speeds and we show the global exponential stability of the pulsating fronts with nonzero speed. Secondly, we investigate the propagation solutions in the larger class of transition fronts. It is shown that, under suitable assumptions, transition fronts are reduced to pulsating fronts with nonzero speed. But we also prove the existence of new types of transition fronts which are not pulsating fronts.
Download or read book Front Propagation for Reaction diffusion Equations of Bistable Type written by G. Barles and published by . This book was released on 1990 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Traveling Front Solutions in Reaction Diffusion Equations written by Masaharu Taniguchi and published by . This book was released on 2021-05-28 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study on traveling fronts in reaction-diffusion equations is the first step to understand various kinds of propagation phenomena in reaction-diffusion models in natural science. One dimensional traveling fronts have been studied from the 1970s, and multidimensional ones have been studied from around 2005. This volume is a text book for graduate students to start their studies on traveling fronts. Using the phase plane analysis, we study the existence of traveling fronts in several kinds of reaction-diffusion equations. For a nonlinear reaction term, a bistable one is a typical one. For a bistable reaction-diffusion equation, we study the existence and stability of two-dimensional V-form fronts, and we also study pyramidal traveling fronts in three or higher space dimensions. The cross section of a pyramidal traveling front forms a convex polygon. It is known that the limit of a pyramidal traveling front gives a new multidimensional traveling front. For the study the multidimensional traveling front, studying properties of pyramidal traveling fronts plays an important role. In this volume, we study the existence, uniqueness and stability of a pyramidal traveling front as clearly as possible for further studies by graduate students. For a help of their studies, we briefly explain and prove the well-posedness of reaction-diffusion equations and the Schauder estimates and the maximum principles of solutions.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets
Download or read book Two Examples of Reaction diffusion Front Propagation in Heterogeneous Media written by Antoine Pauthier and published by . This book was released on 2016 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this thesis is to study two examples of propagation phenomena in heterogeneous reaction-diffusion equations.The purpose of the first part is to understand the effect of nonlocal exchanges between a line of fast diffusion and a two dimensional environment in which reaction-diffusion of KPP type occurs. The initial model was introduced in 2013 by Berestycki, Roquejoffre, and Rossi. In the first chapter we investigate how the nonlocal coupling between the line and the plane enhances the spreading in the direction of the line; we also investigate how different exchange functions may maximize or not the spreading speed.The second chapter is concerned with the singular limit of nonlocal exchanges that tend to Dirac masses. We show the convergence of the dynamics in a rather strong sense. In the third chapter we study the limit of long range exchanges with constant mass. It gives an infimum for the asymptotic speed of spreading for these models that still could be bigger than the usual KPP spreading speed.The second part of this thesis is concerned with entire solutions for heterogeneous bistable equations.We consider a two dimensional domain infinite in one direction, bounded in the other, that converges to a cylinder as x goes to minus infinity. We prove the existence of an entire solution in such a domain which is the bistable wave for t tends to minus infinity. It also lead us to investigate a one dimensional model with a non-homogeneous reaction term,for which we prove the same property.
Download or read book Propagation Phenomena in Reaction advection diffusion Equations written by Christopher Henderson and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Reaction-advection-diffusion (RAD) equations are a class of non-linear parabolic equations which are used to model a diverse range of biological, physical, and chemical phenomena. Originally introduced in the early twentieth century as a model for population dynamics, they have been used in recent years in diverse contexts including climate change, criminal behavior, and combustion. These equations are characterized by the combination of three behaviors: spreading, stirring, and growth/decay. The main focus of mathematical research into RAD equations over the past century has been in characterizing the propagation of solutions. Indeed, these equations are characterized by the invasion of an unstable state by a stable state at a constant rate (for instance, the invasion of empty space by a population until the environmental carrying capacity is reached). In general, this can be characterized by the existence, uniqueness, and stability of traveling wave solutions, or solutions with a fixed profile which move at a constant speed in time. In general, the speed and shape of these traveling waves gives us the speed with which the stable state invades the unstable state. This thesis assumes the following trajectory, investigating two specific RAD equations: the Fisher-KPP equation, used in population dynamics, and a coupled reactive-Boussinesq system, used to model combustion in a fluid. For the former equation, we prove results regarding the precise spreading rate, and for the latter equation, we prove an existence result for a special solution that generalizes the traveling wave. In the first part of this thesis, we prove two results quantifying the precise speed of spreading for solutions to the Cauchy problem of the Fisher-KPP equation. The first of these results, concerning localized initial data, provides intuition for a lower order term obtained non-rigorously in. Specifically, we prove a quantitative convergence-to-equilibrium result in a related model, which has been used as a close approximation of the Fisher-KPP equation. The second of these results, concerning non-localized initial data and building on the work of Hamel and Roques, quantifies the super-linear in time spreading of the population. Here we compute the highest order term in the spreading for a broad class of initial data. In the second part of this thesis, we look at a coupled system that models combustion in a fluid, and we prove a qualitative propagation result. Unlike classical models, this relatively new system accounts for the effect of advection induced by the buoyancy force that results from the evolution of the temperature. Essentially, this means that we take into account the phenomenon that ``hot air rises.'' We exhibit a generalized traveling wave solution of this system, called a pulsating front, in two-dimensional periodic domains. To our knowledge, this is the first result regarding the existence of ``pulsating fronts'' in a coupled system.
Download or read book Fast Propagation in Reaction diffusion Equations with Fractional Diffusion written by Anne-Charline Coulon Chalmin and published by . This book was released on 2014 with total page 175 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis focuses on the long time behaviour, and more precisely on fast propagation, in Fisher-KPP reaction diffusion equations involving fractional diffusion. This type of equation arises, for example, in spreading of biological species. Under some specific assumptions, the population invades the medium and we want to understand at which speed this invasion takes place when fractional diffusion is at stake. To answer this question, we set up a new method and apply it on different models. In a first part, we study two different problems, both including fractional diffusion : Fisher-KPP models in periodic media and cooperative systems. In both cases, we prove, under additional assumptions, that the solution spreads exponentially fast in time and we find the precise exponent of propagation. We also carry out numerical simulations to investigate the dependence of the speed of propagation on the initial condition. In a second part, we deal with a two dimensional environment, where reproduction of Fisher-KPP type and usual diffusion occur, except on a line of the plane, on which fractional diffusion takes place. The plane is referred to as "the field" and the line to "the road", as a reference to the biological situations we have in mind. We prove that the speed of propagation is exponential in time on the road, whereas it depends linearly on time in the field. The expansion shape of the level sets in the field is investigated through numerical simulations.
Download or read book Propagation in Reaction diffusion Equations with Fractional Diffusion written by Anne-Charline Coulon and published by . This book was released on 2014 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis focuses on the long time behaviour of solutions to Fisher-KPP reaction-diffusion equations involving fractional diffusion. This type of equation arises, for example, in spatial propagation or spreading of biological species (rats, insects,...). In population dynamics, the quantity under study stands for the density of the population. It is well-known that, under some specific assumptions, the solution tends to a stable state of the evolution problem, as time goes to infinity. In other words, the population invades the medium, which corresponds to the survival of the species, and we want to understand at which speed this invasion takes place. To answer this question, we set up a new method to study the speed of propagation when fractional diffusion is at stake and apply it on three different problems. Part I of the thesis is devoted to an analysis of the asymptotic location of the level sets of the solution to two different problems : Fisher-KPP models in periodic media and cooperative systems, both including fractional diffusion. On the first model, we prove that, under some assumptions on the periodic medium, the solution spreads exponentially fast in time and we find the precise exponent that appears in this exponential speed of propagation. We also carry out numerical simulations to investigate the dependence of the speed of propagation on the initial condition. On the second model, we prove that the speed of propagation is once again exponential in time, with an exponent depending on the smallest index of the fractional Laplacians at stake and on the reaction term. Part II of the thesis deals with a two dimensional environment, where reproduction of Fisher-KPP type and usual diffusion occur, except on a line of the plane, on which fractional diffusion takes place. The plane is referred to as 'the field' and the line to 'the road', as a reference to the biological situations we have in mind. Indeed, it has long been known that fast diffusion on roads can have a driving effect on the spread of epidemics. We prove that the speed of propagation is exponential in time on the road, whereas it depends linearly on time in the field. Contrary to the precise asymptotics obtained in Part I, for this model, we are not able to give a sharp location of the level sets on the road and in the field. The expansion shape of the level sets in the field is investigated through numerical simulations.
Download or read book Propagation of Reactions in L vy Diffusion written by Tau Shean Lim and published by . This book was released on 2017 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study reaction-diffusion equations u[t] = [L]u + f(u) with homogeneous reactions and Lévy diffusion operators L. These equations are well-studied in classical diffusion L = Laplacian, featuring traveling fronts, wavefront propagation, existence of spreading speeds, etc. The main theme of this thesis is to extend the study of reaction-diffusion equations to a general diffusion model arising from the theory of Lévy processes. In the first part of this thesis, we focus on the one-dimensional theory of reaction-diffusion equations with homogeneous ignition/bistable reactions and symmetric Lévy diffusion operators. Particularly, existence and uniqueness of traveling fronts are established under the assumption that the underlying process {X[t]} admits a finite first moment. This condition for the first moment is also proved to be sharp for ignition media, as no front exists otherwise. In the second part of the thesis, we turn our attention to a class of multi-dimensional reaction-diffusion equations with nonlocal diffusion (a special case of Lévy diffusion) and homogeneous ignition, bistable or monostable reactions. Specifically, we prove some results regarding long time dynamics of their solutions, including spreading, quenching, and asymptotic spreading speeds.
Download or read book Origination and Propagation of Reaction Diffusion Waves in Three Spatial Dimensions written by Philip James Hahn and published by . This book was released on 2004 with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt: Propagation of functional or pathological ionic disturbances in biological systems plays an important role in normal regulatory mechanisms and in disease. Potassium diffusion in brain tissue is involved in spreading excitation. Models of this type of phenomenon often take the form of a reaction-diffusion system in one spatial dimension with continuous dynamic variables. Examined here is propagation in three spatial dimensions through a network of discrete dynamic elements coupled by diffusion. Conditions permissive of pulse origination and propagation can be determined analytically for systems in one spatial dimension. However, in three spatial dimensions or in dynamic systems containing discontinuities, explicit solutions may not exist. Instead, the local dynamics of the excitable system at a point in space are analyzed. The effective diffusive flux or current at a point is interpreted as a slowly varying parameter. The bifurcation structure of the dynamics with respect to this parameter and the effect of waveform on the time course of the parameter are examined. Propagation results when an excursion at a point produces a diffusion current sufficient to move its resting neighbor above some threshold value. The formation of a pulse back depends on the stability of equilibria of the local dynamics. Propagation in some cases may also depend on the geometry of the wavefront. Predictions are verified by numerical simulation using a software package developed by the author for this dissertation. A three dimensional lattice allows for description of the local dynamics at nodal elements and diffusion between elements and throughout the lattice. Three models are studied using the method developed. First, the Fitzhugh-Nagumo equation is used to illustrate the method. Second, the continuous Nelkin-Yaari model, describing spreading excitation in brain tissue, is examined. Third, a novel model of non-synaptic pulse propagation in hippocampal slices is developed and analyzed. Investigation of this new model shows that potassium wave behavior in the CA1 region can be explained using descriptions of only two phenomena: action potential spike dynamics in response to elevated potassium and simple sink functions that allow for the formation of a wave backside and refractory time.
Download or read book An Introduction to Anomalous Diffusion and Relaxation written by Luiz Roberto Evangelista and published by Springer Nature. This book was released on 2023-01-01 with total page 411 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a contemporary treatment of the problems related to anomalous diffusion and anomalous relaxation. It collects and promotes unprecedented applications dealing with diffusion problems and surface effects, adsorption-desorption phenomena, memory effects, reaction-diffusion equations, and relaxation in constrained structures of classical and quantum processes. The topics covered by the book are of current interest and comprehensive range, including concepts in diffusion and stochastic physics, random walks, and elements of fractional calculus. They are accompanied by a detailed exposition of the mathematical techniques intended to serve the reader as a tool to handle modern boundary value problems. This self-contained text can be used as a reference source for graduates and researchers working in applied mathematics, physics of complex systems and fluids, condensed matter physics, statistical physics, chemistry, chemical and electrical engineering, biology, and many others.
Download or read book Threshold Phenomena for a Reaction Diffusion System written by David Terman and published by . This book was released on 1981 with total page 53 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Differential Equations and Control Theory written by Z. Deng and published by CRC Press. This book was released on 2020-11-25 with total page 546 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work presents the proceedings from the International Conference on Differential Equations and Control Theory, held recently in Wuhan, China. It provides an overview of current developments in a range of topics including dynamical systems, optimal control theory, stochastic control, chaos, fractals, wavelets and ordinary, partial, functional and stochastic differential equations.
Download or read book Scientific and Technical Aerospace Reports written by and published by . This book was released on 1992 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Technical Abstract Bulletin written by and published by . This book was released on 1981 with total page 172 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Waves in reaction diffusion written by Joop Pauwelussen and published by . This book was released on 1981 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Mathematical Biology written by James D. Murray and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 783 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics has always benefited from its involvement with developing sciences. Each successive interaction revitalises and enhances the field. Biomedical science is clearly the premier science of the foreseeable future. For the continuing health of their subject mathematicians must become involved with biology. With the example of how mathematics has benefited from and influenced physics, it is clear that if mathematicians do not become involved in the biosciences they will simply not be a part of what are likely to be the most important and exciting scientific discoveries of all time. Mathematical biology is a fast growing, well recognised, albeit not clearly defined, subject and is, to my mind, the most exciting modern application of mathematics. The increasing use of mathematics in biology is inevitable as biol ogy becomes more quantitative. The complexity of the biological sciences makes interdisciplinary involvement essential. For the mathematician, biology opens up new and exciting branches while for the biologist mathematical modelling offers another research tool commmensurate with a new powerful laboratory technique but only if used appropriately and its limitations recognised. However, the use of esoteric mathematics arrogantly applied to biological problems by mathemati cians who know little about the real biology, together with unsubstantiated claims as to how important such theories are, does little to promote the interdisciplinary involvement which is so essential. Mathematical biology research, to be useful and interesting, must be relevant biologically.
