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If we had to formulate in one sentence what this book is about it might be "How partial differential equations can help to understand heat explosion, tumor growth or evolution of biological species". These and many other applications are described by reaction-diffusion equations. The theory of reaction-diffusion equations appeared in the first half of the last century. In the present time, it is widely used in population dynamics, chemical physics, biomedical modelling. The purpose of this book is to present the mathematical theory of reaction-diffusion equations in the context of their numerous applications. We will go from the general mathematical theory to specific equations and then to their applications. Mathematical analysis of reaction-diffusion equations will be based on the theory of Fredholm operators presented in the first volume. Existence, stability and bifurcations of solutions will be studied for bounded domains and in the case of travelling waves. The classical theory of reaction-diffusion equations and new topics such as nonlocal equations and multi-scale models in biology will be considered.
Reaction-diffusion equations --- Differential equations, Elliptic --- Numerical solutions. --- Diffusion-reaction equations --- Equations, Reaction-diffusion --- Differential equations, Parabolic --- Differential equations, partial. --- Partial Differential Equations. --- Partial differential equations --- Partial differential equations. --- Mathematics --- Differential equations, Partial
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Differential equations --- Differential equations, Parabolic --- Équations différentielles paraboliques --- Reaction-diffusion equations. --- Équations de réaction-diffusion --- Asymptotic theory. --- Théorie asymptotique --- Reaction-diffusion equations --- Asymptotic theory of parabolic differential equations --- Asymptotic expansions --- Diffusion-reaction equations --- Equations, Reaction-diffusion --- Asymptotic theory --- Équations de réaction-diffusion. --- Théorie asymptotique.
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Mathematical statistics --- Mathematical physics --- Stochastic processes. --- Reaction-diffusion equations. --- Biology --- Processus stochastiques --- Équations de réaction-diffusion --- Biologie --- Mathematical models. --- Modèles mathématiques --- Reaction-diffusion equations --- Stochastic processes --- Random processes --- Probabilities --- Diffusion-reaction equations --- Equations, Reaction-diffusion --- Differential equations, Parabolic --- Biological models --- Biomathematics --- Mathematical models --- Processus stochastiques. --- Équations de réaction-diffusion. --- Modèles mathématiques.
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Mathematical physics --- Reaction-diffusion equations. --- Perturbation (Mathematics) --- Eigenvalues. --- Équations de réaction-diffusion --- Perturbation (mathématiques) --- Valeurs propres --- Eigenvalues --- Reaction-diffusion equations --- Diffusion-reaction equations --- Equations, Reaction-diffusion --- Differential equations, Parabolic --- Perturbation equations --- Perturbation theory --- Approximation theory --- Dynamics --- Functional analysis --- Matrices --- Équations de réaction-diffusion. --- Valeurs propres.
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Partial differential equations --- Reaction-diffusion equations. --- Bifurcation theory. --- Normal forms (Mathematics) --- Équations de réaction-diffusion --- Bifurcation, Théorie de la --- Formes normales (mathématiques) --- Bifurcation theory --- Reaction-diffusion equations --- Diffusion-reaction equations --- Equations, Reaction-diffusion --- Differential equations, Parabolic --- Forms, Normal (Mathematics) --- Mathematics --- Differential equations, Nonlinear --- Stability --- Numerical solutions --- Équations de réaction-diffusion. --- Bifurcation, Théorie de la.
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This book presents several fundamental results in solving nonlinear reaction-diffusion equations and systems using symmetry-based methods. Reaction-diffusion systems are fundamental modeling tools for mathematical biology with applications to ecology, population dynamics, pattern formation, morphogenesis, enzymatic reactions and chemotaxis. The book discusses the properties of nonlinear reaction-diffusion systems, which are relevant for biological applications, from the symmetry point of view, providing rigorous definitions and constructive algorithms to search for conditional symmetry (a nontrivial generalization of the well-known Lie symmetry) of nonlinear reaction-diffusion systems. In order to present applications to population dynamics, it focuses mainly on two- and three-component diffusive Lotka-Volterra systems. While it is primarily a valuable guide for researchers working with reaction-diffusion systems and those developing the theoretical aspects of conditional symmetry conception, parts of the book can also be used in master’s level mathematical biology courses.
Mathematics. --- Partial differential equations. --- Biomathematics. --- Mathematical physics. --- Mathematical and Computational Biology. --- Partial Differential Equations. --- Mathematical Physics. --- Biology --- Mathematics --- Partial differential equations --- Physical mathematics --- Physics --- Math --- Science --- Differential equations, partial. --- Reaction-diffusion equations. --- Diffusion-reaction equations --- Equations, Reaction-diffusion --- Differential equations, Parabolic
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The book introduces a hot topic of novel and emerging computing paradigms and architectures -computation by travelling waves in reaction-diffusion media. A reaction-diffusion computer is a massively parallel computing device, where the micro-volumes of the chemical medium act as elementary few-bit processors, and chemical species diffuse and react in parallel. In the reaction-diffusion computer both the data and the results of the computation are encoded as concentration profiles of the reagents, or local disturbances of concentrations, whilst the computation per se is performed via the spread
Reaction mechanisms (Chemistry) --- Chemical systems --- Reaction-diffusion equations --- Systems engineering. --- Engineering systems --- System engineering --- Engineering --- Industrial engineering --- System analysis --- Diffusion-reaction equations --- Equations, Reaction-diffusion --- Differential equations, Parabolic --- Systems, Chemical --- Chemistry, Physical and theoretical --- Phase rule and equilibrium --- Inorganic reaction mechanisms --- Mechanisms, Inorganic reaction --- Mechanisms, Reaction (Chemistry) --- Reaction mechanisms, Inorganic --- Chemical reaction, Conditions and laws of --- Data processing. --- Information technology. --- Design and construction
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Numerical solutions of algebraic equations --- Numerical solutions of differential equations --- 519.61 --- Error analysis (Mathematics) --- Numerical calculations --- Reaction-diffusion equations --- Diffusion-reaction equations --- Equations, Reaction-diffusion --- Differential equations, Parabolic --- Numerical analysis --- Errors, Theory of --- Instrumental variables (Statistics) --- Mathematical statistics --- Statistics --- Numerical methods of algebra --- Numerical calculations. --- Reaction-diffusion equations. --- Error analysis (Mathematics). --- 519.61 Numerical methods of algebra --- Équations de réaction-diffusion --- Calculs numériques --- Calcul d'erreur --- Équations de réaction-diffusion. --- Calculs numériques. --- Calcul d'erreur.
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539.219.3 --- Diffusion. Autodiffusion --- Chemical kinetics. --- Reaction-diffusion equations. --- Solid state chemistry. --- 539.219.3 Diffusion. Autodiffusion --- Chemical kinetics --- Reaction-diffusion equations --- Solid state chemistry --- Chemistry, Solid state --- Chemistry, Physical and theoretical --- Diffusion-reaction equations --- Equations, Reaction-diffusion --- Differential equations, Parabolic --- Chemical reaction, Kinetics of --- Chemical reaction, Rate of --- Chemical reaction, Velocity of --- Chemical reaction rate --- Chemical reaction velocity --- Kinetics, Chemical --- Rate of chemical reaction --- Reaction rate (Chemistry) --- Velocity of chemical reaction --- Chemical affinity --- Reactivity (Chemistry)
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This monograph is concerned with the mathematical analysis of patterns which are encountered in biological systems. It summarises, expands and relates results obtained in the field during the last fifteen years. It also links the results to biological applications and highlights their relevance to phenomena in nature. Of particular concern are large-amplitude patterns far from equilibrium in biologically relevant models. The approach adopted in the monograph is based on the following paradigms: • Examine the existence of spiky steady states in reaction-diffusion systems and select as observable patterns only the stable ones • Begin by exploring spatially homogeneous two-component activator-inhibitor systems in one or two space dimensions • Extend the studies by considering extra effects or related systems, each motivated by their specific roles in developmental biology, such as spatial inhomogeneities, large reaction rates, altered boundary conditions, saturation terms, convection, many-component systems. Mathematical Aspects of Pattern Formation in Biological Systems will be of interest to graduate students and researchers who are active in reaction-diffusion systems, pattern formation and mathematical biology.
Pattern formation (Biology) --- Reaction-diffusion equations. --- Diffusion-reaction equations --- Equations, Reaction-diffusion --- Biological pattern formation --- Mathematics. --- Partial differential equations. --- Biomathematics. --- Partial Differential Equations. --- Mathematical and Computational Biology. --- Genetics and Population Dynamics. --- Physiological, Cellular and Medical Topics. --- Biology --- Mathematics --- Partial differential equations --- Math --- Science --- Differential equations, Parabolic --- Developmental biology --- Differential equations, partial. --- Genetics --- Physiology --- Animal physiology --- Animals --- Anatomy --- Embryology --- Mendel's law --- Adaptation (Biology) --- Breeding --- Chromosomes --- Heredity --- Mutation (Biology) --- Variation (Biology) --- Biological systems --- Mathematical models.
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