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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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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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