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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.



Chapter 1. Scalar Reaction-Diffusion Equations: Conditional Symmetry, Exact Solutions and Applications

All the main results on Q-conditional symmetry (nonclassical symmetry) of the general class of nonlinear reaction-diffusion-convection equations are summarized. Although some of them were published about 25 years ago, and the others were derived in the 2000s, it is the first attempt to present an extensive review of this matter. It is shown that several well-known equations arising in applications and their direct generalizations possess conditional symmetry. Notably, the Murray, Fitzhugh–Nagumo, and Huxley equations and their natural generalizations are identified. Moreover, several exact solutions (including travelling fronts) are constructed using the conditional symmetries obtained in order to find exact solutions with a biological interpretation.
Roman Cherniha, Vasyl’ Davydovych

Chapter 2. Q-Conditional Symmetries of Reaction-Diffusion Systems

A recently developed theoretical background for searching Q-conditional (nonclassical) symmetries of systems of evolution partial differential equations is presented. We generalize the standard definition of Q-conditional symmetry by introducing the notion of Q-conditional symmetry of the p-th type and show that different types of symmetry of a given system generate a hierarchy of conditional symmetry operators. It is shown that Q-conditional symmetry of the p-th type possesses some special properties, which distinguish them from the standard conditional symmetry. The general class of two-component nonlinear reaction-diffusion systems is examined in order to find the Q-conditional symmetry operators. The relevant systems of so-called determining equations are solved under additional restrictions. As a result, several reaction-diffusion systems possessing conditional symmetry are constructed. In particular, it is shown that the diffusive Lotka–Volterra system, the Belousov–Zhabotinskii system (with the correctly specified coefficients) and some of their generalizations admit Q-conditional symmetry.
Roman Cherniha, Vasyl’ Davydovych

Chapter 3. Conditional Symmetries and Exact Solutions of Diffusive Lotka–Volterra Systems

Two- and three-component diffusive Lotka–Volterra systems are examined in order to find Q-conditional symmetries, to construct exact solutions and to provide their biological interpretation. An exhaustive description of Q-conditional symmetries of the first type (a special subset of nonclassical symmetries) of these nonlinear systems is derived. An essential part of this chapter is devoted to the construction of exact solutions of the systems in question using the symmetries obtained. Starting from examples of travelling fronts (finding such solutions is important from the applicability point of view), we concentrate mostly on finding exact solutions with a more complicated structure. As a result, a wide range of exact solutions are constructed for the two-component diffusive Lotka–Volterra system and some examples are presented for the three-component diffusive Lotka–Volterra system. Moreover, a realistic interpretation for two and three competing species is provided for some exact solutions.
Roman Cherniha, Vasyl’ Davydovych

Chapter 4. Q-Conditional Symmetries of the First Type and Exact Solutions of Nonlinear Reaction-Diffusion Systems

Two classes of two-component nonlinear reaction-diffusion systems are studied in order to find Q-conditional symmetries of the first type (a special subset of nonclassical symmetries), to construct exact solutions, and to show their applicability. The first class involves systems with constant coefficient of diffusivity, while the second contains systems with variable diffusivities only. The main theoretical results are given in the form of two theorems presenting exhaustive lists (up to the given sets of point transformations) of the reaction-diffusion systems belonging to the above classes and admitting Q-conditional symmetries of the first type. The reaction-diffusion systems obtained allow one to extract specific systems occurring in real-world models. A few examples are presented, including a modification of the classical prey–predator system with diffusivity and a system modelling the gravity-driven flow of thin films of viscous fluid. Exact solutions with attractive properties are found for these nonlinear systems and their possible biological and physical interpretations are presented.
Roman Cherniha, Vasyl’ Davydovych


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