Nonlinear reaction-diffusion systems. Conditional symmetry, exact solutions and their applications in biology (Q2399895)

The aim of this book is to identify particular solutions for a wide range of reaction-diffusion systems, many of which can be motivated from chemistry or mathematical biology (ecology, morphogenesis, chemotaxis\dots). The main idea to construct such particular solutions is to add well-chosen additional constraints that are compatible with the original equation, and so that the problem can be rewritten as an explicitly solvable ordinary differential equation. There are several ways to choose these constraints; in a rough sense, the stronger the constraints, the easier it is to solve the resulting ode but the narrower the range of applicability of the method. The authors focus on the so-called Q-conditional symmetries, which are applicable to several common models though under some restrictions. The first chapter introduces, after a short background on reaction-diffusion equations in mathematical biology, the definition of Q-conditional symmetries in the scalar case. Though it relies on higher order prolongations of differential operators, it is immediately applied in the context of parabolic equations so that general details are omitted. The approach is applied in particular to Fisher-KPP and FitzHugh-Nagumo equations where particular travelling fronts and mixed travelling fronts are constructed. The other three chapters are dedicated to reaction-diffusion systems. Chapter 2 mostly contains generalities: it first extends the definition of Q-conditional symmetries from the first chapter, then classifies among two component reaction-diffusion systems those which do admit such symmetries. Chapter 3 then goes into further details in the case of Lotka-Volterra systems, where these Q-conditional symmetries are used to construct particular solutions. Some short biological interpretations are given. The last chapter 4 then tackles two component systems with nonlinear diffusion. Again, an exhaustive list of systems which admit Q-conditional symmetries is given. This book is primarily adressed to mathematicians working in the field of reaction-diffusion systems. Biological mathematicians can readily use the particular solutions that are listed in this book (rather convenient summary tables are given). These solutions may provide theoretical insight on the dynamics of some reaction-diffusion systems, or be handy in modelling and numerical perspectives. For a deeper understanding, an a priori knowledge is quite recommended.