This book introduces some basic mathematical tools in reaction-diffusion
models, with applications to spatial ecology and evolutionary biology.
It is divided into four parts.
The first part is an introduction to the maximum principle, the theory
of principal eigenvalues for elliptic and periodic-parabolic equations
and systems, and the theory of principal Floquet bundles.
The second part concerns the applications in spatial ecology. We discuss
the dynamics of a single species and two competing species, as well as
some recent progress on N competing species in bounded domains. Some
related results on stream populations and phytoplankton populations are
also included. We also discuss the spreading properties of a single
species in an unbounded spatial domain, as modeled by the Fisher-KPP
equation.
The third part concerns the applications in evolutionary biology. We
describe the basic notions of adaptive dynamics, such as evolutionarily
stable strategies and evolutionary branching points, in the context of a
competition model of stream populations. We also discuss a class of
selection-mutation models describing a population structured along a
continuous phenotypical trait.
The fourth part consists of several appendices, which present a
self-contained treatment of some basic abstract theories in functional
analysis and dynamical systems. Topics include the Krein-Rutman theorem
for linear and nonlinear operators, as well as some elements of monotone
dynamical systems and abstract competition systems.
Most of the book is self-contained and it is aimed at graduate students
and researchers who are interested in the theory and applications of
reaction-diffusion equations.
