  The question addressed here is the long time evolution of the solutions to a
class of one-dimensional reaction-diffusion equations, in which the diffusion
is given by an integral operator. The underlying motivation, discussed in the
first chapter, is the mathematical analysis of models for biological invasions.
The model under study, while simple looking, is of current use in real life
situations. Interestingly, it arises in totally different contexts, such as the
study of branching random walks in probability theory.
  While the model under study has attracted a lot of attention, and while many
partial results about the time asymptotic behaviour of its solutions have been
proved over the last decades, some basic questions on the sharp asymptotics
have remained unanswered. One ambition of this monograph is to close these gaps
and to provide a complete and unified treatment of the equation.
  In some of the situations that we envisage, the level sets organise
themselves into an invasion front that is asymptotically linear in time, up to
a correction that converges exponentially in time to a constant. In other
situations, that constitute the main and newest part of the work, the
correction is asymptotically logarithmic in time. Despite these apparent
different behaviours, there is an underlying common way of thinking in the
study of all these situations.
  The ideas presented in the book apply to more elaborate systems modelling
biological invasions or the spatial propagation of epidemics. The models
themselves may be multidimensional, but they all have in common a mechanism
imposing the propagation in a given direction; examples are presented the
problems that conclude each chapter. These ideas should also be useful in the
treatment of further models that we are not able envisage at the time being.
