  Biological invasion, whereby populations of motile and proliferative
individuals lead to moving fronts that invade into vacant regions, are
routinely studied using partial differential equation (PDE) models based upon
the classical Fisher--KPP model. While the Fisher--KPP model and extensions
have been successfully used to model a range of invasive phenomena, including
ecological and cellular invasion, an often--overlooked limitation of the
Fisher--KPP model is that it cannot be used to model biological recession where
the spatial extent of the population decreases with time. In this work we study
the \textit{Fisher--Stefan} model, which is a generalisation of the Fisher--KPP
model obtained by reformulating the Fisher--KPP model as a moving boundary
problem. The nondimensional Fisher--Stefan model involves just one single
parameter, $\kappa$, which relates the shape of the density front at the moving
boundary to the speed of the associated travelling wave, $c$. Using numerical
simulation, phase plane and perturbation analysis, we construct approximate
solutions of the Fisher--Stefan model for both slowly invading and slowly
receding travelling waves, as well as for rapidly receding travelling waves.
These approximations allow us to determine the relationship between $c$ and
$\kappa$ so that commonly--reported experimental estimates of $c$ can be used
to provide estimates of the unknown parameter $\kappa$. Interestingly, when we
reinterpret the Fisher--KPP model as a moving boundary problem, many
disregarded features of the classical Fisher--KPP phase plane take on a new
interpretation since travelling waves solutions with $c < 2$ are not normally
considered. This means that our analysis of the Fisher--Stefan model has both
practical value and an inherent mathematical value.
