  While there is a long history of employing moving boundary problems in
physics, in particular via Stefan problems for heat conduction accompanied by a
change of phase, more recently such approaches have been adapted to study
biological invasion. For example, when a logistic growth term is added to the
governing partial differential equation in a Stefan problem, one arrives at the
Fisher-Stefan model, a generalisation of the well-known Fisher-KPP model,
characterised by a leakage coefficient $\kappa$ which relates the speed of the
moving boundary to the flux of population there. This Fisher-Stefan model
overcomes one of the well-known limitations of the Fisher-KPP model, since
time-dependent solutions of the Fisher-Stefan model involve a well-defined
front with compact support which is more natural in terms of mathematical
modelling. Almost all of the existing analysis of the standard Fisher-Stefan
model involves setting $\kappa > 0$, which can lead to either invading
travelling wave solutions or complete extinction of the population. Here, we
demonstrate how setting $\kappa < 0$ leads to retreating travelling waves and
an interesting transition to finite-time blow-up. For certain initial
conditions, population extinction is also observed. Our approach involves
studying time-dependent solutions of the governing equations, phase plane and
scaling analysis, leading to new insight into the possibilities of travelling
waves, blow-up and extinction for this moving boundary problem. Matlab software
used to generate the results in this work are available on Github.
