  This review provides open-access computational tools that support a range of
mathematical approaches to analyse three related scalar reaction-diffusion
models used to study biological invasion. Starting with the classic
Fisher-Kolmogorov (Fisher-KPP) model, we illustrate how computational methods
can be used to explore time-dependent partial differential equation (PDE)
solutions in parallel with phase plane and regular perturbation techniques to
explore invading travelling wave solutions moving with dimensionless speed $c
\ge 2$. To overcome the lack of a well-defined sharp front in solutions of the
Fisher-KPP model, we also review two alternative modeling approaches. The first
is the Porous-Fisher model where the linear diffusion term is replaced with a
degenerate nonlinear diffusion term. Using phase plane and regular perturbation
methods, we explore the distinction between sharp- and smooth-fronted invading
travelling waves that move with dimensionless speed $c \ge 1/\sqrt{2}$. The
second alternative approach is to reformulate the Fisher-KPP model as a moving
boundary problem on $0 < x < L(t)$, leading to the Fisher-Stefan model with
sharp-fronted travelling wave solutions arising from a PDE model with a linear
diffusion term. Time-dependent PDE solutions and phase plane methods show that
travelling wave solutions of the Fisher-Stefan model can describe both
biological invasion $(c > 0)$ and biological recession $(c < 0)$. Open source
Julia code to replicate all computational results in this review is available
on GitHub; we encourage researchers to use this code directly or to adapt the
code as required for more complicated models.
