New travelling wave solutions of the Porous-Fisher model with a moving boundary

Abstract

We examine travelling wave solutions of the Porous-Fisher model, ${\partial }_{t}u(x,t)=u(x,t)\left[1-u(x,t)\right]+{\partial }_x\left[u(x,t){\partial }_{x}u(x,t)\right]$, with a Stefan-like condition at the moving front, $x=L(t)$. Travelling wave solutions of this model have several novel characteristics. These travelling wave solutions: (i) move with a speed that is slower than the more standard Porous-Fisher model, $c<1/\sqrt{2}$; (ii) never lead to population extinction; (iii) have compact support and a well-defined moving front, and (iv) the travelling wave profiles have an infinite slope at the moving front. Using asymptotic analysis in two distinct parameter regimes, $c\to 0^{+}$ and $c\to 1/\sqrt{2}{}^{-}$, we obtain closed-form mathematical expressions for the travelling wave shape and speed. These approximations compare well with numerical solutions of the full problem.