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

TL;DR

This paper investigates novel travelling wave solutions of the Porous-Fisher model with a moving boundary, revealing unique properties such as slower speeds, non-extinction, compact support, and infinite slope at the front, supported by asymptotic and numerical analysis.

Contribution

It introduces new travelling wave solutions for the Porous-Fisher model with a moving boundary, including analytical expressions and characterization of their properties.

Findings

Travelling waves move slower than in the standard model.

Solutions never lead to population extinction.

Profiles have infinite slope at the moving front.

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.