New travelling wave solutions of the Porous–Fisher model with a moving boundary
Nabil T. Fadai, Matthew J. Simpson
Abstract
Abstract We examine travelling wave solutions of the Porous–Fisher model, , with a Stefan-like condition at the moving front, . 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, ; (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, and , we obtain closed-form mathematical expressions for the travelling wave shape and speed. These approximations compare well with numerical solutions of the full problem.