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

Nabil T Fadai,Matthew J Simpson

doi:10.1088/1751-8121/ab6d3c

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.

Highlights

Travelling waves arise in many fields, including ecology [1,2,3,4], cell biology[5,6,7,8,9,10,11], and industrial applications involving heat and mass transfer [12,13,14,15]
Smooth fronts do not have compact support, which makes defining the “edge” of the moving front ambiguous [8, 9, 16]. This feature of the FisherKPP model means that it can be hard to apply to practical problems, such as cell invasion [7,8,9], where well-defined fronts are often observed
We consider new travelling wave solutions for the Porous-Fisher model with a moving boundary. This model has certain features that could be considered to be advantageous over other modifications of the Fisher-KPP model

Summary

Introduction

Travelling waves arise in many fields, including ecology [1,2,3,4], cell biology[5,6,7,8,9,10,11], and industrial applications involving heat and mass transfer [12,13,14,15]. While the Fisher-Stefan model has the advantage that it can lead to sharp-fronted travelling waves, the physical or biological explanation of the outward flux at x = L(t) is not obvious Both the Porous-Fisher and the Fisher-Stefan models have the advantage that they lead to travelling wave solutions with a well-defined front (Figure 1b,c). While these two modifications of the Fisher-KPP model have been considered previously, the extension of combining nonlinear degenerate diffusion with a moving boundary condition has yet to be considered. Preliminary numerical solutions of the Por√ous-Fisher-Stefan model suggest that travelling wave solutions exist with 0 ≤ c < 1/ 2 and that these sharp-fronted travelling waves have infinite slope at x = L(t) (Figure 1d) Neither of these properties have been reported or analyzed previously. We determine an approximate form of the travelling wave front that matches numericallycomputed travelling wave solutions with high accuracy

Travelling waves in the Porous-Fisher-Stefan model

Travelling wave solutions for c 1

Comparison of travelling wave solutions

Conclusions