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arXiv · 1910.09135

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

Nabil T. Fadai·Matthew J. Simpson
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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.

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Nabil T. Fadai, Matthew J. Simpson. 2019-10-21. New travelling wave solutions of the Porous-Fisher model with a moving boundary. https://doi.org/10.1088/1751-8121/ab6d3c

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