Critical length for the spreading-vanishing dichotomy in higher dimensions


  • Matthew J. Simpson Queensland University of Technology



reaction–diffusion, Fisher–Kolmogorov model, Fisher’s equation, Stefan condition, moving boundary problem, travelling waves, invasion, extinction


We consider an extension of the classical Fisher–Kolmogorov equation, called the “Fisher–Stefan” model, which is a moving boundary problem on \(0<x<L(t)\) . A key property of the Fisher–Stefan model is the “spreading–vanishing dichotomy”, where solutions with \(L(t) > L_{\textrm{c}}\) will eventually spread as \(t \to \infty\) , whereas solutions where \(L(t) \ngtr L_{\textrm{c}}\) will vanish as \(t \to \infty\) . In one dimension it is well known that the critical length is \(L_{\textrm{c}} = \pi/2\) . In this work, we re-formulate the Fisher–Stefan model in higher dimensions and calculate \(L_{\textrm{c}}\) as a function of spatial dimensions in a radially symmetric coordinate system. Our results show how \(L_{\textrm{c}}\) depends upon the dimension of the problem, and numerical solutions of the governing partial differential equation are consistent with our calculations.

doi: 10.1017/S1446181120000103

Author Biography

Matthew J. Simpson, Queensland University of Technology

School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland 4001, Australia.





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