RT Journal Article
SR Electronic
T1 Revisiting the Fisher-KPP equation to interpret the spreading-extinction dichotomy
JF bioRxiv
FD Cold Spring Harbor Laboratory
SP 673202
DO 10.1101/673202
A1 Maud El-Hachem
A1 Scott W. McCue
A1 Wang Jin
A1 Yihong Du
A1 Matthew J. Simpson
YR 2019
UL http://biorxiv.org/content/early/2019/06/17/673202.abstract
AB The Fisher-KPP model supports travelling wave solutions that are successfully used to model numerous invasive phenomena with applications in biology, ecology, and combustion theory. However, there are certain phenomena that the Fisher-KPP model cannot replicate, such as the extinction of invasive populations. The Fisher-Stefan model is an adaptation of the Fisher-KPP model to include a moving boundary whose evolution is governed by a Stefan condition. The Fisher-Stefan model also supports travelling wave solutions; however, a key additional feature of the Fisher-Stefan model is that it is able to simulate population extinction, giving rise to a spreading-extinction dichotomy. In this work, we revisit travelling wave solutions of the Fisher-KPP model and show that these results provide new insight into travelling wave solutions of the Fisher-Stefan model and the spreading-extinction dichotomy. Using a combination of phase plane analysis, perturbation analysis and linearisation, we establish a concrete relationship between travelling wave solutions of the Fisher-Stefan model and often-neglected travelling wave solutions of the Fisher-KPP model. Furthermore, we give closed-form approximate expressions for the shape of the travelling wave solutions of the Fisher-Stefan model in the limit of slow travelling wave speeds, c ≪ 1.