Revisiting the Fisher–Kolmogorov–Petrovsky–Piskunov equation to interpret the spreading–extinction dichotomy

Abstract

The Fisher–Kolmogorov–Petrovsky–Piskunov model, also known as 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 linearization, 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.

Critical Length for the Spreading–vanishing Dichotomy in Higher Dimensions

“…The solutions of equation (13) with κ = 1 are given in Figure 1(a)-(d), showing that the initial density evolves into a constant speed, constant shape travelling wave. Insights into such travelling waves can be obtained by phase plane analysis [15]. The evolution of L(t) is given in Figure 1…”

“…As we will show, the adoption of equation (12) alleviates both the shortcomings of the classical Fisher-Kolmogorov model, since this moving boundary analogue leads to solutions with well-defined fronts, as well as permitting certain initial conditions to become extinct. The moving boundary analogue of the Fisher-Kolmogorov model has been called the "Fisher-Stefan model" [15]. To simplify our analysis, we re-scale the variables: x = x/ √ D/λ, t = tλ and u = u/K.…”

A free boundary mechanobiological model of epithelial tissues

“…We express Equations (18)- (20) in terms of q(x, t) and f (x, t) through a change of variables from (ī, t) to (x, t) [5,8]. The change of variables gives…”

“…Many free boundary models use a classical one-phase Stefan condition to describe the evolution of the free boundary [20,21]. In these models, the speed of the free boundary is proportional to the spatial gradient of the density at the moving boundary [20,21]. Other free boundary models, particularly those used to study biological development, prespecify the rate of tissue elongation to match experimental observations [22][23][24][25][26][27][28].…”

Semi-infinite travelling waves arising in a general reaction–diffusion Stefan model

“…One particular reaction-diffusion modelling framework giving rise to semi-infinite travelling waves over a range of wavespeeds is via the incorporation of a moving boundary [8][9][10][11]24], whereby x ∈ (−∞, L(t)] and L(t) evolves based on a Stefan-like condition at the edge of travelling wave. For particular choices of linear [8][9][10]24] and degenerate diffusivities [11], D(u), semi-infinite travelling waves exist for all wavespeeds c ∈ [0, c * ], where the value of the critical wavespeed c * depends on D(u). However, generalisations of the models presented in [10,11] for a broader class of reaction functions, R(u) and nonlinear diffusivities, D (u), has yet to be considered.…”