Propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion

Abstract

The propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion coefficients is studied. Using coordinate changes, WKB approximations, and multiple scales analysis, we provide an analytic framework that describes propagation of the front up to the minimum of the diffusion coefficient. We also present results showing the behavior of the front after it passes the minimum. In each case, we show that standard traveling coordinate frames do not properly describe front propagation. Last, we provide numerical simulations to support our analysis and to show, that around the minimum, the motion of the front is arrested on asymptotically significant time scales.

Mathematical models for cell migration with real-time cell cycle dynamics

“…We observe that the numericallyestimated wave speed is consistent with Eq (26) over the range of κ we consider. Therefore, we have numerical evidence to strongly support the claim that the minimum speed is given by Eq (26).…”

“…Therefore, understanding the relationship between the parameters in the mathematical model and the speed of the travelling wave solution is a useful way to help parameterise the mathematical model to match experimental observations. Travelling wave solutions have also been observed in other mathematical models of cell migration [16,19,21,22], as well as other reaction-diffusion models related to biological processes [23,24,25,26]. Most travelling wave solutions take the form of moving wavefronts, which have a monotone profile.…”

Investigating the Reproducibility of In Vitro Cell Biology Assays Using Mathematical Models

“…Most continuum [14,22,76,98,100] and discrete [15,109,111] models of collective cell spreading are associated with a logistic growth source term to model cell proliferation.…”

“…Mean field analysis of this traditional proliferation mechanism leads to the logistic source term in the partial differential equation (PDE) description of the model [15,109,111]. In terms of continuum models, carrying-capacity limited proliferation is often represented using the logistic equation, dC / dt = λ C(1 − C) [14,22,76,98,100,102], where λ is the proliferation rate, and the density has been scaled relative to the carrying capacity. However, there is some awareness that the logistic model does not always match experimental data.…”

“…Firstly, a continuum reaction-diffusion equation can be applied to mimic certain features of the experiment [76]. Most previous continuum models represent cell migration with a diffusiontype mechanism, and a logistic source term to represent carrying capacity-limited proliferation [14,22,76,98,100,102]. Secondly, a discrete random walk model can be used to mimic certain features of the experiment [19,46].…”