Invading and Receding Sharp-Fronted Travelling Waves

Revisiting Fisher-KPP model to interpret the spatial spreading of invasive cell population in biology

In this paper, the homotopy analysis method, a powerful analytical technique, is applied to obtain analytical solutions to the Fisher-KPP equation in studying the spatial spreading of invasive species in ecology and to extract the nature of the spatial spreading of invasive cell populations in biology. The effect of the proliferation rate of the model of interest on the entire population is studied. It is observed that the invasive cell or the invasive population is decreased within a short time with the minimum proliferation rate. The homotopy analysis method is found to be superior to other analytical methods, namely the Adomian decomposition method, the homotopy perturbation method, etc. because of containing an auxiliary parameter, which provides us with a convenient way to adjust and control the region of convergence of the series solution. Graphical representation of the approximate series solutions obtained by the homotopy analysis method, the Adomian decomposition method, and the Homotopy perturbation method is illustrated, which shows the superiority of the homotopy analysis method. The method is examined on several examples, which reveal the ingenuousness and the effectiveness of the method of interest.

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Closed-form solutions are obtained for the Fisher-KPP equation through the Homotopy analysis method.

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The effect of the proliferation rate of the model of interest on the entire population is studied.

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The invasive cell or the invasive population decreases in short time with the minimum proliferation rate.

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The Homotopy analysis method is found superior over other analytical methods.

Non-vanishing sharp-fronted travelling wave solutions of the Fisher-Kolmogorov model

The Fisher-Kolmogorov-Petrovsky-Piskunov (KPP) model, and generalizations thereof, involves simple reaction-diffusion equations for biological invasion that assume individuals in the population undergo linear diffusion with diffusivity $D$, and logistic proliferation with rate $\lambda $. For the Fisher-KPP model, biologically relevant initial conditions lead to long-time travelling wave solutions that move with speed $c=2\sqrt {\lambda D}$. Despite these attractive features, there are several biological limitations of travelling wave solutions of the Fisher-KPP model. First, these travelling wave solutions do not predict a well-defined invasion front. Second, biologically relevant initial conditions lead to travelling waves that move with speed $c=2\sqrt {\lambda D}> 0$. This means that, for biologically relevant initial data, the Fisher-KPP model cannot be used to study invasion with $c \ne 2\sqrt {\lambda D}$, or retreating travelling waves with $c < 0$. Here, we reformulate the Fisher-KPP model as a moving boundary problem and show that this reformulated model alleviates the key limitations of the Fisher-KPP model. Travelling wave solutions of the moving boundary problem predict a well-defined front that can propagate with any wave speed, $-\infty < c < \infty $. Here, we establish these results using a combination of high-accuracy numerical simulations of the time-dependent partial differential equation, phase plane analysis and perturbation methods. All software required to replicate this work is available on GitHub.

A Continuum Mathematical Model of Substrate-Mediated Tissue Growth

We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{u}}(\hat{{\mathbf {x}}},{\hat{t}})$$\end{document}u^(x^,t^) that is coupled to the concentration of an immobile extracellular substrate, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{s}}(\hat{{\mathbf {x}}},{\hat{t}})$$\end{document}s^(x^,t^). Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{s}}$$\end{document}s^, while a logistic growth term models cell proliferation. The extracellular substrate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{s}}$$\end{document}s^ is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c = c_{\mathrm{min}}$$\end{document}c=cmin, as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c > c_{\mathrm{min}}$$\end{document}c>cmin. We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub.