Critical Length for the Spreading–vanishing Dichotomy in Higher Dimensions

AbstractIn this study, we couple intracellular signalling and cell–based mechanical properties to develop a novel free boundary mechanobiological model of epithelial tissue dynamics. Mechanobiological coupling is introduced at the cell level in a discrete modelling framework, and new reaction–diffusion equations are derived to describe tissue–level outcomes. The free boundary evolves as a result of the underlying biological mechanisms included in the discrete model. To demonstrate the accuracy of the continuum model, we compare numerical solutions of the discrete and continuum models for two different signalling pathways. First, we study the Rac–Rho pathway where cell- and tissue–level mechanics are directly related to intracellular signalling. Second, we study an activator–inhibitor system which gives rise to spatial and temporal patterning related to Turing patterns. In all cases, the continuum model and free boundary condition accurately reflect the cell–level processes included in the discrete model.

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“…Equations (32)- (37) are solved numerically using a boundary fixing transformation [44]. In doing so, Equations 32…”

Section: Continuum Modelmentioning

confidence: 99%

A free boundary mechanobiological model of epithelial tissues

Tambyah

Murphy

Buenzli

et al. 2020

Preprint

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“…Du and Lin 16 refer to this situation as the spreading‐vanishing dichotomy. Aspects of these phenomena have been studied rigorously and numerically by Du and coworkers, 16,28–31,50 us, 27,46–49 and others.…”

Section: Discussionmentioning

confidence: 99%

“…In very recent times, a variety of extensions have been documented, including a rigorous results for different boundary conditions, nonlinear or nonlocal diffusion, generalized reaction terms, with one or two phases, as well as motivation in terms of applications in ecology 32–45 . Formal results and numerical simulations have been described by us 27,46–49 and others 50 …”

Section: Introductionmentioning

confidence: 99%

Traveling waves, blow‐up, and extinction in the Fisher–Stefan model

2021

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While there is a long history of employing moving boundary problems in physics, in particular via Stefan problems for heat conduction accompanied by a change of phase, more recently such approaches have been adapted to study biological invasion. For example, when a logistic growth term is added to the governing partial differential equation in a Stefan problem, one arrives at the Fisher–Stefan model, a generalization of the well‐known Fisher–KPP model, characterized by a leakage coefficient κ$\kappa$ which relates the speed of the moving boundary to the flux of population there. This Fisher–Stefan model overcomes one of the well‐known limitations of the Fisher–KPP model, since time‐dependent solutions of the Fisher–Stefan model involve a well‐defined front which is more natural in terms of mathematical modeling. Almost all of the existing analysis of the standard Fisher–Stefan model involves setting κ>0$\kappa > 0$, which can lead to either invading traveling wave solutions or complete extinction of the population. Here, we demonstrate how setting κ<0$\kappa < 0$ leads to retreating traveling waves and an interesting transition to finite‐time blow‐up. For certain initial conditions, population extinction is also observed. Our approach involves studying time‐dependent solutions of the governing equations, phase plane, and asymptotic analysis, leading to new insight into the possibilities of traveling waves, blow‐up, and extinction for this moving boundary problem. MATLAB software used to generate the results in this work is available on Github.

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Section: Introductionmentioning

confidence: 99%

“…Simpson [42] extended the work of Du and Lin [37] by considering the Fisher-Stefan model on an n-sphere. By replacing the second derivative term in (1.4a) with the ndimensional radially-symmetric Laplacian operator, Simpson [42] showed that a critical radius, R c , governs survival and extinction analogously to the critical length. This critical radius depends on the dimension n. In the two-dimensional spreading-disc problem, the critical radius is R c = α 01 ≈ 2.4048, where α 01 is the first zero of J 0 (x), the zeroth-order Bessel function of the first kind [42].…”

Section: Introductionmentioning

confidence: 99%

The Effect of Geometry on Survival and Extinction in a Moving-Boundary Problem Motivated by the Fisher-KPP Equation

Tam,

Simpson

2022

Preprint

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The Fisher-Stefan model involves solving the Fisher-KPP equation on a domain whose boundary evolves according to a Stefan-like condition. The Fisher-Stefan model alleviates two practical limitations of the standard Fisher-KPP model when applied to biological invasion. First, unlike the Fisher-KPP equation, solutions to the Fisher-Stefan model have compact support, enabling one to define the interface between occupied and unoccupied regions unambiguously. Second, the Fisher-Stefan model admits solutions for which the population becomes extinct, which is not possible in the Fisher-KPP equation. Previous research showed that population survival or extinction in the Fisher-Stefan model depends on a critical length in one-dimensional Cartesian or radially-symmetric geometry. However, the survival and extinction behaviour for general two-dimensional regions remains unexplored. We combine analysis and level-set numerical simulations of the Fisher-Stefan model to investigate the survival-extinction conditions for rectangular-shaped initial conditions. We show that it is insufficient to generalise the critical length conditions to critical area in two-dimensions. Instead, knowledge of the region geometry is required to determine whether a population will survive or become extinct.

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Critical Length for the Spreading–vanishing Dichotomy in Higher Dimensions

Cited by 9 publications

References 58 publications

A free boundary mechanobiological model of epithelial tissues

A free boundary mechanobiological model of epithelial tissues

Traveling waves, blow‐up, and extinction in the Fisher–Stefan model

The Effect of Geometry on Survival and Extinction in a Moving-Boundary Problem Motivated by the Fisher-KPP Equation

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