New travelling wave solutions of the Porous–Fisher model with a moving boundary

Fadai, Nabil T.; Simpson, Matthew J.

doi:10.1088/1751-8121/ab6d3c

Cited by 37 publications

(65 citation statements)

References 26 publications

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“…This generalization is often motivated by the desire to seek solutions with a welldefined sharp front. While it is certainly possible to generalize equation (2.4) to include degenerate nonlinear diffusion [16], an important consequence of this in the moving boundary context is that this extension does not lead to a spreadingvanishing dichotomy. This is because the flux at the moving boundary x = L(t), where u(L(t), t) = 0, is always zero, thereby always preventing extinction.…”

Section: Discussionmentioning

confidence: 99%

Critical Length for the Spreading–vanishing Dichotomy in Higher Dimensions

Simpson

2020

ANZIAM J.

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We consider an extension of the classical Fisher–Kolmogorov equation, called the “Fisher–Stefan” model, which is a moving boundary problem on $0<x<L(t)$ . A key property of the Fisher–Stefan model is the “spreading–vanishing dichotomy”, where solutions with $L(t)>L_{\text{c}}$ will eventually spread as $t\rightarrow \infty$ , whereas solutions where $L(t)\ngtr L_{\text{c}}$ will vanish as $t\rightarrow \infty$ . In one dimension it is well known that the critical length is $L_{\text{c}}=\unicode[STIX]{x1D70B}/2$ . In this work, we re-formulate the Fisher–Stefan model in higher dimensions and calculate $L_{\text{c}}$ as a function of spatial dimensions in a radially symmetric coordinate system. Our results show how $L_{\text{c}}$ depends upon the dimension of the problem, and numerical solutions of the governing partial differential equation are consistent with our calculations.

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Section: Discussionmentioning

confidence: 99%

Critical Length for the Spreading–vanishing Dichotomy in Higher Dimensions

Simpson

2020

ANZIAM J.

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“…We solve Equations ( 26)-( 28) by introducing a boundary fixing transformation x = l(t)ξ (Fadai and Simpson, 2020), so that the model becomes…”

Section: Appendix C: Numerical Methods For Model Describing Tumour Spheroids With Fucci Labellingmentioning

confidence: 99%

Mathematical Model of Tumour Spheroid Experiments with Real-Time Cell Cycle Imaging

et al. 2021

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Three-dimensional (3D) in vitro tumour spheroid experiments are an important tool for studying cancer progression and potential drug therapies. Standard experiments involve growing and imaging spheroids to explore how different experimental conditions lead to different rates of spheroid growth. These kinds of experiments, however, do not reveal any information about the spatial distribution of the cell cycle within the expanding spheroid. Since 2008, a new experimental technology called fluorescent ubiquitination-based cell cycle indicator (FUCCI), has enabled

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“…One way to overcome this limitation is to work with the Porous-Fisher model where the linear diffusion term is generalised to a degenerate nonlinear diffusion term with a power law diffusivity (Fadai and Simpson 2020 ; Sanchez and Maini 1994 ; Sengers et al. 2007 ; Witeslki 1994 ; Witelski 1995 ).…”

Section: Introductionmentioning

confidence: 99%

A Continuum Mathematical Model of Substrate-Mediated Tissue Growth

2022

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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, $${\hat{u}}(\hat{{\mathbf {x}}},{\hat{t}})$$ u ^ ( x ^ , t ^ ) that is coupled to the concentration of an immobile extracellular substrate, $${\hat{s}}(\hat{{\mathbf {x}}},{\hat{t}})$$ s ^ ( x ^ , t ^ ) . Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to $${\hat{s}}$$ s ^ , while a logistic growth term models cell proliferation. The extracellular substrate $${\hat{s}}$$ 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, $$c = c_{\mathrm{min}}$$ c = c min , as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, $$c > c_{\mathrm{min}}$$ c > c min . 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.

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New travelling wave solutions of the Porous–Fisher model with a moving boundary

Cited by 37 publications

References 26 publications

Critical Length for the Spreading–vanishing Dichotomy in Higher Dimensions

Critical Length for the Spreading–vanishing Dichotomy in Higher Dimensions

Mathematical Model of Tumour Spheroid Experiments with Real-Time Cell Cycle Imaging

A Continuum Mathematical Model of Substrate-Mediated Tissue Growth

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