We are analyzing https://link.springer.com/article/10.1007/s00285-025-02223-y.

Title:
Phenotype structuring in collective cell migration: a tutorial of mathematical models and methods | Journal of Mathematical Biology
Description:
Populations are heterogeneous, deviating in numerous ways. Phenotypic diversity refers to the range of traits or characteristics across a population, where for cells this could be the levels of signalling, movement and growth activity, etc. Clearly, the phenotypic distribution – and how this changes over time and space – could be a major determinant of population-level dynamics. For instance, across a cancerous population, variations in movement, growth, and ability to evade death may determine its growth trajectory and response to therapy. In this review, we discuss how classical partial differential equation (PDE) approaches for modelling cellular systems and collective cell migration can be extended to include phenotypic structuring. The resulting non-local models – which we refer to as phenotype-structured partial differential equations (PS-PDEs) – form a sophisticated class of models with rich dynamics. We set the scene through a brief history of structured population modelling, and then review the extension of several classic movement models – including the Fisher-KPP and Keller-Segel equations – into a PS-PDE form. We proceed with a tutorial-style section on derivation, analysis, and simulation techniques. First, we show a method to formally derive these models from underlying agent-based models. Second, we recount travelling waves in PDE models of spatial spread dynamics and concentration phenomena in non-local PDE models of evolutionary dynamics, and combine the two to deduce phenotypic structuring across travelling waves in PS-PDE models. Third, we discuss numerical methods to simulate PS-PDEs, illustrating with a simple scheme based on the method of lines and noting the finer points of consideration. We conclude with a discussion of future modelling and mathematical challenges.
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Keywords {🔍}

google, scholar, models, rho, population, cell, model, mathematical, mathscinet, phenotypic, mathbb, left, endaligned, pspde, beginaligned, dynamics, journal, form, phenotype, partial, cells, varepsilon, populations, equiv, growth, quad, biology, pde, equation, lorenzi, time, spatial, travelling, displaystyle, equations, state, nonlocal, rightarrow, function, space, mathcal, solutions, analysis, modelling, movement, waves, varvecy, varvecx, proliferation, wave,

Topics {✒️}

$$\begin{aligned} \partial _t uk/ons/guide-method/census/2011/census-history/200-years $$\begin{aligned} n_{\varepsilon } $$\begin{aligned} \rho _\varepsilon time-dependent advection-diffusion-reaction equations rescaled taxis-based ps-pde model \partial ^2_{yy} partial ^2_{yy} }{\partial ^2_{yy} } \partial ^2_{yy} dipartimento di scienze $$\begin{aligned} \psi _x big ]}_{\begin{array}{ exp \left[ -\dfrac{\left displaystyle {\rho _\varepsilon basic reaction-advection-diffusion framework $$\begin{aligned} n_{ grant prin2022-pnrr project partial differential equations exp \left[ -\frac{\left }_{\begin{array}{ age-structured reaction-diffusion model partial differential equation monotonic upstream-centered scheme nonlocal lotka-volterra equations patlak-keller-segel chemotaxis model vector-host epidemic model lotka-volterra parabolic equations multiscale dti-based model equation $$\begin{aligned} $$\begin{aligned} a_y incorporate pressure-dependent inhibition reaction-advection-diffusion equation nonlocal reaction-diffusion equation nonlocal reaction-diffusion model hamilton-jacobi equation subject exploring ligand-receptor dynamics $$\begin{aligned} a_{ local reaction-diffusion equations adding sink/source terms age-structured models arising dfrac{h_x}{\tau } \left relation $$\begin{aligned} cell-cell recognition phenomenon relations $$\begin{aligned} rescaled fisher-kpp equation stochastic birth-death process reaching ps-pde models validate ps-pde models form $$\begin{aligned}

Questions {❓}

Can similar detail be absorbed within continuous models?
Discrete, continuous, or both?
How can we form a bridge, spanning discrete to continuous populations according to phenotypic state?
Surveys served purposes from economic (how much tax can we collect?
Theveneau E, Linker C (2017) Leaders in collective migration: are front cells really endowed with a particular set of skills?
A population continuously structured across some phenotypic space?

Schema {🗺️}

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         description:Populations are heterogeneous, deviating in numerous ways. Phenotypic diversity refers to the range of traits or characteristics across a population, where for cells this could be the levels of signalling, movement and growth activity, etc. Clearly, the phenotypic distribution – and how this changes over time and space – could be a major determinant of population-level dynamics. For instance, across a cancerous population, variations in movement, growth, and ability to evade death may determine its growth trajectory and response to therapy. In this review, we discuss how classical partial differential equation (PDE) approaches for modelling cellular systems and collective cell migration can be extended to include phenotypic structuring. The resulting non-local models – which we refer to as phenotype-structured partial differential equations (PS-PDEs) – form a sophisticated class of models with rich dynamics. We set the scene through a brief history of structured population modelling, and then review the extension of several classic movement models – including the Fisher-KPP and Keller-Segel equations – into a PS-PDE form. We proceed with a tutorial-style section on derivation, analysis, and simulation techniques. First, we show a method to formally derive these models from underlying agent-based models. Second, we recount travelling waves in PDE models of spatial spread dynamics and concentration phenomena in non-local PDE models of evolutionary dynamics, and combine the two to deduce phenotypic structuring across travelling waves in PS-PDE models. Third, we discuss numerical methods to simulate PS-PDEs, illustrating with a simple scheme based on the method of lines and noting the finer points of consideration. We conclude with a discussion of future modelling and mathematical challenges.
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