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Invasion fronts and adaptive dynamics in a model for the growth of cell populations with heterogeneous mobility

Part of: General first-order equations and systems Miscellaneous topics - Partial differential equations Representations of solutions Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  08 July 2021

T. LORENZI Open the ORCID record for T. LORENZI [Opens in a new window] ,
B. PERTHAME  and
X. RUAN
T. LORENZI
Affiliation:
Department of Mathematical Sciences “G. L. Lagrange”, Dipartimento di Eccellenza 2018-2022, Politecnico di Torino, 10129 Torino, Italy email: tommaso.lorenzi@polito.it
B. PERTHAME
Affiliation:
Sorbonne Universite, CNRS, Universite de Paris, Inria, Laboratoire Jacques-Louis Lions UMR7598, F-75005 Paris, France emails: Benoit.Perthame@sorbonne-universite.fr; xinran.ruan@polito.it
X. RUAN
Affiliation:
Department of Mathematical Sciences “G. L. Lagrange”, Dipartimento di Eccellenza 2018-2022, Politecnico di Torino, 10129 Torino, Italy email: tommaso.lorenzi@polito.it Sorbonne Universite, CNRS, Universite de Paris, Inria, Laboratoire Jacques-Louis Lions UMR7598, F-75005 Paris, France emails: Benoit.Perthame@sorbonne-universite.fr; xinran.ruan@polito.it
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Abstract

We consider a model for the dynamics of growing cell populations with heterogeneous mobility and proliferation rate. The cell phenotypic state is described by a continuous structuring variable and the evolution of the local cell population density function (i.e. the cell phenotypic distribution at each spatial position) is governed by a non-local advection–reaction–diffusion equation. We report on the results of numerical simulations showing that, in the case where the cell mobility is bounded, compactly supported travelling fronts emerge. More mobile phenotypic variants occupy the front edge, whereas more proliferative phenotypic variants are selected at the back of the front. In order to explain such numerical results, we carry out formal asymptotic analysis of the model equation using a Hamilton–Jacobi approach. In summary, we show that the locally dominant phenotypic trait (i.e. the maximum point of the local cell population density function along the phenotypic dimension) satisfies a generalised Burgers’ equation with source term, we construct travelling-front solutions of such transport equation and characterise the corresponding minimal speed. Moreover, we show that, when the cell mobility is unbounded, front edge acceleration and formation of stretching fronts may occur. We briefly discuss the implications of our results in the context of glioma growth.

Keywords

Heterogeneous cell populationsheterogeneous mobilityinvasion frontsadaptive dynamicsnon-local advection–reaction–diffusion equations

MSC classification

Primary: 35Q92: PDEs in connection with biology and other natural sciences 35B40: Asymptotic behavior of solutions 35C07: Traveling wave solutions
Secondary: 35R09: Integro-partial differential equations 35F21: Hamilton-Jacobi equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 4 , August 2022 , pp. 766 - 783
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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