Nonlocal and local models for taxis in cell migration: a rigorous limit procedure
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Mathematical Biology
Nonlocal and local models for taxis in cell migration: a rigorous limit procedure Maria Eckardt1 · Kevin J. Painter2 · Christina Surulescu1 · Anna Zhigun3 Received: 25 November 2019 / Revised: 18 August 2020 © The Author(s) 2020
Abstract A rigorous limit procedure is presented which links nonlocal models involving adhesion or nonlocal chemotaxis to their local counterparts featuring haptotaxis and classical chemotaxis, respectively. It relies on a novel reformulation of the involved nonlocalities in terms of integral operators applied directly to the gradients of signaldependent quantities. The proposed approach handles both model types in a unified way and extends the previous mathematical framework to settings that allow for general solution-dependent coefficient functions. The previous forms of nonlocal operators are compared with the new ones introduced in this paper and the advantages of the latter are highlighted by concrete examples. Numerical simulations in 1D provide an illustration of some of the theoretical findings. Keywords Cell–cell and cell–tissue adhesion · Nonlocal and local chemotaxis · Haptotaxis · Integro-differential equations · Unified approach · Global existence · Rigorous limit behaviour · Weak solutions Mathematics Subject Classification 35Q92 · 92C17 · 35K55 · 35R09 · 47G20 · 35B45 · 35D30
1 Introduction Macroscopic equations and systems describing the evolution of populations in response to soluble and insoluble environmental cues have been intensively studied and the palette of such reaction-diffusion-taxis models is continuously expanding. Models of such form are motivated by problems arising in various contexts, a large part related to cell migration and proliferation connected to tumor invasion, embryonal
ME and CS were supported in part by the research initiative Mathematics Applied to Real World Challenges (MathApp) of the TU Kaiserslautern. CS also acknowledges funding by the Federal Ministry of Education and Research (BMBF) in the project GlioMaTh 05M2016. Extended author information available on the last page of the article
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development, wound healing, biofilm formation, insect behavior in response to chemical cues, etc. We refer, e.g. to Bellomo et al. (2015) for a recent review also containing some deduction methods for taxis equations based on kinetic transport equations. Apart from such purely local PDE systems with taxis, several spatially nonlocal models have been introduced over the last two decades and are attracting ever increasing interest. They involve integro-differential operators in one or several terms of the featured reaction-diffusion-drift equations. Their aim is to characterize interactions between individuals or signal perception happening not only at a specific location, but over a whole set (usually a ball) containing (centered at) that location. In the context of cell populations, for instance, this seems to be a more realistic modeling assumption, as cells are able to extend various protrusions (such as lam
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