Dynamical spike solutions in a nonlocal model of pattern formation

Marciniak–Czochra, Anna; Härting, Steffen; Karch, Grzegorz; Suzuki, Kanako

doi:10.48550/arxiv.1307.6236

Cited by 1 publication

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“…For example, in the Alt and Lauffenberger's [2] model of leucocytes reacting to a bacterial invasion by moving up a gradient of some chemical attractant produced by the bacteria (see Section 13.4.2 in [47]) a system of three PDEs is reduced to one equation provided bacterial diffusion is much smaller than the diffusion of leukocytes or of chemoattractants (which is typically the case). Similarly, in the early carcinogenesis model of Marcinak-Czochra and Kimmel [16,53,54,55], a system of two ODEs coupled with a single diffusion equation (involving Neumann boundary conditions) is replaced by a socalled shadow system of integro-differential equations with ordinary differentiation, provided diffusion may be assumed fast.…”

Section: Discussionmentioning

confidence: 99%

An averaging principle for fast diffusions in domains separated by semi-permeable membranes

Bobrowski

Kaźmierczak

Kunze

2017

Math. Models Methods Appl. Sci.

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Abstract. We prove an averaging principle which asserts convergence of diffusion processes on domains separated by semi-permeable membranes, when diffusion coefficients tend to infinity while the flux through the membranes remains constant. In the limit, points in each domain are lumped into a single state of a limit Markov chain. The limit chain's intensities are proportional to the membranes' permeability and inversely proportional to the domains' sizes. Analytically, the limit is an example of a singular perturbation in which boundary and transmission conditions play a crucial role. This averaging principle is strongly motivated by recent signaling pathways models of mathematical biology, which are discussed towards the end of the paper.

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Section: Discussionmentioning

confidence: 99%