Workshop on Pattern Formation and Functional Morphology, Tue, 08 Jan, 2008
Speaker: Tony Bracken
Abstract
A model for the self-organisation of zones of differing enzymatic activity in liver capillaries was developed twenty years ago by Bass, Bracken, Holmaker and Jefferies (Journal of the Australian Mathematical Society 29(1987), 156-194). The existence of such liver zones is now well-established experimentally. The model can be described as an extension of Volterra's model of competitive exclusion to include spatial as well as temporal variations. Two different cell types compete for sites on the wall of each capillary. An effectively nonlocal interaction between the cells arises from local consumption of oxygen from blood flowing through the capillary, which gives rise to gradients of oxygen concentration in turn influencing rates of division and death of the two competing cell types. The process is modelled by a pair of coupled nonlinear integro-differential equations for the cell densities as functions of time and position along the capillary. For certain ranges of the model parameters and initial conditions, solutions approach an asymptotically stable stationary solution with a zonal structure as time increases, with cells of one type located entirely upstream of the other type, and with jump discontinuities in the cell densities at a certain distance along the capillary. More recent work will be described, involving competition between more than two species and possible application to other biological situations.
URL: www.ricam.oeaw.ac.at/specsem/ssqbm/participants/abstracts/index.php
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