Velocity-jump processes with proliferation

Author： Treloar Katrina K. Simpson Matthew J. McCue Scott W.

ISSN： 1751-8121

Source： Journal of Physics A: Mathematical and Theoretical, Vol.46, Iss.1, 2013-01, pp. : 15003-15019

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Abstract

Cell invasion involves a population of cells that migrate along a substrate and proliferate to a carrying capacity density. These two processes, combined, lead to invasion fronts that move into unoccupied tissues. Traditional modelling approaches based on reaction-diffusion equations cannot incorporate individual-level observations of cell velocity, as information propagates with infinite velocity according to these parabolic models. In contrast, velocity-jump processes allow us to explicitly incorporate individual-level observations of cell velocity, thus providing an alternative framework for modelling cell invasion. Here, we introduce proliferation into a standard velocity-jump process and show that the standard model does not support invasion fronts. Instead, we find that crowding effects must be explicitly incorporated into a proliferative velocity-jump process before invasion fronts can be observed. Our observations are supported by numerical and analytical solutions of a novel coupled system of partial differential equations, including travelling wave solutions, and associated random walk simulations.