UBC Mathematics Department

Colloquium Abstract: Dr.Daniel Grunbaum, Dept. of Mathematics, Univ. of Utah

Advection-diffusion equations for random walks with internal state dynamics

Random walks are important elements of cell migration in immune responses, wound healing, and morphogenesis; they also are the basis of many foraging, dispersal, and reproductive behaviors in micro-organisms and higher animals. In most biological examples, movement depends on internal state dynamics (physiological or cognitive states, signal transduction mechanics, etc.). This state-dependence is essential to the function of the behavior and to understanding its medical, ecological, and evolutionary implications.

Distributions of random walkers in time and space are described by Boltzmann-type hyperbolic partial integro-differential equations governing probability density of individual state and position. A mathematical literature going back to the 1950's has developed methods of deriving tractable parabolic approximations (in the form of advection-diffusion equations) to the complete hyperbolic models. However, no derivation has yet incorporated internal state dynamics; consequently, this literature is of limited usefulness in generating mechanistic models of random walks in the vast majority of biologically important applications.

In this talk, I derive parabolic approximations for more general and realistic behaviors, in which behavior is explicitly tied to internal state variables, whose dynamics in turn are determined by attractants, resources, or other spatially-varying environmental factors. The analysis employs a Hilbert-Schmidt eigenfunction expansion of the Boltzmann equation in velocity and state variables, and takes advantage of a difference in time- and space-scales between individual responses and environmental variations. A singular perturbation expansion results in a general advection-diffusion equation that may be solved analytically or numerically to investigate the effects of various behaviors and substrate distributions. I show how the results of these equations can be cast in terms of measurable and biologically relevant parameters, and can give useful insights in ecological, evolutionary, and medical applications.

NOTE: The speaker is a candidate for a position in the
Mathematics Department. All faculty are urged to attend this talk.

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