**UBC Mathematics Department**

*http://www.math.ubc.ca*

## 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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