  The movement of many organisms can be described as a random walk at either or
both the individual and population level. The rules for this random walk are
based on complex biological processes and it may be difficult to develop a
tractable, quantitatively-accurate, individual-level model. However, important
problems in areas ranging from ecology to medicine involve large collections of
individuals, and a further intellectual challenge is to model population-level
behavior based on a detailed individual-level model. Because of the large
number of interacting individuals and because the individual-level model is
complex, classical direct Monte Carlo simulations can be very slow, and often
of little practical use. In this case, an equation-free approach may provide
effective methods for the analysis and simulation of individual-based models.
In this paper we analyze equation-free coarse projective integration. For
analytical purposes, we start with known partial differential equations
describing biological random walks and we study the projective integration of
these equations. In particular, we illustrate how to accelerate explicit
numerical methods for solving these equations. Then we present illustrative
kinetic Monte Carlo simulations of these random walks and show a decrease in
computational time by as much as a factor of a thousand can be obtained by
exploiting the ideas developed by analysis of the closed form PDEs. The
illustrative biological example here is chemotaxis, but it could be any random
walker which biases its movement in response to environmental cues.
