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 [I.G. Kevrekidis, C.W. Gear, J.M. Hyman, P. Kevrekidis, O. Runborg, K. Theodoropoulos, Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis, Commun. Math. Sci. 1 (4) (2003) 715-762] 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 that 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 that biases its movement in response to environmental cues.
Bibliographical noteFunding Information:
The first author’s research was supported in part by NSF grant DMS 0317372 and the Minnesota Supercomputing Institute. The second author’s research was supported in part by an NSF/ITR grant (CTS 0205484). The third author’s research was supported in part by NIH grant GM 29123, NSF grant DMS 0317372, and the Minnesota Supercomputing Institute.
- Coarse integration
- Individual-based models
- Projective integration