THE UNIVERSITY OF CALGARY Stochastic Simulation of Pattern Formation: An Application of L-systems by Mikolaj Cieslak A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE
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چکیده
Turing proposed a mathematical model based on chemical reactions and diffusion of substances throughout the xy-plane, called reaction-diffusion, to study the emergence of patterns from a homogeneous medium [65]. Hammel and Prusinkiewicz used L-systems and reaction-diffusion in a one-dimensional medium to model the formation of sea shell patterns, and in an expanding one-dimensional medium to model heterocyst spacing in the cyanobacterium Anabaena catenula [25]. They considered the formation of these two patterns as a continuous deterministic process neglecting the noise which is inherent to such a process. The approach taken in this work is to stochastically model reaction-diffusion in a spatial, and possibly expanding, linear structure using L-systems. The stochastic simulation method for chemical reaction kinetics which was developed by Gillespie [20] is used to study this model. On the basis of theoretical considerations, the Lsystem modelling language L+C [31] is extended to include a stochastic rewriting strategy based on Gillespie’s algorithm.
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تاریخ انتشار 2006