1211.6494 (Christopher N. Angstmann et al.)

Pattern Formation on Networks with Reactions: A Continuous Time Random
Walk Approach    [PDF]

Christopher N. Angstmann, Isaac C. Donnelly, Bruce I. Henry

We derive the generalised master equation for reaction-diffusion on networks from an underlying stochastic process, the continuous time random walk (CTRW). Using this model we have investigated different types of pattern formation across the vertices on a range of networks. Importantly, the CTRW defines the Laplacian operator on the network in a non ad-hoc manner and the pattern formation depends on the structure of this Laplacian. Here we focus attention on CTRWs with exponential waiting times for two cases; one in which where the rate parameter is constant for all vertices and the other where the rate parameter is proportional to the vertex degree. This results in non-symmetric and symmetric CTRW Laplacians respectively. In the case of symmetric Laplacians, pattern formation follows from the Turing instability. However in non-symmetric Laplacians, pattern formation may be possible with or without a Turing instability.

View original: http://arxiv.org/abs/1211.6494