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We investigate the dynamics of a broad class of stochastic copying processes on a network that includes examples from population genetics (spatially-structured Wright-Fisher models), ecology (Hubbell-type models), linguistics (the utterance selection model) and opinion dynamics (the voter model) as special cases. These models all have absorbing states of fixation where all the nodes are in the same state. Earlier studies of these models showed that the mean time when this occurs can be made to grow as different powers of the network size by varying the the degree distribution of the network. Here we demonstrate that this effect can also arise if one varies the asymmetry of the copying dynamics whilst holding the degree distribution constant. In particular, we show that the mean time to fixation can be accelerated even on homogeneous networks when certain nodes are very much more likely to be copied from than copied to. We further show that there is a complex interplay between degree distribution and asymmetry when they may co-vary; and that the results are robust to correlations in the network or the initial condition.
Diffusion of a two component fluid is studied in the framework of differential equations, but where these equations are systematically derived from a well-defined microscopic model. The model has a finite carrying capacity imposed upon it at the meso scopic level and this is shown to lead to non-linear cross diffusion terms that modify the conventional Fickean picture. After reviewing the derivation of the model, the experiments carried out to test the model are described. It is found that it can adequately explain the dynamics of two dense ink drops simultaneously evolving in a container filled with water. The experiment shows that molecular crowding results in the formation of a dynamical barrier that prevents the mixing of the drops. This phenomenon is successfully captured by the model. This suggests that the proposed model can be justifiably viewed as a generalization of standard diffusion to a multispecies setting, where crowding and steric interferences are taken into account.
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