We consider population dynamics on a network of patches, each of which has a the same local dynamics, with different population scales (carrying capacities). It is reasonable to assume that if the patches are coupled by very fast migration the whole
system will look like an individual patch with a large effective carrying capacity. This is called a well-mixed system. We show that, in general, it is not true that the well-mixed system has the same dynamics as each local patch. Different global dynamics can emerge from coupling, and usually must be figured out for each individual case. We give a general condition which must be satisfied for well-mixed systems to have the same dynamics as the constituent patches.
In many dynamical systems there is a large separation of time scales between typical events and rare events which can be the cases of interest. Rare-event rates are quite difficult to compute numerically, but they are of considerable practical import
ance in many fields: for example transition times in chemical physics and extinction times in epidemiology can be very long, but are quite important. We present a very fast numerical technique that can be used to find long transition times (very small rates) in low-dimensional systems, even if they lack detailed balance. We illustrate the method for a bistable non-equilibrium system introduced by Maier and Stein and a two-dimensional (in parameter space) epidemiology model.
Fortuin-Kastelyn clusters in the critical $Q$-state Potts model are conformally invariant fractals. We obtain simulation results for the fractal dimension of the complete and external (accessible) hulls for Q=1, 2, 3, and 4, on clusters that wrap aro
und a cylindrical system. We find excellent agreement between these results and theoretical predictions. We also obtain the probability distributions of the hull lengths and maximal heights of the clusters in this geometry and provide a conjecture for their form.
Dispersal of species to find a more favorable habitat is important in population dynamics. Dispersal rates evolve in response to the relative success of different dispersal strategies. In a simplified deterministic treatment (J. Dockery, V. Hutson, K
. Mischaikow, et al., J. Math. Bio. 37, 61 (1998)) of two species which differ only in their dispersal rates the slow species always dominates. We demonstrate that fluctuations can change this conclusion and can lead to dominance by the fast species or to coexistence, depending on parameters. We discuss two different effects of fluctuations, and show that our results are consistent with more complex treatments that find that selected dispersal rates are not monotonic with the cost of migration.
We present a technique, which we call etching, which we use to study the harmonic measure of Fortuin-Kasteleyn clusters in the Q-state Potts model for Q=1-4. The harmonic measure is the probability distribution of random walkers diffusing onto the pe
rimeter of a cluster. We use etching to study regions of clusters which are extremely unlikely to be hit by random walkers, having hitting probabilities down to 10^(-4600). We find good agreement between the theoretical predictions of Duplantier and our numerical results for the generalized dimension D(q), including regions of small and negative q.
We study a generalization of the voter model on complex networks, focusing on the scaling of mean exit time. Previous work has defined the voter model in terms of an initially chosen node and a randomly chosen neighbor, which makes it difficult to di
sentangle the effects of the stochastic process itself relative to the network structure. We introduce a process with two steps, one that selects a pair of interacting nodes and one that determines the direction of interaction as a function of the degrees of the two nodes and a parameter $alpha$ which sets the likelihood of the higher degree node giving its state. Traditional voter model behavior can be recovered within the model. We find that on a complete bipartite network, the traditional voter model is the fastest process. On a random network with power law degree distribution, we observe two regimes. For modest values of $alpha$, exit time is dominated by diffusive drift of the system state, but as the high nodes become more influential, the exit time becomes becomes dominated by frustration effects. For certain selection processes, a short intermediate regime occurs where exit occurs after exponential mixing.
We obtain the harmonic measure of the hulls of critical percolation clusters and Ising-model Fortuin-Kastelyn clusters using a biased random-walk sampling technique which allows us to measure probabilities as small as 10^{-300}. We find the multifrac
tal D(q) spectrum including regions of small and negative q. Our results for external hulls agree with Duplantiers theoretical predictions for D(q) and his exponent -23/24 for the harmonic measure probability distribution. For the complete hull, we find the probability decays with an exponent of -1 for both systems.
Recently we considered a stochastic discrete model which describes fronts of cells invading a wound cite{KSS}. In the model cells can move, proliferate, and experience cell-cell adhesion. In this work we focus on a continuum description of this pheno
menon by means of a generalized Cahn-Hilliard equation (GCH) with a proliferation term. As in the discrete model, there are two interesting regimes. For subcritical adhesion, there are propagating pulled fronts, similarly to those of Fisher-Kolmogorov equation. The problem of front velocity selection is examined, and our theoretical predictions are in a good agreement with a numerical solution of the GCH equation. For supercritical adhesion, there is a nontrivial transient behavior, where density profile exhibits a secondary peak. To analyze this regime, we investigated relaxation dynamics for the Cahn-Hilliard equation without proliferation. We found that the relaxation process exhibits self-similar behavior. The results of continuum and discrete models are in a good agreement with each other for the different regimes we analyzed.
