Fisher waves and front roughening in a two-species invasion model with preemptive competition.

We study front propagation when an invading species competes with a resident; we assume nearest-neighbor preemptive competition for resources in an individual-based, two-dimensional lattice model. The asymptotic front velocity exhibits an effective power-law dependence on the difference between the two species' clonal propagation rates (key ecological parameters). The mean-field approximation behaves similarly, but the power law's exponent slightly differs from the individual-based model's result. We also study roughening of the front, using the framework of nonequilibrium interface growth. Our analysis indicates that initially flat, linear invading fronts exhibit Kardar-Parisi-Zhang (KPZ) roughening in one transverse dimension. Further, this finding implies, and is also confirmed by simulations, that the temporal correction to the asymptotic front velocity is of O(t(-2/3)).

Roughness scaling for Edwards-Wilkinson relaxation in small-world networks.

Motivated by a fundamental synchronization problem in scalable parallel computing and by a recent criterion for "mean-field" synchronizability in interacting systems, we study the Edwards-Wilkinson model on two variations of a small-world network. In the first version each site has exactly one random link of strength p, while in the second one each site on average has p links of unit strength. We construct a perturbative description for the width of the stationary-state surface (a measure of synchronization), in the weak- and sparse-coupling limits, respectively, and verify the results by performing exact numerical diagonalization. The width remains finite in the limit of infinite system size for both cases, but exhibits anomalous scaling with p in the latter for d< or =2.