The breakdown of biogeographic barriers allows some invasive species to reshape ecological communities and threaten local biodiversity. Most introductions of exotic species fail to generate an invasion. However, once introduction succeeds, invader density increases rapidly. We apply nucleation theory to describe spatio-temporal patterns of the invasion process under preemptive competition. The predictions of the theory are confirmed by Monte Carlo simulations of the underlying discrete spatial stochastic dynamics. In particular, for large enough spatial regions, invasion occurs through the nucleation and subsequent growth of many clusters of the invasive species, and the global densities are well approximated by
Avrami's law for homogeneous nucleation. For smaller systems or very small introduction rates, invasion typically occurs through a single cluster, whose appearance is inherently stochastic.
We considered diffusion driven processes on power-law small-world networks: random walk on randomly folded polymers and surface growth related to synchronization problems. We found a rich phase diagram, with different transient and recurrent phases. The calculations were done in two limiting cases: the annealed case, when the rearrangement of the random links is fast (the configuration of the polymer changes fast) and the quenched case, when the link rearrangement is slow (the polymer configuration is static) compared to the motion of the random walker. In the quenched case, the random links introduced in small-world networks often lead to mean-filed coupling (i.e., the random links can be treated in an annealed fashion) but in some systems mean-field predictions break down, such as for diffusion in one dimension. This break-down can be understood treating the random links perturbatively where the mean field prediction appears as the lowest order term of a naive perturbation expansion. Our results were obtained using self-consisten perturbation theory. Numerical results will also be shown as a confirmation of the theory.
Synchronization is a fundamental problem in natural and artificial
coupled multi-component systems. We investigate to what extent
small-world couplings (extending the original local relaxational dynamics through the random links) lead to the suppression of extreme
fluctuations in such systems. We use the framework of non-equilibrium surface growth to study and characterize the degree of synchronization in the system. In the absence of the random links, the surface in the steady state is "rough" (strongly de-synchronized state) and the average and the extreme height fluctuations diverge in the same power-law fashion with the system size (number of nodes). With small-world links present, the average size of the fluctuations becomes finite (synchronized state) and the extreme heights diverge only logarithmically in the large system-size limit. This latter property ensures synchronization in a practical sense in coupled multi-component autonomous systems with short-tailed noise and effective relaxation through the links. The statistics of the extreme heights is governed by the Fisher-Tippett-Gumbel distribution. We illustrate our findings through an actual synchronization problem in parallel discrete-event simulations.
Conference Committee Involvement (1)
Noise in Complex Systems and Stochastic Dynamics III