Session D7: Fluctuations and Critical Phenomena in Population Dynamics

Scaling Laws, Fluctuations and Pattern Formation in Ecosystems

Ecology is fundamentally concerned with the relationship of organisms to space and time. Thus, it is natural for statistical physicists to ask if there are universal phenomena, and if so, how they can be separated from purely idiosyncratic features of special systems. This talk will focus primarily on fluctuations in ecosystems, showing how certain well-documented scaling laws, such as the species-area law, follow from rather general properties of ecosystems. Fluctuations, especially demographic noise, drive patterns in space and time, and we will review how this solves the predator-prey limit cycle paradox, and can account for the robust occurrence of patchiness in ecosystems.   

WKB theory of stochastic epidemics in well-mixed populations

Stochastic effects in disease transmission can eliminate the disease from small communities. In one scenario this happens after the disease already reached an endemic state. In another scenario the disease can fade out immediately after an epidemic outbreak. I will review a recent progress in theoretical analysis of these and related problems [1-3]. In a probabilistic language, spread of a disease can be described by a master equation with specified transition rates. When the population size is sufficiently large, one can use a dissipative version of WKB approximation thus reducing the problem to that of finding ``optimal paths'' to disease extinction. These are special instanton-like trajectories of an underlying classical Hamiltonian. Further analytical progress in multi-population systems is possible when either disparity of transition rates, or proximity to a bifurcation, causes time scale separation. I will illustrate these points on the examples of the stochastic SI (Susceptible-Infected) and SIS (Susceptible-Infected-Susceptible) models of epidemiology. \\[4pt] [1] M.I. Dykman, I.B. Schwartz, and A.S. Landsman, Phys. Rev. Lett. \textbf{101}, 078101 (2008).\\[0pt] [2] A. Kamenev and B. Meerson, Phys. Rev. E \textbf{77}, 061107 (2008).\\[0pt] [3] B. Meerson and P.V. Sasorov, Phys. Rev. E \textbf{80}, 041130 (2009).   

Dynamics of Voting Models

The voter model provides a paradigmatic description of consensus formation in a population of interacting agents. Each voter can be in one of two opinion states and continuously updates its opinion at a rate proportional to the fraction of neighbors of the opposite opinion. This model has been completely solved when the voters are situated on the nodes of a regular graph. This talk will discuss several extensions of the basic voter model: (i) on complex graphs, consensus is generally achieved quickly, (ii) multi-state and strategic voting leads opinion evolution with multiple time scales, (iii) heterogeneous voters can lead to ultra-slow evolution.   

Consensus formation in social networks

In social networks, friendships emerge and fade, as individuals develop and change their opinions. Here, we discuss a simple model of such a network, in which the agents (``individuals'') are modeled by Ising spins on the nodes of the network, while their connections (``friendships'') are modeled by the presence or absence of edges. Nodes evolve according to a simple majority rule, and links are established or removed between pairs of nodes, depending on their spin content. Thus, both nodes and links become dynamic variables, correlated with each other, and the network is termed ``adaptive.'' Using simulations and exact solutions, we look at an extensive vs an intensive version of the model: In the former, the average degree scales with the number of nodes while remaining fixed in the latter. We analyze the long-time behavior of these two versions, both for finite systems and in the thermodynamic limit. We find significant differences, both with regards to the number of phases found in the thermodynamic limit, and with regards to the life times of metastable states in finite systems. Consequences for social networks with spatial structure will be discussed.