Effective multidimensional crossover behavior in a one-dimensional voter model with long-range probabilistic interactions

Ordering kinetics of the two-dimensional voter model with long-range interactions

We study analytically the ordering kinetics of the two-dimensional long-range voter model on a two-dimensional lattice, where agents on each vertex take the opinion of others at distance r with probability P(r)∝r^{-α}. The model is characterized by different regimes, as α is varied. For α>4, the behavior is similar to that of the nearest-neighbor model, with the formation of ordered domains of a typical size growing as L(t)∝sqrt[t], until consensus is reached in a time of the order of NlnN, with N being the number of agents. Dynamical scaling is violated due to an excess of interfacial sites whose density decays as slowly as ρ(t)∝1/lnt. Sizable finite-time corrections are also present, which are absent in the case of nearest-neighbor interactions. For 0<α≤4, standard scaling is reinstated and the correlation length increases algebraically as L(t)∝t^{1/z}, with 1/z=2/α for 3<α<4 and 1/z=2/3 for 0<α<3. In addition, for α≤3, L(t) depends on N at any time t>0. Such coarsening, however, only leads the system to a partially ordered metastable state where correlations decay algebraically with distance, and whose lifetime diverges in the N→∞ limit. In finite systems, consensus is reached in a time of the order of N for any α<4.

Species exclusion and coexistence in a noisy voter model with a competition-colonization tradeoff

We introduce an asymmetric noisy voter model to study the joint effect of immigration and a competition-dispersal tradeoff in the dynamics of two species competing for space in regular lattices. Individuals of one species can invade a nearest-neighbor site in the lattice, while individuals of the other species are able to invade sites at any distance but are less competitive locally, i.e., they establish with a probability g≤1. The model also accounts for immigration, modeled as an external noise that may spontaneously replace an individual at a lattice site by another individual of the other species. This combination of mechanisms gives rise to a rich variety of outcomes for species competition, including exclusion of either species, monostable coexistence of both species at different population proportions, and bistable coexistence with proportions of populations that depend on the initial condition. Remarkably, in the bistable phase, the system undergoes a discontinuous transition as the intensity of immigration overcomes a threshold, leading to a half loop dynamics associated to a cusp catastrophe, which causes the irreversible loss of the species with the shortest dispersal range.

Competition between local erasure and long-range spreading of a single biochemical mark leads to epigenetic bistability

The mechanism through which cells determine their fate is intimately related to the spreading of certain biochemical (so-called epigenetic) marks along their genome. The mechanisms behind mark spreading and maintenance are not yet fully understood, and current models often assume a long-range infectionlike process for the dynamics of marks, due to the polymeric nature of the chromatin fiber which allows looping between distant sites. While these existing models typically consider antagonizing marks, here we propose a qualitatively different scenario which analyses the spreading of a single mark. We define a one-dimensional stochastic model in which mark spreading/infection occurs as a long-range process whereas mark erasure/recovery is a local process, with an enhanced rate at boundaries of infected domains. In the limiting case where our model exhibits absorbing states, we find a first-order-like transition separating the marked/infected phase from the unmarked/recovered phase. This suggests that our model, in this limit, belongs to the long-range compact directed percolation universality class. The abrupt nature of the transition is retained in a more biophysically realistic situation when a basal infection/recovery rate is introduced (thereby removing absorbing states). Close to the transition there is a range of bistability where both the marked/infected and unmarked/recovered states are metastable and long lived, which provides a possible avenue for controlling fate decisions in cells. Increasing the basal infection/recovery rate, we find a second transition between a coherent (marked or unmarked) phase, and a mixed, or random, one.