Stochastic drift in discrete waves of non-locally interacting-particles

Abstract

In this paper, we investigate a generalised model of $N$ particles undergoing second-order non-local interactions on a lattice. Our results have applications across many research areas, including the modelling of migration, information dynamics and Muller's ratchet -- the irreversible accumulation of deleterious mutations in an evolving population. Strikingly, numerical simulations of the model are observed to deviate significantly from its mean-field approximation even for large population sizes. We show that the disagreement between deterministic and stochastic solutions stems from finite-size effects that change the propagation speed and cause the position of the wave to fluctuate. These effects are shown to decay anomalously as $(\log N{)}^{-2}$ and $(\log N{)}^{-3}$, respectively -- much slower than the usual $N^{-1/2}$ factor. Our results suggest that the accumulation of deleterious mutations in a Muller's ratchet and the loss of awareness in a population may occur much faster than predicted by the corresponding deterministic models. The general applicability of our model suggests that this unexpected scaling could be important in a wide range of real-world applications.

arXiv comment14 pages, 9 figures

References in corpus (16)

• Shift in the velocity of a front due to a cut-off
• Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review
• Non-local Interaction Effects on Pattern Formation in Population Dynamics
• Effects of Fluctuations on Propagating Fronts
• Noise-Induced Bistable States and Their Mean Switching Time in Foraging Colonies
• Demographic noise can reverse the direction of deterministic selection
• Front Propagation: Precursors, Cutoffs and Structural Stability
• Interfacial Velocity Corrections due to Multiplicative Noise
• Noisy traveling waves: effect of selection on genealogies
• Internal Fluctuations Effects on Fisher Waves
• Demographic noise can lead to the spontaneous formation of species
• Existence and spatial limit theorems for lattice and continuum particle systems
• The Weakly Pushed Nature of "Pulled" Fronts with a Cutoff
• Asymptotic Scaling of the Diffusion Coefficient of Fluctuating "Pulled" Fronts
• Fronts with a Growth Cutoff but Speed Higher than $v^{\ast }$
• Fluctuating "Pulled" Fronts: the Origin and the Effects of a Finite Particle Cutoff