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 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 $sqrt{N}$ factor. As a result, the accumulation of deleterious mutations in a Muller’s ratchet and the loss of awareness in a population are processes that 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.
Andrei Sontag
Research Fellow
My research interests include stochastic processes applied to complex systems with a focus on ecology, evolution, financial markets and collective behaviour.