Stochastic drift in discrete waves of nonlocally interacting particles #

Andrei Sontag, Tim Rogers, Christian A Yates

14:50 Tuesday in 3Q16.

Part of the Stochastic processes and random walks session.

Abstract #

Many processes of interest in biology such as the modelling of migration, information dynamics in epidemics, and Muller’s ratchet — the irreversible accumulation of deleterious mutations in an evolving population — can be modelled as a stochastic $N$ particle system undergoing second-order nonlocal interactions on a lattice. Strikingly, the average of numerous numerical simulations of the stochastic model is observed to deviate significantly from its corresponding deterministic solution, even for large population sizes. In this talk, based on our recent work [1], I will 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 stochastic wave to fluctuate. These effects are shown to decay anomalously as $(ln N)^{-2}$ and $(ln N)^{-3}$, 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.

[1] A. S., T. R., C. A. Y. Physical Review E, vol. 107, no. 1. American Physical Society (APS), Jan. 18, 2023. doi: 10.1103/physreve.107.014128.