Discreteness Effects in Population Dynamics

TL;DR

This paper investigates how small initial populations affect the accuracy of large deviation function estimations in stochastic population dynamics and proposes a time delay method to mitigate discreteness effects.

Contribution

It introduces a realization-dependent time delay approach to reduce discreteness effects in population-based large deviation calculations.

Findings

The time delay method improves large deviation function estimates.

Discreteness effects are significant in initial transient regimes.

The approach enhances numerical accuracy in population dynamics simulations.

Abstract

We analyse numerically the effects of small population size in the initial transient regime of a simple example population dynamics. These effects play an important role for the numerical determination of large deviation functions of additive observables for stochastic processes. A method commonly used in order to determine such functions is the so-called cloning algorithm which in its non-constant population version essentially reduces to the determination of the growth rate of a population, averaged over many realizations of the dynamics. However, the averaging of populations is highly dependent not only on the number of realizations of the population dynamics, and on the initial population size but also on the cut-off time (or population) considered to stop their numerical evolution. This may result in an over-influence of discreteness effects at initial times, caused by small population size. We overcome these effects by introducing a (realization-dependent) time delay in the evolution of populations, additional to the discarding of the initial transient regime of the population growth where these discreteness effects are strong. We show that the improvement in the estimation of the large deviation function comes precisely from these two main contributions.