Binomial maps: stochastically evolving iterated integer maps for finite populations

TL;DR

This paper introduces Binomial maps as a stochastic extension of deterministic population models, explicitly connecting them to agent-based models and analyzing finite population effects like demographic noise.

Contribution

It formulates Binomial maps for population dynamics, linking deterministic maps to stochastic agent-based models and deriving conditions for their equivalence.

Findings

Binomial maps incorporate stochasticity into population models.

Explicit connection between deterministic maps and agent-based models.

Conditions for equivalence between stochastic and deterministic models.

Abstract

Many models of population dynamics are formulated as deterministic iterated maps although real populations are stochastic. This is justifiable in the limit of large population sizes, as the stochastic fluctuations are negligible then. However, this also makes it challenging to use the same models for small populations where finite size effects like demographic noise and extinction cannot be ignored. Moreover, adding noise to the equations does not solve this problem as it can only represent the environmental stochasticity. An approach, sometimes used in ecological literature, but surprisingly uncommon in dynamical systems community, is \emph{Binomial maps}, which allow stochastic evolution of deterministic iterated map models of population. Here we present their formulation in a way so as to make their connection to the agent-based models explicit, and demonstrate it for the Logistic and Ricker maps. We also show that the Binomial maps are not completely equivalent to their deterministic counterparts, and derive sufficient conditions under which the equivalence holds. This approach enables rigorous finite-population analysis within familiar map-based models, bridging the deterministic map models and stochastic agent-based models.