Population size in stochastic discrete-time ecological dynamics

TL;DR

This paper investigates how environmental stochasticity affects long-term population sizes in ecological models, revealing that fluctuations can either increase, decrease, or not change population sizes depending on model specifics and parameters.

Contribution

It introduces methods to analyze invariant measures in stochastic ecological models and demonstrates how small noise impacts population sizes in various scenarios.

Findings

Environmental fluctuations can alter population sizes in different ways.

The effect of noise depends on model parameters and the baseline deterministic model.

Using recent mathematical results, the study links growth rates at stationarity to population outcomes.

Abstract

We study how environmental stochasticity influences the long-term population size in certain one- and two-species models. The difficulty is that even when one can prove that there is persistence, it is usually impossible to say anything about the invariant probability measure which describes the persistent species. We are able to circumvent this problem for some important ecological models by noticing that the per-capita growth rates at stationarity are zero, something which can sometimes yield information about the invariant probability measure. For more complicated models we use a recent result by Cuello to explore how small noise influences the population size. We are able to show that environmental fluctuations can decrease, increase, or leave unchanged the expected population size. The results change according to the dynamical model and, within a fixed model, also according to which parameters (growth rate, carrying capacity, etc) are affected by environmental fluctuations. Moreover, we show that not only do things change if we introduce noise differently in a model, but it also matters what one takes as the deterministic `no-noise' baseline for comparison.