Finding analytical approximations for discrete, stochastic, individual-based models of ecology

TL;DR

This paper develops new analytical approximations for complex individual-based ecological models, revealing how local interactions and dispersal influence population stability and extinction risks.

Contribution

It introduces two novel approximation methods for spatially explicit stochastic models, bridging local and long-range interactions in ecological dynamics.

Findings

Dispersal stabilizes population dynamics.

Local approximations effectively capture demographic stochasticity.

Convergence shown between local and global models as dispersal and population size increase.

Abstract

Discrete time, spatially extended models play an important role in ecology, modelling population dynamics of species ranging from micro-organisms to birds. An important question is how 'bottom up', individual-based models can be approximated by 'top down' models of dynamics. Here, we study a class of spatially explicit individual-based models with contest competition: where species compete for space in local cells and then disperse to nearby cells. We start by describing simulations of the model, which exhibit large-scale discrete oscillations and characterise these oscillations by measuring spatial correlations. We then develop two new approximate descriptions of the resulting spatial population dynamics. The first is based on local interactions of the individuals and allows us to give a difference equation approximation of the system over small dispersal distances. The second approximates the long-range interactions of the individual-based model. These approximations capture demographic stochasticity from the individual-based model and show that dispersal stabilizes population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the non-spatial global approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Our results provide new approximate analytical descriptions of a complex bottom-up model and deepen understanding of spatial population dynamics.