Mathematical and simulation methods for deriving extinction thresholds in spatial and stochastic models of interacting agents

Abstract

Abstract In ecology, one of the most fundamental questions relates to the persistence of populations, or conversely to the probability of their extinction. Deriving extinction thresholds and characterizing other critical phenomena in spatial and stochastic models is highly challenging, with few mathematically rigorous results being available for discrete‐space models such as the contact process. For continuous‐space models of interacting agents, to our knowledge no analytical results are available concerning critical phenomena, even if continuous‐space models can arguably be considered to be more natural descriptions of many ecological systems than lattice‐based models. Here we present both mathematical and simulation‐based methods for deriving extinction thresholds and other critical phenomena in a broad class of agent‐based models called spatiotemporal point processes. The mathematical methods are based on a perturbation expansion around the so‐called mean‐field model, which is obtained at the limit of large‐scale interactions. The simulation methods are based on examining how the mean time to extinction scales with the domain size used in the simulation. By utilizing a constrained Gaussian process fitted to the simulated data, the critical parameter value can be identified by asking when the scaling between logarithms of the time to extinction and the domain size switches from sublinear to superlinear. As a case study, we derive the extinction threshold for the spatial and stochastic logistic model. The mathematical technique yields rigorous approximation of the extinction threshold at the limit of long‐ranged interactions. The asymptotic validity of the approximation is illustrated by comparing it to simulation experiments. In particular, we show that species persistence is facilitated by either short or long spatial scale of the competition kernel, whereas an intermediate scale makes the species vulnerable to extinction. Both the mathematical and simulation methods developed here are of very general nature, and thus we expect them to be valuable for predicting many kinds of critical phenomena in continuous‐space stochastic models of interacting agents, and thus to be of broad interest for research in theoretical ecology and evolutionary biology.