Ecological modeling and parameter estimation for predator–prey dynamics in a closed habitat: A case study of Isle Royale

Abstract

Translating ecological observations into predictive mathematical models presents significant difficulties, which include the complexity of accurately capturing biological processes through simplified models, limited data availability, data measurement noise, and parameter estimation for such models. This paper introduces a stochastic framework to facilitate model selection and regularize the parameter estimation problem for continuous-time predator–prey ecological systems in a closed habitat given scarce discrete-time measurements. We use the historical moose–wolf population data on Isle Royale as a case study to illustrate the modeling process. Ecological studies of the moose–wolf relationship in Isle Royale National Park have been reported annually since 1959. Based on aerial surveys during winter, fluctuations in the abundance of wolves and moose were estimated from 1959 to present. We propose a model-based Markov chain Monte Carlo (MCMC) method and a sequential Monte Carlo (SMC) method, a.k.a. particle filter, to estimate this time series. We start with the classic constant-coefficient Lotka–Volterra model. While the model captures the oscillatory behavior of the data with the MCMC algorithm, it fails to capture finer-scale population changes and dynamics due to the constant parameters. To increase the predictive precision, we pose the time series estimation as an evolution–observation process, where the process function is a varying-coefficient Lotka–Volterra equation. We estimate the stochastic process using a particle filter, which simultaneously estimates the states and coefficients. We also introduce a local optimization state predictor using a least-square optimization method to further improve the estimation accuracy. Compared with the stochastic mean-particle estimator, we show that this local predictor is robust and effective, regardless of the complexity of the state model and the presence of noises. Finally, we explore the framework’s ability to interpret parameters.