Modeling and prediction of phase shifts in noisy two-cycle oscillations

Understanding and predicting ecological dynamics in the presence of noise remains a substantial and important challenge. This is particularly true in light of the poor quality of much ecological data and the imprecision of many ecological models. As a first approach to this problem, we focus here on a simple system expressed as a discrete time model with 2-cycle behavior, reflecting alternating high and low population sizes. Such dynamics naturally arise in ecological systems with overcompensatory density dependence. We ask how the amount of detail included in the population estimates affects the ability to forecast the likelihood of changes in the phase of oscillation, meaning whether high populations occur in odd or in even years. We adjust the level of detail by converting continuous population levels to simple, coarse-grained descriptions using two-state and four-state models. We also consider a cubic noisy over-compensatory model with three parameters. The focus on phase changes is what distinguishes the question we are asking and the methods we use from more standard time series approaches. Obviously, adding observation states improves the ability to forecast phase shifts. In particular, the four-state model and cubic model outperform the two-state model because they include a transition state, through which the dynamics typically pass during a phase change. Nonetheless, at high noise levels the improvement in forecast skill is relatively modest. Additionally, the frequency of phase changes depends strongly on the noise level, and is much less affected by the parameter determining amplitude in the population model, so phase shift frequencies could possibly be used to infer noise levels.