Distinguishing intrinsic limit cycles from forced oscillations in ecological time series

No sensitivity to functional forms in the Rosenzweig-MacArthur model with strong environmental stochasticity

The classic Rosenzweig-MacArthur predator-prey model has been shown to exhibit, like other coupled nonlinear ordinary differential equations (ODEs) from ecology, worrying sensitivity to model structure. This sensitivity manifests as markedly different community dynamics arising from saturating functional responses with nearly identical shapes but different mathematical expressions. Using a stochastic differential equation (SDE) version of the Rosenzweig-MacArthur model with the three functional responses considered by Fussmann & Blasius (2005), I show that such sensitivity seems to be solely a property of ODEs or stochastic systems with weak noise. SDEs with strong environmental noise have by contrast very similar fluctuations patterns, irrespective of the mathematical formula used. Although eigenvalues of linearized predator-prey models have been used as an argument for structural sensitivity, they can also be an argument against structural sensitivity. While the sign of the eigenvalues' real part is sensitive to model structure, its magnitude and the presence of imaginary parts are not, which suggests noise-driven oscillations for a broad range of carrying capacities. I then discuss multiple other ways to evaluate structural sensitivity in a stochastic setting, for predator-prey or other ecological systems.

Noise-induced versus intrinsic oscillation in ecological systems

Studies of oscillatory populations have a long history in ecology. A first-principles understanding of these dynamics can provide insights into causes of population regulation and help with selecting detailed predictive models. A particularly difficult challenge is determining the relative role of deterministic versus stochastic forces in producing oscillations. We employ statistical physics concepts, including measures of spatial synchrony, that incorporate patterns at all scales and are novel to ecology, to show that spatial patterns can, under broad and well-defined circumstances, elucidate drivers of population dynamics. We find that when neighbours are coupled (e.g. by dispersal), noisy intrinsic oscillations become distinguishable from noise-induced oscillations at a transition point related to synchronisation that is distinct from the deterministic bifurcation point. We derive this transition point and show that it diverges from the deterministic bifurcation point as stochasticity increases. The concept of universality suggests that the results are robust and widely applicable.

Fitting stochastic predator-prey models using both population density and kill rate data

Most mechanistic predator-prey modelling has involved either parameterization from process rate data or inverse modelling. Here, we take a median road: we aim at identifying the potential benefits of combining datasets, when both population growth and predation processes are viewed as stochastic. We fit a discrete-time, stochastic predator-prey model of the Leslie type to simulated time series of densities and kill rate data. Our model has both environmental stochasticity in the growth rates and interaction stochasticity, i.e., a stochastic functional response. We examine what the kill rate data brings to the quality of the estimates, and whether estimation is possible (for various time series lengths) solely with time series of population counts or biomass data. Both Bayesian and frequentist estimation are performed, providing multiple ways to check model identifiability. The Fisher Information Matrix suggests that models with and without kill rate data are all identifiable, although correlations remain between parameters that belong to the same functional form. However, our results show that if the attractor is a fixed point in the absence of stochasticity, identifying parameters in practice requires kill rate data as a complement to the time series of population densities, due to the relatively flat likelihood. Only noisy limit cycle attractors can be identified directly from population count data (as in inverse modelling), although even in this case, adding kill rate data - including in small amounts - can make the estimates much more precise. Overall, we show that under process stochasticity in interaction rates, interaction data might be essential to obtain identifiable dynamical models for multiple species. These results may extend to other biotic interactions than predation, for which similar models combining interaction rates and population counts could be developed.

The relative strengths of rapid and delayed density dependence acting on a terrestrial herbivore change along a pollution gradient

Animal populations vary in response to a combination of density-dependent and density-independent forces, which interact to drive their population dynamics. Understanding how abiotic forces mediate the form and strength of density-dependent processes remains a central goal of ecology, and is of increasing urgency in a rapidly changing world. Here, we report for the first time that industrial pollution determines the relative strength of rapid and delayed density dependence operating on an animal population. We explored the impacts of pollution and climate on the population dynamics of an eruptive leafmining moth, Phyllonorycter strigulatella, around a coal-fired power plant near Apatity, north-western Russia. Populations were monitored at 14 sites over 26 years. The relative strengths of rapid and delayed density dependence varied with distance from the power plant. Specifically, the strength of rapid density dependence increased while the strength of delayed density dependence decreased with increasing distance from the pollution source. Paralleling the increasing strength of rapid density dependence, we observed declines in the densities of P. strigulatella, increases in predation pressure from birds and ants, and declines in an unknown source of mortality (perhaps plant antibiosis) with increasing distance from the power plant. In contrast to the associations with pollution, associations between climate change and leafminer population densities were negligible. Our results may help to explain the outbreaks of insect herbivores that are frequently observed in polluted environments. We show that they can result from the weakening of rapid (stabilizing) density dependence relative to the effects of destabilizing delayed density dependence. Moreover, our results may explain some of the variation reported in published studies of animal populations in polluted habitats. Variable results may emerge in part because of the location of the study sites on different parts of pollution gradients. Finally, in a rapidly changing world, effects of anthropogenic pollution may be as, or more, important than are effects of climate change on the future dynamics of animal populations.