Elevated nonlinearity as an indicator of shifts in the dynamics of populations under stress.

Populations occasionally experience abrupt changes, such as local extinctions, strong declines in abundance or transitions from stable dynamics to strongly irregular fluctuations. Although most of these changes have important ecological and at times economic implications, they remain notoriously difficult to detect in advance. Here, we study changes in the stability of populations under stress across a variety of transitions. Using a Ricker-type model, we simulate shifts from stable point equilibrium dynamics to cyclic and irregular boom-bust oscillations as well as abrupt shifts between alternative attractors. Our aim is to infer the loss of population stability before such shifts based on changes in nonlinearity of population dynamics. We measure nonlinearity by comparing forecast performance between linear and nonlinear models fitted on reconstructed attractors directly from observed time series. We compare nonlinearity to other suggested leading indicators of instability (variance and autocorrelation). We find that nonlinearity and variance increase in a similar way prior to the shifts. By contrast, autocorrelation is strongly affected by oscillations. Finally, we test these theoretical patterns in datasets of fisheries populations. Our results suggest that elevated nonlinearity could be used as an additional indicator to infer changes in the dynamics of populations under stress.

Stock assessment and end-to-end ecosystem models alter dynamics of fisheries data.

Although all models are simplified approximations of reality, they remain useful tools for understanding, predicting, and managing populations and ecosystems. However, a model's utility is contingent on its suitability for a given task. Here, we examine two model types: single-species fishery stock assessment and multispecies marine ecosystem models. Both are efforts to predict trajectories of populations and ecosystems to inform fisheries management and conceptual understanding. However, many of these ecosystems exhibit nonlinear dynamics, which may not be represented in the models. As a result, model outputs may underestimate variability and overestimate stability. Using nonlinear forecasting methods, we compare predictability and nonlinearity of model outputs against model inputs using data and models for the California Current System. Compared with model inputs, time series of model-processed outputs show more predictability but a higher prevalence of linearity, suggesting that the models misrepresent the actual predictability of the modeled systems. Thus, caution is warranted: using such models for management or scenario exploration may produce unforeseen consequences, especially in the context of unknown future impacts.

Equation-free mechanistic ecosystem forecasting using empirical dynamic modeling.

It is well known that current equilibrium-based models fall short as predictive descriptions of natural ecosystems, and particularly of fisheries systems that exhibit nonlinear dynamics. For example, model parameters assumed to be fixed constants may actually vary in time, models may fit well to existing data but lack out-of-sample predictive skill, and key driving variables may be misidentified due to transient (mirage) correlations that are common in nonlinear systems. With these frailties, it is somewhat surprising that static equilibrium models continue to be widely used. Here, we examine empirical dynamic modeling (EDM) as an alternative to imposed model equations and that accommodates both nonequilibrium dynamics and nonlinearity. Using time series from nine stocks of sockeye salmon (Oncorhynchus nerka) from the Fraser River system in British Columbia, Canada, we perform, for the the first time to our knowledge, real-data comparison of contemporary fisheries models with equivalent EDM formulations that explicitly use spawning stock and environmental variables to forecast recruitment. We find that EDM models produce more accurate and precise forecasts, and unlike extensions of the classic Ricker spawner-recruit equation, they show significant improvements when environmental factors are included. Our analysis demonstrates the strategic utility of EDM for incorporating environmental influences into fisheries forecasts and, more generally, for providing insight into how environmental factors can operate in forecast models, thus paving the way for equation-free mechanistic forecasting to be applied in management contexts.

Distinguishing random environmental fluctuations from ecological catastrophes for the North Pacific Ocean.

The prospect of rapid dynamic changes in the environment is a pressing concern that has profound management and public policy implications. Worries over sudden climate change and irreversible changes in ecosystems are rooted in the potential that nonlinear systems have for complex and 'pathological' behaviours. Nonlinear behaviours have been shown in model systems and in some natural systems, but their occurrence in large-scale marine environments remains controversial. Here we show that time series observations of key physical variables for the North Pacific Ocean that seem to show these behaviours are not deterministically nonlinear, and are best described as linear stochastic. In contrast, we find that time series for biological variables having similar properties exhibit a low-dimensional nonlinear signature. To our knowledge, this is the first direct test for nonlinearity in large-scale physical and biological data for the marine environment. These results address a continuing debate over the origin of rapid shifts in certain key marine observations as coming from essentially stochastic processes or from dominant nonlinear mechanisms. Our measurements suggest that large-scale marine ecosystems are dynamically nonlinear, and as such have the capacity for dramatic change in response to stochastic fluctuations in basin-scale physical states.