Nonlinear Tools for a Nonlinear World: Applications of Empirical Dynamic Modeling to Marine Ecosystems

A fundamental objective in the study of dynamic systems is to understand and predict their behavior. The research presented in this thesis addresses this goal using the general framework of empirical dynamic modeling (EDM). In the classical approach, system behavior is described using fixed mathematical equations, and multiple effects are often treated as linearly separable (i.e. in a reductionist framework). In contrast, EDM applies Takens' Theorem and the method of time delay embeddings to reconstruct system dynamics from time series data. This gives EDM the flexibility to model nonlinear, state-dependent interactions that are otherwise challenging for traditionally linear mathematical models.The first part of this thesis applies EDM towards the study of sockeye salmon populations from the Fraser River in British Columbia, Canada in order to understand the factors that affect recruitment and to produce better models for the annual returns. Whereas classical (linear) fisheries models do not improve when incorporating the environment, I show that Fraser River sockeye salmon actually exhibit nonlinear dynamics, and therefore are not amenable to these methods. Instead, EDM models that can account for nonlinearity show improved forecasts, and moreover, benefit greatly from the incorporation of state-dependent environmental effects. In addition, I demonstrate that the abrupt changes in the salmon populations, correlated with North Pacific climate indices can be explained as state-dependent nonlinear behavior. Whereas classical fisheries models or linear correlations would suggest sudden shifts in behavior associated with climate regimes, an appropriate nonlinear lens indicates that environmental effects are state-dependent, and that aggregation of data at the regional level produces the apparent linear patterns.The second part of this thesis involves the development of new methods in the EDM framework to distill data (i.e. time series) into information (i.e. inferences and conclusions). I show that a lagged form of convergent cross mapping (CCM), a method to infer causation in time series, can greatly enhance its capabilities, by quantifying the time delay associated with causation. This new method can be used to distinguish between direct and indirect, transitive, effects as well as produce more reliable estimates of interaction strength. I also develop Multiview Embedding (MVE) to address the issues of noise and short time series length in high-dimensional complex systems. By using a multimodel approach that leverages the ``equation-free'' framework of EDM, MVE combines multiple reconstructions of system behavior, producing more accurate and precise forecasts, and demonstrating that complexity can be an asset, because of how information about the system dynamics is duplicated across interacting variables.Finally, these methods are included in a software package for EDM, developed for the R statistical language. A user guide for this software package, including installation instructions and examples, is included as an appendix.

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