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State Space Reconstruction to Forecast Ecological Dynamics

By Jillian Kunze

Policymakers often make decisions that have broad ecological and economic impacts. When managing fisheries and wildlife communities, leaders must balance concerns regarding revenue and jobs while maintaining a sustainable wild population. Politicians must also respond to various ecological extremes—such as pest outbreaks, harmful algae blooms, and wildfires—which collectively cost about $70 billion a year globally. Unfortunately, these political decisions often reflect the ecological state at that moment in time and may not be the best choices for the future of the environment. To best inform policymakers about the ramifications of their environmental decisions, scientists need to be able to create forecasts of ecological systems. 

During a minisymposium presentation at the 2020 SIAM Annual Meeting, Bethany Johnson of the University of California, Santa Cruz described a model to forecast ecological dynamics across a spatially extended system. Johnson explained how her Southern California upbringing, complete with frequent trips to the coast, inspired her fascination with the marine sciences. This project allowed her to combine her ecological interests with her love for mathematics. 

The dynamics of any species depends on many complicated factors, including the population of predators and prey, competition from other species, human activities, and climate. Unfortunately, these factors are inherently uncertain and often lack concrete data, which makes it difficult for researchers to model a species’ dynamics. To tackle these challenges, Johnson and her collaborators used state space reconstruction (SSR). The method is based on Takens’ Embedding Theorem, which states that one can utilize the time lags of a single variable in a dynamical system to reconstruct an image of the full system. SSR is useful for modelling species because it only requires the time series of one variable, instead of a multitude of complicated factors; the technique’s basic goal is to identify a function that will take a variable’s history to the future of that variable. 

An inherent challenge of this approach is the fact that ecological data often presents in the form of short time series, which is not ideal for SSR. However, the data does frequently have a spatial reference of time series of the same variable that is taken in different places, which one can incorporate into the model. Johnson and her team tried two approaches to include this spatial information. The first approach leveraged spatial information for spatial SSR by using both time lags and spatial lags, thus compiling a library of training data with the time series. The second approach employed concatenated libraries from different locations as one large set of training data. 

Johnson compared the effectiveness of different strategies when creating the SSR model.
 Johnson and her colleagues used a Bayesian Gaussian process regression to approximate the function that governs the species dynamics. They found it helpful to train the Gaussian process with a physically-informed prior instead of a naïve prior, as doing so created much more consistency between different trials. To test the simulation, the team modelled the dynamics across space and time for species in several different types of relationships, such as predator-prey and host-parasitoid-parasitoid. 

The model performed well, even with incomplete observations. Johnson and her colleagues investigated the effect of time series length on the model by generating chaotic dynamics with three different training time series lengths, testing them on 20 time points, and repeating the process 100 times. They found that with shorter time periods, library concatenation produced better results than utilization of a library from just one location. In addition, their varied attempts demonstrated that concatenation’s benefit decreased as heterogeneity increased, likely because it is more reasonable to share information when dynamics between different spaces are similar. 

To further test the model, the team applied it to data from NOAA’s Northeast Fisheries Science Center, which performs a bottom trawl survey every autumn to count fish, weigh them, and sort them by species. Johnson and her collaborators focused on three widely-distributed, short-lived species: longfin squid, silver hake, and butterfish. The results from this application were fairly similar to those from the simulations; the library concatenation again did not significantly improve the model’s output at a wide spatial range, as too much heterogeneity existed. 

In the future, Johnson intends to compare her research team’s SSR simulation with other data-driven methods. In addition, she would like to find a way to circumvent the requirement that the model’s time series input be taken at constant intervals, as that is not always realistic for ecological data. But even in its current state, data-driven forecasting with SSR allowed her team to overcome the challenges of incomplete observation and structural uncertainty. Johnson and her collaborators hope that this work will eventually lead to a robust control scheme for fisheries, pest control, and other ecological systems. 

  Jillian Kunze is the associate editor of SIAM News