Stage Structured Model Measures Distribution Among Transient Fish Species

By Lina Sorg

Over the last century, the mean global annual temperature has risen by roughly 1.0 degree Celsius. If the current warming rate persists, scientists expect the global temperature to increase by an additional 1.5 degrees between 2030 and 2052. This phenomenon has inspired the migration of multiple species to different locations that are more comfortable, like deeper waters and higher latitudes and elevations. These locational shifts will persist as long as temperatures continue to rise, potentially impacting the surrounding communities of fauna and humans. For example, the relationship between fishermen and the species of fish to which they are accustomed will drastically change. Since 1970, the mean latitude of summer flounder and red hake populations has gradually increased, affecting the success and locations of fishery landings. “What will become of these species and the communities and people that rely on them for food?” Jude Kong of Princeton University asked. “What will become of the fisherman that rely on these products? Unless you introduce the public to these new species, they will not be used to them.”

During a minisymposium presentation at the 2020 SIAM Conference on the Life Sciences, which took place virtually last month, Kong employed a species distribution model to account for the complex physical and biological processes involved in the dynamics of migrating fish species. The goal of his work is to understand, forecast, and project species distribution under potential future ecosystem conditions to support resource managers and resource waters. “Which species should we rely on?” he asked. “Which species will move into our waters? Which species will move away?”

Researchers typically rely on two main types of models in this kind of scenario: correlational and process-based models. As indicated by their name, correlational models assume a correlational relationship between species, in that species’ distributions are in equilibrium with their environments. However, these models have certain drawbacks. For example, they neglect spatial population dynamics, transient dynamics, source-sink dynamics, and dispersal limitations. In reality, a species’ dispersal or population growth might constrain its ability to spread into a newly suitable habitat, thus creating time lags that would likely be important for range dynamics. Process-based models are thus a crucial alternative, since they account for the population dynamics and dispersal processes that underline range dynamics. The challenge of this model type pertains to parameter identifiability, i.e., the difficulty of distinguishing whether rapid colonization rates are due to swift population growth or extensive dispersal. This complication has significantly limited researchers’ utilization of process-based models to predict species distribution and abundance.

A potential solution to this problem involves examining stage structure within populations. For instance, size data facilitates the identification of processes that would have typically been lumped over, thus preventing identification of the separate rates. Kong and his team hypothesize that the use of available stage or size data for populations on the move may help disentangle growth and dispersal process that occur on different timescales. “With stage structure we should be able to infer all these processes,” Kong said.

He began by developing a stage structured, process-based dynamic range model for a fish population that accounts for the detailed physical and biological processes involved in species dynamics. This model, which focuses on temperature, assumes that each latitude is a patch and is thus divided into several latitudes. Kong and his team also assume that the fish life cycle is comprised of three stages: juveniles, large juveniles, and adults. Adults reproduce juveniles, and a fraction of those new juveniles get displaced into another patch. In addition, both large juveniles and adults can become dispersed into another patch or die.

Next, Kong tested the model with virtual data using a virtual observer. He simulated the dynamics of a hypothetical species and a virtual scientist who collected data during the first 70 years of temperature rise. The intricacies of these dynamics necessitated an approximate Bayesian computation (ABC) framework. “ABC is very useful when you have complex systems such as this,” Kong said. He employed the ABC algorithm to estimate the parameters of his stage structured model using the first 40 years of temperature rise, and then tested the model forecast skill by comparing it with an additional 30 years of simulations that were not used for the purposes of fitting. For simplicity, Kong assumed that the virtual scientist sampled from 10 habitat patches—spread across a temperature gradient— and recognized three length classes.

Kong compared his stage structured model with existing models (a single species model and a correlational model) to validate its ability to estimate species abundance. Finally, he fit the model to Atlantic cod (Gadus Morhua) data from the Northeast U.S. region from 1968 to 2015. Ultimately, Kong hopes to utilize his findings to advise policymakers and communities. “It’s very important for us to be able to inform these people of which species they can rely on, which species will go extinct, and which species will go away,” he said. “If we are informed, we will know what to do.”

Lina Sorg is the managing editor of SIAM News.