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Dynamical Systems Modeling in Fisheries Management: Keeping It Simple

By Jillian Kunze

Fisheries management relies on mathematical biology modeling to balance conservation with economic concerns (see Figure 1), but some common population dynamic models may produce overly extractive fishing policies. “How do we build the most useful models?” Matthew Holden of the University of Queensland asked. One wants a model to be simple, but not too simple — and it can be difficult to figure out where to draw the line. Simple models may be unable to predict unusual negative outcomes, while complex models may produce inaccurate predictions due to overfitting and sensitivity to small perturbations.

Figure 1. A tank with several species of fish, including coral trout (the orange fish). Matthew Holden performed his first government-commissioned stock assessment on this species using traditional dynamic fisheries management models, which was the first time that this species was assessed in the Torres Strait in Australia. Photo courtesy of Matthew Holden.
During a minisymposium presentation at the 2023 SIAM Conference on Applications of Dynamical Systems, which took place in Portland, Ore., this May, Holden described several models for fisheries management with varying levels of complexity and compared their performances. He began with a linear model, which is the simplest model of population growth. “If my goal is to remove the largest number of fish from the ocean, what should be my harvesting strategy?” he asked. For the linear case with a set time period, finding the optimal harvesting strategy is easy: one would simply harvest nothing throughout the time period as the population continually grows, and then harvest all of the fish at the end. Obviously, this representation is not particularly helpful for wildlife managers. 

The most common population dynamic model in fisheries management is the Beverton-Holt model. Reframing the question as finding the optimal number of fish to leave in the ocean when harvesting—i.e., the optimal escapement—the answer for this model is 45 percent of the carrying capacity, given the parameters that Holden was using. For the similarly common Ricker model, the optimal escapement is 47 percent.

“I’d like to propose a model that will make some mathematicians and fisheries managers squirm,” Holden continued. “That is the hockey stick model.” This population model increases in a linear fashion up to a certain point, and then becomes flat (a sort of hockey stick shape). Holden noted that mathematicians are generally not huge fans of models that are not smooth, and ecologists do not like the linear growth rate with no density dependence. However, the model does produce some interesting insights.

The optimal escapement can be found at the corner where the graph transitions from a linear increase to a flat line, and indicates that 67 percent of the carrying capacity should be allowed to escape harvest. “That’s quite high!” Holden said. “What’s going on here?” Most fisheries management scenarios use a reference point of leaving 60 percent of the carrying capacity in the ocean, but if the hockey stick model is correct, that is not optimal. Furthermore, Holden found that while the optimal escapement for the Beverton-Holt model is very insensitive to the growth rate, the optimal escapement for the hockey stick model is quite sensitive to this parameter and declines with increasing growth rates (see Figure 2). 

Figure 2. The optimal escapement—that is, the percentage of the carrying capacity for a specific species that should be left in the ocean in fisheries management—versus the growth rate. The black dashed line represents the outcome from the hockey stick model, and the red line represents a common fishery stock assessment model. The hockey stick model recommends leaving a much larger portion of fish in the ocean at slower growth rates. Figure courtesy of Matthew Holden.
To investigate which model was able to fit best to actual fisheries data, Holden and student Vince Cattoni looked at the Ram Legacy Stock Assessment Database. This database provides population dynamics datasets for 476 species of fish, though the researchers reduced this number to around 300 through data cleaning. They then fit each model to the data with the nonlinear least squares method. While the Beverton-Holt model produced the best fit, all of the approaches produced similar \(R^2\) values across the datasets, indicating that the simple hockey stick model was on par with more common and complex models. Specifically, the hockey stick model actually had a higher \(R^2\) value for 31 percent of the datasets.

“One thing that’s fascinating is plotting the cumulative number of species with the best fit [for each model] over the growth rate,” Holden said. The hockey stick model creates the best fits for species that have low growth rates, for which good fisheries management is especially essential. This model also tends to recommend less permissive fishing policies with higher optimal escapements—especially for species with lower growth rates—leading to a higher standing biomass in the ocean for 72 percent of the species datasets.

While it is still hard to say which model is best overall, Holden’s efforts demonstrate that there is more to learn about modeling in fisheries management. “Maybe we should not blindly do this 60 percent of biomass,” he said. Since employing the wrong model could lead to fisheries management policies that overexploit marine ecosystems, Holden supports a more cautious approach with the model that generates the most conservative fishing policies. This tactic could help preserve marine ecosystems while still maintaining economic output.

  Jillian Kunze is the associate editor of SIAM News