Fishery management—the active protection of fishery resources for sustainable exploitation and harvest—is increasingly common in the commercial fishing industry as researchers seek to conserve and sustain a dwindling natural and renewable food source. They must reconcile these ecological preservation efforts with employment, quality of life, and socio-economic demands of those who live in and profit from fishery communities.
One such community resides on the southern Turkish coast of the Black Sea and harvests the European anchovy, a small forage fish characterized by a forked tail, single dorsal fin, and pointed snout. The anchovy—of family Engraulidae—comprises over 10 percent of global fish landings (catches of marine fish in foreign and domestic ports). The European anchovy is the third most widely-harvested species, and 40 percent of landings come from the Black Sea alone. This tiny fish is both a predator and prey of many other marine creatures, and thus a crucial part of the pelagic food web.
In the last 30 years, multiple threats have compromised the European anchovy’s sustainability. Persistent overfishing has significantly reduced population numbers, and lack of profit has effectively ended commercial fishing of the anchovy in the northern waters of the Black Sea. The Mnemiopsis leidyi, an invasive type of cone jellyfish, poses yet another threat. In 1982, scientists first discovered the M. leidyi in the Black Sea as a likely product of merchant ships. It flourished in the salty environment; by 1989, upwards of 400 individual jellyfish could be found in a single cubic meter of water. Its success triggered a dramatic decline in the European anchovy population in 1990, as M. leidyi consumes anchovy eggs and larvae. Both species also compete for the same food source (zooplankton).
Due to the Mnemiopsis leidyi, the European anchovy experienced a severe population decline in 1990.
As a consequence of this dropoff, anchovies currently exist exclusively in southern waters. Thus, nearly the entirety of anchovy landings now come from the Turkish coast. Seasonal fishing on the coast occurs between September 1 and April 14. But in the interest of population conservation, commercial fishing of anchovies is permitted for only three months. If the population experiences another significant drop, the anchovy might no longer be a product of the Black Sea.
Researchers have acknowledged the lack of optimal and ecosystem-based fishery management in the Black Sea as a significant factor in European anchovy decline. During the 2018 SIAM Conference on Mathematics of Planet Earth, currently taking place in Philadelphia, Pa., Mahir Demir of the University of Tennessee used a spatial food chain model to investigate harvest strategies for anchovy fisheries on the Black Sea’s Turkish coast. His model, which has three trophic levels and accounts for seasonal harvest, calculates an optimal harvesting approach that maximizes the anchovy population’s discounted net profit. “The fishing effort for the Black Sea anchovy is high and not optimal,” Demir said. “Tracking the effect of the fishery on the corresponding food web and considering interactions between species could be helpful in avoiding overfishing and collapse.”
Around 1950, Russian mathematician Lev Pontryagin and his collaborators developed optimal control theory for ordinary differential equations (ODEs). He introduced adjoint variables that attach differential equations to the objective functional. Now, Demir channels Pontryagin’s ideas to employ optimal control theory in his own study.
Diagram of anchovy consumption in the seasonal harvest system.
First, Demir identified the model’s initial conditions. Variables include anchovy, predator, and zooplankton biomass; harvest, predation, and intrinsic growth rate; and carrying capacity. He then introduced the objective functional, which accounts for time intervals of seasonal harvest, number of fishing seasons (in years), discount rate, fishery yield, and harvest cost. Demir proceeded to present the Hamiltonian, characterize optimal control, and derive the adjoint equations. “We use the forward-backward sweep method as a numerical method for solving the optimality system numerically,” he said. Using annual landing data of anchovy populations in the southern Black Sea—courtesy of the Scientific, Technical and Economic Committee for Fisheries—Demir estimated parameters in MATLAB and approximated solutions for his ODE model.
After introducing his model, Demir tested three harvest strategies with seasonal fishery for 14 years: (1) constant harvest rates for each season, (2) different harvest rates for each season (similar to the current harvesting strategy), and (3) optimal control rates. Upon comparing the resulting graphs under the assumption of 50 percent net profit, he concluded that optimal control produced the best graph shape. “Total landing and discounted net profit are increased by using optimal control and constant effort for the anchovy fishery on the southern part of the Black Sea,” he said.
Demir also noted that traditional fishery management frequently fails to actively maintain and protect biodiversity, marine ecosystems, and sustainable fishing practices. Therefore, he used optimal control theory to compare traditional and ecosystem-based fishery management. “Traditional fishery management focuses on a single species in an ecosystem and ignores interactions between species and environmental effects on the ecosystem,” Demir said. “Ecosystem-based fishery management focuses on the whole food web and the ecosystem of a target fish population to obtain a healthy, sustainable, and diverse ecosystem.” Unsurprisingly, he found that accounting for the food web of the European anchovy offers more reliable information about fishery management.
In conclusion, Demir’s food chain model identifies optimal control as the most effective anchovy harvest strategy, as it offers a better structure for the related food web. “In order to protect the ecosystem and biodiversity of the Black Sea, it is important to observe the structure of the food web via food chain models before applying any harvesting strategy,” he said. “The optimal control strategy offers 40 percent more profit than the current strategy when we apply the approximate harvest rate.”
|| Lina Sorg is the associate editor of SIAM News.