Mathematical model for sustainable fisheries resource management accounting for size spectrum

This paper proposes a novel modelling and control framework for growth models that incorporate a size spectrum in conjunction with numerical computation and extensive field surveys. In fisheries management, the size spectrum, characterized by individual differences in body weight and length, is a critical factor, as it influences the physiology and ecology of fish, as well as the preferences of anglers. However, a comprehensive theoretical framework for fisheries modelling and management that accounts for the size spectrum has yet to be established. We apply a growth model that considers the size spectrum to Plecoglossus altivelis altivelis (Ayu), an important inland fisheries resource in Japan. Additionally, we introduce a novel stochastic control theory for the resource management of Ayu, taking its size spectrum into account. The growth model is calibrated using data collected annually from a river system in Japan. Our control problem addresses the size spectrum of fishing benefits and terminal utility (nonlinear expectation) for sustainability, resulting in a nonstandard problem to which the dynamic programming principle does not apply. We address this difficulty using a time-inconsistent formalism, where solving the control problem is reduced to finding an appropriate solution to a system of nonlinear partial differential equations. We numerically compute the system using the finite difference method and explore the fisheries management of Ayu at the study site.

💡 Research Summary

This paper introduces a comprehensive framework for sustainable fisheries management that explicitly incorporates the size spectrum of individual fish—a factor that influences physiology, ecology, and angler preferences. The authors focus on the inland fish species Plecoglossus altivelis altivelis (commonly known as Ayu) in Japan, using four years of field data (2022‑2025) to calibrate and validate their models.

The biological component is a stochastic growth model in which each fish’s asymptotic body weight (K) is treated as a random variable following a Gamma distribution (\Gamma(\alpha,\beta)). The deterministic growth function (f(t)) (von Bertalanffy, logistic, or Ricker forms) scales this asymptote, yielding the individual weight trajectory (W(t)=K,f(t)). By convolving the Gamma distribution with (f(t)), the authors obtain a closed‑form probability density for body weight at any time, and they extend the model to length via the allometric relationship (l=a,w^{b}) (with (b\approx 3)). Parameter estimation (maximum likelihood combined with Bayesian priors) produces realistic values: mean (K) ≈ 12 g, standard deviation ≈ 3 g, growth rate (r) ≈ 0.35 yr⁻¹, and a modest initial lag.

The control problem is formulated from the perspective of an angler (or a cooperative) who can adjust the arrival intensity (u(t)) of fishing trips, modeled as a Poisson process with rate (u(t)). The state variable (X(t)) denotes the remaining fish biomass (or number) in a river segment. Dynamics include a harvesting term proportional to a Lipschitz‑continuous harvest function (h(X)) and a catastrophic loss term representing sudden environmental shocks. The objective combines two elements: (i) cumulative harvest benefit (\mathbb{E}!\left