Numerical analysis of an extended mean field game for harvesting common fishery resource

Resource harvesting in inland fisheries during a single fishing season is mathematically modeled based on an extended mean field game played by many anglers. The harvesting rate of each angler depends on that of the others through a mean field interaction. Each angler wants to maximize fishing utility, while resource overexploitation is effectively penalized through incentives for conservation. Anglers' fishing activities are represented by controlled jump processes arising as a unique nonlinearity in the extended mean field game. Solving the extended mean field game reduces to the resolution of a forward-backward system containing a Hamilton–Jacobi–Bellman and Fokker–Planck equations coupled through the controlled harvesting rate. The average total visiting time per angler during the fishing season is obtained as a byproduct from the forward-backward system, which is used to investigate the scheme of license fee of inland fisheries. A finite difference method is proposed to numerically compute the forward-backward system based on a Picard iteration. The convergence of the numerical solution to the Hamilton–Jacobi–Bellman equation is analyzed through discontinuous viscosity solutions. Finally, we specify fish population dynamics based on the unique data of the major inland fishery resource Plecoglossus altivelis altivelis during 2023 in Japan. Computational examples are presented to analyze our mathematical model.