A primitive model for stochastic regular‐impulse population control and its application to ecological problems

• Published in: Advanced control for applications 2019-12, Vol.1 (1), p.n/a
• Main Authors: Yoshioka, Hidekazu, Yaegashi, Yuta, Yoshioka, Yumi, Hamagami, Kunihiko, Fujihara, Masayuki
• Format: Article
• Language: English
• Subjects: deteriorating items; finite difference scheme; Fisheries; Hamilton‐Jacobi‐Bellman equation; inland fisheries management; Mathematical analysis; Mathematical models; mixed stochastic control; Optimal control; Population control; Stochastic processes
• ISSN: 2578-0727; 2578-0727
• DOI: 10.1002/adc2.16

A primitive mathematical model for population control in natural environment is formulated under a unified regular‐impulse stochastic control formalism. This kind of mathematical models, although they are candidates of the models of population control, seem not to be well studied. In this paper, the impulse control uses deteriorating items and is scheduled only at predetermined times as in many management problems. Finding the optimal control reduces to solving a system of recursive Hamilton‐Jacobi‐Bellman equations. Mathematical analysis on the system provides a viscosity characterization of its solution. The model is applied to management problems of the predatory bird and harmful benthic alga, both of which are concerns in recent inland fisheries. We show that the optimal regular controls for the two problems have qualitatively different behavior with each other, which is due to different interactions between the regular and the impulse controls.