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

Abstract

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.

Mathematical and Computational Approaches for Stochastic Control of River Environment and Ecology: From Fisheries Viewpoint

“…Stochastic optimal control as a branch of modern mathematical sciences has been playing a central role in analysis and management of noise-driven dynamical systems (Øksendal and Sulem 2019). The noises in the context of environmental and ecological management arise from stochastic river water flows (Song et al 2020) and highly nonlinear and possibly unresolved biological phenomena such as the biological growth phenomena (Yoshioka et al 2019a). Stochastic differential equations (SDEs) (Øksendal and Sulem 2019), which are formally seen as ordinary differential equations (ODEs) driven by noises, serve as an efficient mathematical tool for modeling and controlling the noisy system dynamics.…”