Non-Local Fokker-Planck Equation of Imperfect Impulsive Interventions and its Effectively Super-Convergent Numerical Discretization

Abstract

Human interventions to control environmental and ecological system dynamics are efficiently described as impulsive interventions by which the system state suddenly transits. Such interventions in applications are imperfect in the sense that the state transition is not exactly controllable and thus uncertain. Mathematical description of the imperfect impulsive interventions, despite relevance in practical problems of environmental and ecological engineering, has not been addressed so far to the best of the authors’ knowledge. The objectives and contributions of this research are formulation and numerical computation of single-species population dynamics controlled through imperfect impulsive interventions. We focus on a management problem of a waterfowl population as a model problem where the population dynamics follows a stochastic differential equation subject to impulsive harvesting. We show that the stationary probability density function of the population dynamics is governed by a 1-D Fokker-Planck equation with a special non-locality, which potentially becomes an obstacle in analyzing the equation. We demonstrate that the equation is analytically solvable under a simplified condition, which is validated through a Monte-Carlo simulation result. We also demonstrate that a simple finite volume scheme can approximate its solution in a stable, conservative, and super-convergent manner.