Non-Local Fokker-Planck Equation of Imperfect Impulsive Interventions and its Effectively Super-Convergent Numerical Discretization
Non-Local Fokker-Planck Equation of Imperfect Impulsive Interventions and its Effectively Super-Convergent Numerical Discretization
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DOI：
10.1007/978-981-15-1078-6_7
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发表时间：
2019-10
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影响因子：
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通讯作者：
H. Yoshioka;Y. Yaegashi;M. Tsujimura;M. Fujihara
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文献类型：
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作者：
H. Yoshioka;Y. Yaegashi;M. Tsujimura;M. Fujihara
控制环境和生态系统动态的人类干预有效地描述为系统状态突然过渡的脉冲干预。在应用程序中的这种干预是不完美的，因为状态转换不是完全可控的，因此是不确定的。数学描述的不完美的脉冲干预，尽管在环境和生态工程的实际问题的相关性，尚未解决到目前为止，以最好的作者的知识。本研究的目标和贡献是通过不完美脉冲干预控制的单种群动力学的制定和数值计算。我们专注于管理问题的水禽种群作为一个模型问题，人口动态如下随机微分方程脉冲收获。我们发现，定态概率密度函数的人口动力学是由一个1-D的Fokker-Planck方程具有特殊的非局域性，这可能成为一个障碍，在分析方程。通过蒙特-卡罗模拟验证了该方程在简化条件下的解析解。我们还证明了一个简单的有限体积格式可以近似其解决方案，在一个稳定的，保守的，超收敛的方式。
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.