A simplified stochastic optimization model for logistic dynamics with control-dependent carrying capacity

Abstract

A simplified stochastic control model for optimization of logistic dynamics with the control-dependent carrying capacity, which is motivated by a recent algae population management problem in the river environment, is presented. Solving the optimization problem reduces to finding a solution to a non-local first-order differential equation called tt-Jacobi-Bellman (HJB) equation. It is shown that the HJB equation has a unique viscosity solution and that the solution can be approximated with a finite difference scheme. Asymptotic estimates of the solution and the optimal control are derived and compared with numerical solutions. Finally, parameter dependence of the optimal control is examined numerically with implications to river environmental management.

A hybrid stochastic river environmental restoration modeling with discrete and costly observations

“…Their least‐squares fitting results are presented in Table 1. An important finding is the estimate 0 < a < 1, implying that the conventional ( a = 1) 45,85 with a constant D does not apply. Figure 2 and Table 1 imply that the model parameters a and p are uncertain, even under the same experimental conditions.…”

“…In this regime, the algae population stochastically grows up following a logistic‐like dynamics 60‐62 . The detachment coefficient D is often considered to be independent of the population dynamics but to be flow dependent 45,63 . An experimental function is used in our application.…”

“…We assume that there is an increasing and nonnegative continuous function c f : [0, 1] → [0, +∞) with c f (0) = 0 such that. This assumption allows us to use continuous functions not only the previous Lipschitz and Hölder continuous functions as the disutility, 45,56,69 but also more irregular continuous.…”

“…A finite difference scheme for discretization of the PDEs ) and () is briefly explained. The presentation of the scheme itself is simple and the underlying discretization techniques are found in the authors' previous research, 45,56,58 and is therefore not presented in this article. The numerical techniques in Yoshioka and Tsujimura 46,58 can handle the nonlinear terms.…”

“…Modeling and control of algae and macrophytes dynamics in river environment have been studied from the viewpoint of continuous‐time observations of continuous‐time processes 45 . However, as Yoshioka and Tsujimura 46 and Yoshioka et al 47 pointed out, continuous observations of environmental and ecological processes are often impossible due to labor and technical reasons, and have usually been carried out in a discrete‐time manner like once in a month, in a week, or in several days.…”