A minimal stochastic generalization of a deterministic open-ended logistic growth model is proposed for efficiently describing the biological growth of individual organisms under natural environment. The model is a system of stochastic differential equations. Its unique solvability in a strong sense is proven, and the behaviour of the solution is analysed. The presented model is then applied to the migratory fish Plecoglossus altivelis altivelis (P. altivelis, Ayu) having a one-year life history based on the data sets collected in 2017 and 2018.

Summary A recent river environmental restoration problem is approached from a standpoint of stochastic control of hybrid regime‐switching diffusion processes with discrete and costly observations. This setting harmonizes with real problems because continuously obtaining environmental and ecological information is difficult, and is often costly. The main problem is to decide when and how much of the sediment should be supplied into a river environment to effectively suppress bloom of benthic algae. The interventions are allowed only at observation times. Finding the optimal river restoration policy ultimately reduces to solving an optimality equation in an unconventional form due to the observation cost and discrete observations. We show its unique solvability and present a connection with degenerate parabolic partial differential equations, with which we can construct an effective algorithm for its approximation. An uncertainty‐averse optimization problem is also considered as an advanced problem. Coefficients and parameter values are identified from experimental and observation results to numerically compute the optimal restoration policy of an existing river environment with and without uncertainty.

A new mathematical model for finding the optimal harvesting policy of an inland fishery resource under incomplete information is proposed in this paper. The model is based on a stochastic control formalism in a regime‐switching environment. The incompleteness of information is due to uncertainties involved in the body growth rate of the fishery resource: a key biological parameter. Finding the most cost‐effective harvesting policy of the fishery resource ultimately reduces to solving a terminal and boundary value problem of a Hamilton‐Jacobi‐Bellman equation: a nonlinear and degenerate parabolic partial differential equation. A simple finite difference scheme for solving the equation is then presented, which turns out to be convergent and generates numerical solutions that comply with certain theoretical upper and lower bounds. The model is finally applied to the management of Plecoglossus altivelis, a major inland fishery resource in Japan. The regime switching in this case is due to the temporal dynamics of benthic algae, the main food of the fish. Model parameter values are identified from field measurement results in 2017. Our computational results clearly show the dependence of the optimal harvesting policy on the river environmental and biological conditions. The proposed model would serve as a mathematical tool for fishery resource management under uncertainties.

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An optimization problem of controlling a dam installed in a river is analyzed based on a stochastic control formalism of a diffusion process under model ambiguity: a new mathematical approach to this issue. The diffusion process is a pathwise unique solution to a water balance equation considering the inflow, outflow, water loss in the reservoir, and direct rainfall. Finding the optimal reservoir operation policy reduces to solving a degenerate parabolic partial differential equation: a Hamilton-Jacobi-Bellman-Isaacs equation. A monotone finite difference scheme is constructed for discretization of the equation, successfully generating nonoscillatory and reasonably accurate numerical solutions. Stability analysis of the resulting water balance dynamics is finally carried out for both environmentally friendly and not friendly reservoir operations. KEYWORDS finite difference scheme, Hamilton-Jacobi-Bellman-Isaacs equation, reservoir operation, stochastic control, viscosity solution INTRODUCTIONThis paper contributes to providing a mathematical framework for modeling and control of reservoirs created in rivers.Since the problem has an engineering background, it is first presented, and then, the issues to be addressed and our contributions are explained.Many rivers have dams serving as central infrastructures to supply essential water resources for human lives. 1-3 The stored water in the reservoir created by a dam is utilized for multiple purposes, such as drinking water, irrigation, and hydropower generation. 4 Several dams are used for flood mitigation as well. 5,6 On the other hand, controlling a dam critically affects its downstream river environment because the original flow regime is altered. 7-9 Sediment transport and attached algae population dynamics in dam downstream experience significant changes because dams trap sediment and the algae growth is highly flow-dependent. 10 Dam operation is thus desired to be environmentally friendly such that its impacts on the downstream are minimized, while it should be fit-for-purpose at the same time.Management of environment and ecology in rivers has been a long-standing engineering problem, 11-13 and mathematical tools and concepts for their modeling, analysis, and control are still developing. Several researchers considered river management problems based on stochastic process models such as stochastic differential equations (SDEs) 14 that can naturally consider stochasticity involved in environmental and ecological processes. 15 Interactions between aquatic species, such as riparian vegetation and fishes, and river discharges have been described with SDEs. 16 Miyamoto and Kimura 17 analyzed seedling, growth, and mortality dynamics of riparian trees subject to stochastic flood disturbances. Vesipa et al 18 764

We identify stochastic process models describing the time series of inflow and outflow discharges of Obara Dam in Hii River, Japan. These models are based on tempered stable Ornstein–Uhlenbeck (TSOU) processes that have not been utilized in hydrological analysis but can capture both large and small fluctuation of the time series data. In addition, the models can be exactly simulated in a statistical sense by utilizing a recent tailored discretization algorithm, serving as efficient stochastic tools. We show that the identified models accurately reproduce statistical moments of the time series data and probability density functions. Based on the mathematical framework of backward stochastic differential equation (BSDE), the identified model is applied to a unique dynamic stochastic analysis of dissolved silicon (DSi) load flowing into the reservoir associated with Obara Dam. We thus contribute to the first application of TSOU processes and a BSDE to hydrological analysis.