Singular stochastic control model for algae growth management in dam downstream

Oscillation Flow Dam Operation Method for Algal Bloom Mitigation

Green algae play an important role in ecosystems as primary producers, but they can cause algal blooms, which are socio-environmental burdens as responding to them requires water resources from dam reservoirs. This study proposes an alternative for reducing algal blooms through dam operation without using additional water resources. A novel oscillation flow concept was suggested: oscillating discharge of dam for irregular flow. To examine its effect, the Environmental Fluid Dynamics Code—National Institute of Environment Research (EFDC-NIER) model was constructed and calibrated for the Namhan River, South Korea, from downstream of the Chungju Dam to downstream of Gangcheon Weir. The water quality in the study area were simulated and analyzed for August 2019, which is when the largest number of harmful cyanobacteria had been reported in recent years. Our results showed that the oscillation flow produced significant variance of flow velocity, and algal bloom density in the Namhan River was reduced by 20–30% through the operation of the Chungju Dam. However, further study and investigation are required before practical application.

Stochastic control of single-species population dynamics model subject to jump ambiguity

A logistic type stochastic control model for cost-effective single-species population management subject to an ambiguous jump intensity is presented based on the modern multiplier-robust formulation. The Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation for finding the optimal control is then derived. Mathematical analysis of the HJBI equation from the viewpoint of viscosity solutions is carried out with an emphasis on the non-linear and non-local term, which is a key term arising due to the jump ambiguity. We show that this term can be efficiently handled in the framework of viscosity solutions by utilizing its monotonicity property. A numerical scheme to discretize the HJBI equation is presented as well. Our model is finally applied to management of algae population in river environment. Optimal management policies ranging from the short-term to long-term viewpoints are numerically investigated.

Harvesting a remote renewable resource

In standard models of spatial harvesting, a resource is distributed over a continuous domain with an agent who may harvest everywhere all the time. For some cases though (e.g., fruits, mushrooms, algae), it is more realistic to assume that the resource is located at a fixed point within that domain so that an agent has to travel in order to be able to harvest. This creates a combined travelling–and–harvesting problem where slower travel implies a lower travelling cost and, due to a later arrival, a higher abundance of the resource at the beginning of the harvesting period; this, though, has to be traded off against less time left for harvesting, given a fixed planning horizon. Possible bounds on the controls render the problem even more intricate. We scrutinise this bioeconomic setting using a two-stage optimal control approach, and find that the agent economises on the travelling cost and thus avoids to arrive at the location of the resource too early. More specifically, the agent adjusts the travelling time so as to be able to harvest with maximum intensity at the beginning and the end of the harvesting period, but may also find it optimal to harvest at a sustainable level, where the harvesting and the growth rate of the stock coincide, in an intermediate time interval.

Stochastic optimal switching model for migrating population dynamics

An optimal switching control formalism combined with the stochastic dynamic programming is, for the first time, applied to modelling life cycle of migrating population dynamics with non-overlapping generations. The migration behaviour between habitats is efficiently described as impulsive switching based on stochastic differential equations, which is a new standpoint for modelling the biological phenomenon. The population dynamics is assumed to occur so that the reproductive success is maximized under an expectation. Finding the optimal migration strategy ultimately reduces to solving an optimality equation of the quasi-variational type. We show an effective linkage between our optimality equation and the basic reproduction number. Our model is applied to numerical computation of optimal migration strategy and basic reproduction number of an amphidromous fish Plecoglossus altivelis altivelis in Japan as a target species.

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

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.

Stochastic differential game for management of non-renewable fishery resource under model ambiguity

A new bio-economic model for managing population of non-renewable inland fishery resource in uncertain environment is presented. Population dynamics of the resource is described with stochastic differential equations (SDEs) having ambiguous growth and mortality rates. The performance index to be maximized by the manager of the resource while minimized by nature is presented in the context of differential game theory. The dynamic programming principle leads to a Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation that governs the optimal resource management strategy under the ambiguity. The main contribution of this paper is a series of theoretical analysis on the reduced HJBI equation for non-renewable fishery resources in a broad sense, indicating that the ambiguity critically affects the resulting optimal controls. Practical implications of the theoretical analysis results are also presented focusing on artificially hatched Plecoglossus altivelis (Ayu), an important inland fishery resource in Japan.