Yuta Yaegashi

An optimal management strategy for stochastic population dynamics of released Plecoglossus altivelis in rivers

Excessive predation pressure from the waterfowl Phalacrocorax carbo (Great Cormorant) on Plecoglossus altivelis (Ayu) has recently been a severe problem of river environment in Japan. Local fishery cooperatives are currently suffering from economic difficulties due to decrease of the fish catch of P. altivelis. Local fishery cooperatives and municipalities have been enthusiastically trying to develop countermeasures that can effectively reduce the predation pressure; however, their effectiveness and efficiency have not been systematically quantified well. This aim can be achieved with the help of an appropriate mathematical model. In this paper, based on a pure death process, a practical stochastic control model for population dynamics of released P. altivelis in river environment under predation pressure from P. carbo, harvesting by human, and environmental fluctuations is proposed. Finding an optimal management strategy ultimately reduces to solving a 2D Hamilton–Jacobi–Bellman equation, which is performed with a finite element scheme. Its application to a Japanese river environment successfully computes the optimal management strategy that is consistent with the reality. Numerical sensitivity analysis of the presented mathematical model is also performed for comprehension of dependence of the optimal strategy on the model parameters.

Numerical Simulation of a Hamilton-Jacobi-Bellman Equation for Optimal Management Strategy of Released Plecoglossus Altivelis in River Systems

A stochastic differential equation model for population dynamics of released Plecoglossus altivelis (Ayu) in a river system subject to feeding damage by Phalacrocorax carbo (Great Cormorant) and fishing activity by human is proposed. A stochastic optimal control problem to maximize the sum of the cost of countermeasure to prevent the feeding damage and the benefit of harvesting the fish is formulated, which ultimately reduces to solving a Hamilton-Jacobi-Bellman equation. Application of a Petrov-Galerkin finite element scheme to the equation successfully computes the optimal management strategies for the population dynamics of P. altivelis in a real river system and ecological and economical indices to verify them.

Application of Stochastic Control Theory to Biophysics of Fish Migration Around a Weir Equipped with Fishways

A weir installed along a river cross-section potentially serves as a physical barrier that prevents fishes from migrating toward upstream. Many rivers in the world encounter this severe and ubiquitous ecological issue. The objective of this paper is to present a biophysical application of stochastic control theory to upstream fish migration in river reaches where movements of individual fishes are considered as horizontally 2-D controlled processes. Identifying the biological and ecological objective function to be maximized with the dynamic programming principle leads to a 2-D nonlinear elliptic equation referred to as the Hamilton-Jacobi-Bellman Equation (HJBE). Solving the HJBE leads to an optimal swimming velocity field of individual fishes in water flows. Utilizing appropriate differential equations associated with the HJBE enables us to efficiently and consistently compute attraction ability of fishways installed at a weir from a statistical viewpoint. An application of the present mathematical model to upstream migration of juvenile Plecoglossus altivelis (Ayu) around a recently renovated weir in Hii River, San-in area, Japan is carried out in order to assess attraction ability of its associated fishways.

Singular stochastic control model for algae growth management in dam downstream

ABSTRACT A stochastic control model for finding an ecologically sound, fit-for-purpose dam operation policy to suppress bloom of attached algae in its downstream is presented. A singular exactly solvable and a more realistic regular-singular cases are analysed in terms of a Hamilton–Jacobi–Bellman equation. Regularity and consistency of the value function are analysed and its classical verification theorem is established. Practical implications of the mathematical analysis results are discussed focusing on parameter dependence of the optimal controls. An asymptotic analysis with a numerical computation reveals solution behaviour of the Hamilton–Jacobi–Bellman equation near the origin, namely at the early stage of algae growth.

A finite difference scheme for variational inequalities arising in stochastic control problems with several singular control variables

Abstract A finite difference scheme is developed for solving 1-D variational inequalities arising in stochastic control problems with several singular control variables. The scheme guarantees the uniqueness of numerical solutions. A policy iteration algorithm is then proposed to solve the discretized problem. The present approach is applied to solving variational inequalities associated to cost-effective management problems of benthic algae on the riverbed downstream of a dam: an urgent environmental problem. Accuracy of the scheme is verified to be first-order for both the solution and its free boundaries. An advanced problem that involves a max–min differential game structure is also examined. The scheme then computes reasonably accurate numerical solutions which are consistent with the theoretical asymptotic estimates.

An optimal stopping approach for onset of fish migration

Comprehending life history of migratory fish, onset of migration in particular, is a key biological and ecological research topic that still has not been clarified. In this paper, we propose a simple mathematical model for the onset of fish migration in the context of a stochastic optimal stopping theory, which is a new attempt to our knowledge. Finding the criteria of the onset of migration reduces to solving a variational inequality of a degenerate elliptic type. As a first step of the new mathematical modeling, mathematical and numerical analyses with particular emphasis on whether the model is consistent with the past observation results of fish migration are examined, demonstrating reasonable agreement between the theory and observation results. The present mathematical model thus potentially serves as a simple basis for analyzing onset of fish migration.

Unique solvability of a singular stochastic control model for population management

Abstract We present and mathematically analyze a singular stochastic control model for cost-effective and ecologically-sound indirect population control strategy for fish-eating birds. Finding the optimal strategy of a threshold type reduces to solving a variational inequality. We prove the unique existence of its viscosity solution that is neither convex nor concave, which turns out to be a classical solution. Comparative statics on the optimal threshold is performed as well to demonstrate practical implications of the model.

A Stochastic Impulse Control Model for Population Management of Fish-Eating Bird Phalacrocorax Carbo and Its Numerical Computation

Feeding damage from a fish-eating bird Phalacrocorax carbo to a fish Plecoglossus altivelis is severe in Japan. A stochastic impulse control model is introduced for finding the cost-effective and ecologically conscious population management policy of the bird. The optimal management policy is of a threshold type; if the population reaches an upper threshold, then taking a countermeasure to immediately reduce the bird to a lower threshold. This optimal policy is found through solving a Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI). We propose a numerical method for HJBQVIs based on a policy iteration approach. Its accuracy on numerical solutions and the associated free boundaries for the management thresholds of the population, is investigated against an exact solution. The computational results indicate that the proposed numerical scheme can successfully solve the HJBQVI with the first-order computational accuracy. In addition, it is shown that the scheme captures the free boundaries subject to errors smaller than element lengths.

Wise-Use of Sediment for River Restoration: Numerical Approach via HJBQVI

A stochastic differential game for cost-effective restoration of river environment based on sediment wise-use, an urgent environmental issue, is formulated. River restoration here means extermination of harmful algae in dam downstream. The algae population has weak tolerance against turbid river water flow, which is why the sediment is focused on in this paper. Finding the optimal strategy of the sediment transport reduces to solving a spatio-temporally 4-D Hamilton-Jacobi-Bellman Quasi-Variational Inequality (HJBQVI): a degenerate nonlinear and nonlocal parabolic problem. Solving the HJBQVI is carried out with a specialized finite difference scheme based on an exponentially-fitted discretization with penalization, which generates stable numerical solutions. An algorithm for solving the discretized HJBQVI without resorting to the conventional iterative matrix inversion methods is then presented. The HJBQVI is applied to a real problem in a Japanese river where local fishery cooperatives and local government have been continuing to debate the way of using some stored sediment in a diversion channel for flood mitigation. Our computational results indicate when and how much amount of the sediment should be applied to the river restoration, which can be useful for their decision-making.

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

ABSTRACT 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.