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

Abstract

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.