A dynamic reaction model of a two-species fishery with taxation as a control instrument: a capital theoretic analysis

Abstract

In a fully dynamic model of an open-access fishery, the level of fishing effort expands or contracts according as the perceived rent (i.e., the net economic revenue to the fishermen) is positive or negative. A model reflecting this dynamic interaction between the perceived rent and the effort in a fishery, is called a dynamic reaction model. The present paper deals with a dynamic reaction model of a fishery consisting of two competing species, each of which obeys the logistic law of growth. A regulatory agency controls exploitation of the fishery by imposing a tax per unit biomass of the landed fish. It is also assumed that the gross rate of investment of capital in the fishery is proportional to the perceived rent. With this capital theoretic approach, the dynamical system consisting of the growth equations of the two-species and also of the fishing effort is formulated. The existence of its steady states and their stability are then studied using eigen value analysis. The optimal harvest policy is discussed next with the help of Pontryagin’s maximum principle. The results are then illustrated using numerical examples.