A Theoretical Consideration on the Management of a Stock-Fishery System by Catch Quota and on Its Dynamical Properties

Abstract

A stock-fishey system model is set up and its dynamical properties are studied. The stock grows according to a logistic type model with a minimum limit on the stock size which can be sustained. The fishery is regulated through the catch quota, which is increased or decreased depending upon the present level of the stock size relative to the target level and its rate of change. A time lag between the observation and the enforcement of the regulation is incorporatd into the model. Simultaneous finite difference equations are derived and a stability analysis around the equilibrium point nd some simulations are conducted. The following is suggested: (1) the stability is high when the target level is larger than the MSY level, (2) incorporating a time lag until the enforcement of regulations occurs reduces the stability considerably, (3) for regulating the quota, more weight should be placed on the rate of change than on the present level of the stock size, and (4) a locally unstable equilibrium point may have a limit cycle around it. The only essential information on the stock needed to run this management system is the relative stock size, and hence if we have a reliable index of the rlative stock size, we would be able to manage the stock and to approach the target point.