Could a mathematics student have prevented the collapse of the Atlanto-Scandian herring?

Abstract

Herring in the ocean between Iceland and Norway was one of the largest fish stocks in the world until the fishery crashed in the late 1960s. The catch in 1971 was only 20 thousand metric tons in contrast with the record of 2 million tons in 1966 and the spawning stock declined from 10 million tons to 10 thousand tons in 20 years. After 25 years of almost no fishing the stock finally recovered. With hindsight the cause of this dramatic change was a combination of biological, technological, ecological and economic factors, but the question raised here is whether mathematical foresight might have hindered the mismanagement of this vital resource. In the mid 1960s the International Council for the Exploration of the Sea (ICES) had assessed the size of the stock from the early 1950s by gathering information from tagging experiments, acoustic measurements and underwater photography. After specifying an exponential-trigonometric time series model: S = AeBt[1 + Csin(Dt)], where S is the stock size and t time, the four parameters are either derived analytically or estimated statistically by the Gaussian least squares method using data from 1953 to 1963. Then the development of the stock is predicted until the end of the decade and compared to what actually happened with surprising similarities. More accurate stock assessment by virtual population analysis (VPA), not available until several years after the collapse, showed the same trend with some modifications. During the exercise we are confronted with some matters of opinion in the teaching of mathematics in the age of electronic calculators and computers: (i) What are the minimum analytical skills required at high school or college level, e.g. in differentiating functions like the stock model above or solving trigonometric equations manually such as asin(2x) + bcos(2x) = c? (ii) How should we divide available time between traditional problem solving, model building, computing and critical interpretation of numerical results? (iii) Should we propose realistic or idealistic problems?