Growth of fish biomass: An important model for commercial fisheries is that of Beverton and Holt.31 It begins with the study of a single cohort of fish— that is, all the fish in the study are born at the same time. For a cohort of the North Sea plaice (a type of flatfish), the number N = N (t) of fish in the population is given by
Here w is measured in pounds and t in years. The variable t measures the so-called recruitment age, which we refer to simply as the age. The biomass B = B(t) of the fish cohort is defined to be the total weight of the cohort, so it is obtained by multiplying the population size by the weight of a fish.
a. If a plaice weighs 3 pounds, how old is it?
b. Use the formulas for N and w given above to find a formula for B = B(t), and then make a graph of B against t. (Include ages through 20 years.)
c. At what age is the biomass the largest?
d. In practice, fish below a certain size can’t be caught, so the biomass function becomes relevant only at a certain age.
i. Suppose we want to harvest the plaice population at the largest biomass possible, but a plaice has to weigh 3 pounds before we can catch it. At what age should we harvest?
ii. Work part i under the assumption that we can catch plaice weighing at least 2 pounds.