さあ！諸君！捕鯨問題だ！

Fixing the position of the curve is the essential for
determining such trends. Consider an example in which
there are three years of samples and take the first 5
age-classes. Following de la Mare (1989), a system of linear
simultaneous equations relate the logarithms of
catch-at -age data to a scaling factor, a recruitment factor,
and natural mortality schedule as follows:
g5 + q5 = b1,5
g5 + q4 - M1 = b2,5
g5 + q3 - M1 - M2 = b3,5
g5 + q2 - M1 - M2 - M3 = b4,5
g5 + q5 - M1 - M2 - M3 - M4 = b5,5
g6 + q6 = b1,6
g6 + q5 - M1 = b2,6
g6 + q4 - M1 - M2 = b3,6
g6 + q3 - M1 - M2 - M3 = b4,6
g6 + q2 - Ml - M2 - M3 - M4 = b5,6
g7 + q7 = b1,7
g7 + q6 - M1 = b2,7
g7 + q5 - M1 - M2 = b3,7
g7 + q4 - M1 - M2 - M3 = b4,7
g7 + q3 - M1 - M2 - M3 - M4 = b5,7 (6)

where gt is the logarithm of the proportion of the
population represented by the sample collected in year t
(the scaling factor), qi is the logarithm of the number of
animals born in year i (the recruitment factor), and Ma is
the natural mortality rate of animals aged a.
It is shown in de la Mare (1989) that there is no unique
solution to sets of equations such as these because there are
always two more unknown parameters than independent
equations. Therefore, a unique solution requires two
unknown parameters to be fixed independently. The
catch-at-age data can be used to estimate the
age-dependent pattern of mortality if one age-dependent
mortality value is fixed. This is equivalent to fixing the
average level of mortality by independent means. In
addition, to obtain a unique solution, the recruitment in
one year also needs to be fixed. However, the level can be
arbitrary, in which case a relative trend in recruitment is
obtained.
（つづく）