262
WILLIAM STREIFER
The integral over a' is
&a' = I
and the integral over m' is
m' e-m'l,mdm' = 1.1e-Pl
1.0
(A8)
1" .Im, m i
-
Thus N ( t ) x N ,
obtain the total birth rate, 6. Eqn (27),
To determine K we integrate ~ ( 0 ,
m, t ) as given by Eqn (34) over m to
The results (see Peirce, 1929, formula 402) are
so that
K = -In a more realistic example K would follow directly from the submodels. It could be determined from Eqn ( A l l ) and the knowledge of
A, T, and dN/dt
I births '
Appendix B
EXTENSIONS O F T H E A G E - S I Z E SPECIFIC MODEL
In sections IVD, E and F extensions of the age-size specific model
were presented without derivation. A rigorous derivation of the equations is quite lengthy and complicated, just as is that of Eqn (20)
(Sinko and Streifer, 1967). Instead, a less rigorous more intuitive
derivation is presented.
x2, ..., xm, t ) of m variables xl, x2,
..., xm, and t. The symbols 21, i = 1, ..., m could represent age, mass,
location etc. Now visualize a space of m dimensions with orthogonal
axis xl, x2, ..., xm. A small m-dimensional "cube" in that space with
edges Axs has a volume A V given by the product
Consider a density function
A V = Ax1 AX, ... Axm
(Bl)