266
WILLIAM STREIFER
To illustrate the procedure in evaluating the birth rate,
we consider the particular submodel Eqn (30),
q(0, m, t ) = JOm JOm b(a’; m; m’, t)q(a‘, m‘, t)da‘dm’
B(t) = N(t) Jo Jo
JO*
( C W
By substituting Eqn (Cl2) in ( C l l ) and integrating over m, we obtain
p ( d , m’, t)q(u’, m’, t)da’dm’
((313)
1
m
m
where
,B(a‘, m’, t ) =
b(a’; m, m’; t)dm
((314)
Equation (C13) is evaluated by performing a Taylor expansion of p
about 6, rii, and retraining only terms through the second derivative,
/3(af, m’, t ) = p(G, 6, t)+/3a(B’-G)+/3m(m‘-rii)
++(paB,,(u’-G)2+ 2pam(u’-G)(m‘-rii) + ~ m m ( m ‘ - r i i ) 2 } ((215)
Here too the subscripts refer to partial derivatives evaluated at G, rii.
Upon substituting (C15) in (C13) and evaluating the integrals, we
obtain
= B” + +(.Pas,, + 2vpam + PBmm)
(C16)
where 8‘0) = /3(G, 6, t). More complicated birth submodels than (C12)
result in more complex expressions than (Cl6).