264
WILLIAM STREIFER
boundary condition. Upon equating (Be) and (B7) and eliminating A V ,
we obtain a partial differential equation for 7,
The variables x( in Eqn (BS) can be the characteristic c discussed in
section IVD, the time from conception in section IVE, or location as in
section IVF. In each case dxi/dt is the rate a t which xi changes for
individuals characterized by xl, x2, . . . , xm at t . If xi never changes with
t, dxt/dt always equals zero and the ith term does not appear in Eqn (BS).
Note too that if some of the xg are location variables, as discussed in
section IVF, then migration across the physical boundaries will not be
included in W and 9, but will provide boundary conditions. If location
variables were not included among the xi, 9 l and 9 would represent
immigration and emigration respectively. Finally, it is possible to
incorporate boundary conditions in W and 9, but this requires the use of
Dirac &functions and will not be pursued further here.
Appendix C
THE CRITICAL VARIABLE EQUATIONS
To derive Eqn (67a) for dN/dt, we differentiate Eqn (62a)
N ( t ) = 1; s. " .I(a, m, t)dadm
with respect to t to obtain
Upon substituting Eqn (63) for aq/at, the result is
- = I1+12+13
dN
at
where
and
WILLIAM STREIFER
boundary condition. Upon equating (Be) and (B7) and eliminating A V ,
we obtain a partial differential equation for 7,
The variables x( in Eqn (BS) can be the characteristic c discussed in
section IVD, the time from conception in section IVE, or location as in
section IVF. In each case dxi/dt is the rate a t which xi changes for
individuals characterized by xl, x2, . . . , xm at t . If xi never changes with
t, dxt/dt always equals zero and the ith term does not appear in Eqn (BS).
Note too that if some of the xg are location variables, as discussed in
section IVF, then migration across the physical boundaries will not be
included in W and 9, but will provide boundary conditions. If location
variables were not included among the xi, 9 l and 9 would represent
immigration and emigration respectively. Finally, it is possible to
incorporate boundary conditions in W and 9, but this requires the use of
Dirac &functions and will not be pursued further here.
Appendix C
THE CRITICAL VARIABLE EQUATIONS
To derive Eqn (67a) for dN/dt, we differentiate Eqn (62a)
N ( t ) = 1; s. " .I(a, m, t)dadm
with respect to t to obtain
Upon substituting Eqn (63) for aq/at, the result is
- = I1+12+13
dN
at
where
and