REALISTIC MODELS IN POPULATION ECOLOQY
225
some specific time (say to), r],(a,, m,, u, to) and r],(a,, m,, 0, t ) for t 2 to is
all that is required to solve Eqn (52) for all t 2 to. At every t > t o ,
r],(a,, m,, t,, t ) determines the age-mass distribution function of females
who are at term; here as above t, is the gestation time. These females
leave population 7, by the process of giving birth and re-enter the
population of nonpregnant females r], at reduced mass. The reduction in
mass is the mass lost in parturition. The birth rates are given by
and
r ] Z ( O 7 m27 t , = J
: J : B2('3; m2, m3; t)r]3(a3, m37 t,> t ) d a , d n t 3 (54b)
in analogy with Eqns (51a) and (51b).
The process of removing females from population 7, when they
become pregnant and returning them to r], after parturition must be
taken into consideration in the equation for 7,. Since the rate at which
females become pregnant and leave population r], is given by 5, this
term is equivalent to a partial death rate for population ys. Similarly,
the term r],(u,, m3-m,, t,, t ) is the rate at which females return to r ] ,
after parturition. Here m3 has been reduced by m,, the mass lost in
childbirth, and m, can be a function of a, and m,, the mother's age and
mass. The equation for r], is thus
where a, = a, and m, = m3-mo in 77,. The set of equations, viz. (52) to
(55), together with
%+-+- arl1 a (3,r],) = - 9 , r ] 1
at
aa, am,
and the values of q1, r], and r], at an initial time constitute a sufficient
set of equations for this bisexual population. It should be noted that the
knowledge of r],(a,, m,, u, t ) at t-initial obviates the need to know r], and
7, during a prior time interval of length to, since 7, contains all information about conception in that prior period.
The bisexual reproduction model described above does not take into
account marriage^'^. Fredrickson (1971) uses age-specific models and
defines three density functions, for males, females and couples. At the
expense of substantial complication, his density function for couplea
225
some specific time (say to), r],(a,, m,, u, to) and r],(a,, m,, 0, t ) for t 2 to is
all that is required to solve Eqn (52) for all t 2 to. At every t > t o ,
r],(a,, m,, t,, t ) determines the age-mass distribution function of females
who are at term; here as above t, is the gestation time. These females
leave population 7, by the process of giving birth and re-enter the
population of nonpregnant females r], at reduced mass. The reduction in
mass is the mass lost in parturition. The birth rates are given by
and
r ] Z ( O 7 m27 t , = J
: J : B2('3; m2, m3; t)r]3(a3, m37 t,> t ) d a , d n t 3 (54b)
in analogy with Eqns (51a) and (51b).
The process of removing females from population 7, when they
become pregnant and returning them to r], after parturition must be
taken into consideration in the equation for 7,. Since the rate at which
females become pregnant and leave population r], is given by 5, this
term is equivalent to a partial death rate for population ys. Similarly,
the term r],(u,, m3-m,, t,, t ) is the rate at which females return to r ] ,
after parturition. Here m3 has been reduced by m,, the mass lost in
childbirth, and m, can be a function of a, and m,, the mother's age and
mass. The equation for r], is thus
where a, = a, and m, = m3-mo in 77,. The set of equations, viz. (52) to
(55), together with
%+-+- arl1 a (3,r],) = - 9 , r ] 1
at
aa, am,
and the values of q1, r], and r], at an initial time constitute a sufficient
set of equations for this bisexual population. It should be noted that the
knowledge of r],(a,, m,, u, t ) at t-initial obviates the need to know r], and
7, during a prior time interval of length to, since 7, contains all information about conception in that prior period.
The bisexual reproduction model described above does not take into
account marriage^'^. Fredrickson (1971) uses age-specific models and
defines three density functions, for males, females and couples. At the
expense of substantial complication, his density function for couplea