REALISTIC MODELS IN POPULATION ECOLOGY
223
The growth and death submodels for males 9, and 9, and those for
females 9, and 9, need not be identical, except for species in which the
sexes were sufficiently similar biologically. I n any case, each growth
function depends on both density functions, since both sexes deplete the
food supply. For example, 9, and 9, could both be inversely proportional
to ( N , + N , ) , where N , and N , are respectively the total male and
female subpopulations. If the male and female subpopulations influenced
each other's death rates, that dependence should be included in the
death submodels.
The birth submodel is more complicated, and I will discuss only the
case of live births. Let the probability of a male in the age interval
(a,, a, + da,) and mass interval (m,, m, + dm,) encountering a female in
the age interval (a,, a, + da,) and mass interval (m,, m, + dm,) within the
time interval (t, t + d t ) and causing that female to conceive be
Wa,, m,; a,, m,; t ; T,, ~,)da,dm,da&m,dt
Thus
I
5(a,, m2, t ) = { I ; 1; m,, m,; a,, m,; t ; 9 1 9 %)?Jl(~l, m,, t)da,dm,
%(a,, 7 % t ) (50)
is the rate a t which females of age az and mass m, become pregnant at t.
The total rate a t which all females become pregnant is
and the total number of females which become pregnant between t and
t+Atis
L"*' sb"J; .$(a,, m2, wa,dm,dt
I assume temporarily that pregnant and nonpregnant females are
biologically similar so that both have the same growth and death submodels. Thus, knowing ~,(a,, m,, t ) for all t between t - t, and t, where t,
is the gestation period, specifies the percentage of females for all ages
a2 and masses m,, impregnated at t-t,, which survive the pregnancy
period to t. Of course the females age by an amount t, in that period.
Furthermore, this knowledge of 7, relates the mass of the mother mi
at t with her mass m, at t-ts. Thus, .$(a,, m,, t-t,) determines
[(a, + t,, mi, t ) which is the rate at which females of age a, + t, and mass
mi attain term at t.
In general, male and female neonates are produced at different rates
and with different mass distributions. Two expressions are therefore
223
The growth and death submodels for males 9, and 9, and those for
females 9, and 9, need not be identical, except for species in which the
sexes were sufficiently similar biologically. I n any case, each growth
function depends on both density functions, since both sexes deplete the
food supply. For example, 9, and 9, could both be inversely proportional
to ( N , + N , ) , where N , and N , are respectively the total male and
female subpopulations. If the male and female subpopulations influenced
each other's death rates, that dependence should be included in the
death submodels.
The birth submodel is more complicated, and I will discuss only the
case of live births. Let the probability of a male in the age interval
(a,, a, + da,) and mass interval (m,, m, + dm,) encountering a female in
the age interval (a,, a, + da,) and mass interval (m,, m, + dm,) within the
time interval (t, t + d t ) and causing that female to conceive be
Wa,, m,; a,, m,; t ; T,, ~,)da,dm,da&m,dt
Thus
I
5(a,, m2, t ) = { I ; 1; m,, m,; a,, m,; t ; 9 1 9 %)?Jl(~l, m,, t)da,dm,
%(a,, 7 % t ) (50)
is the rate a t which females of age az and mass m, become pregnant at t.
The total rate a t which all females become pregnant is
and the total number of females which become pregnant between t and
t+Atis
L"*' sb"J; .$(a,, m2, wa,dm,dt
I assume temporarily that pregnant and nonpregnant females are
biologically similar so that both have the same growth and death submodels. Thus, knowing ~,(a,, m,, t ) for all t between t - t, and t, where t,
is the gestation period, specifies the percentage of females for all ages
a2 and masses m,, impregnated at t-t,, which survive the pregnancy
period to t. Of course the females age by an amount t, in that period.
Furthermore, this knowledge of 7, relates the mass of the mother mi
at t with her mass m, at t-ts. Thus, .$(a,, m,, t-t,) determines
[(a, + t,, mi, t ) which is the rate at which females of age a, + t, and mass
mi attain term at t.
In general, male and female neonates are produced at different rates
and with different mass distributions. Two expressions are therefore