REALISTIC MODELS IN POPULATION ECOLOQY
243
those with crowding of eight animalslml, i.e. d&). We further assume
that animals of mass m = s(a)/2 experience a death rate equal to those
with crowding of 32 animalslml, d3,/(a), since these animals in Frank’s
experiment appeared to be approximately half the volume of corresponding animals a t eightlml. For very small animals, m 5 s(a)/4, we
set the death rate at 2d3,, since no data are available. In Sinko and
Streifer (1969), death rates for intermediate masses were obtained by
linear interpolation. Smoother curves recently obtained by Flint ( 1972)
utilize quadratic interpolation with d,, d3,, and data for d, (four
animals/ml). His results are illustrated in Fig. 9, which shows two views
of the surface 9 ( a , m ) vs a, m.
3. Solution
Routine numerical methods described by Sinko (1969) were employed
to solve the equations. The initial age distribution was taken from
Frank’s survivorship data for 32 animals/ml. The initial mass was taken
as a triangular distribution between Richman’s s(a) and sl(a), which are,
respectively, the masses of individuals of age a fed with 25 000 cells/ml/
day and 100000 cells/ml/day. Furthermore, we assumed that each
animal produces two neonates during the first four days. In all cases,
after the first cycle, the initial conditions had virtually no effect on the
results.
Early in the course of the numerical computations we found that all
animals of a particular age a t a particular time had the same mass. This
does not imply that animals of a particular age a, at t, have the same
mass as animals of that age a, at t,. Nevertheless, a substantial simplification was possible in that Eqn (82) was replaced by the system
-
= -D(a, t)n(a, t )
an(a, t ) + an@, t )
at
aa
-
and
where
n(a, t ) =
~ ( a ,
m, t)dm
r
is Von Foerster’s age-specific density function and
The solution of Eqn (91b) determines m(a, t ) .
243
those with crowding of eight animalslml, i.e. d&). We further assume
that animals of mass m = s(a)/2 experience a death rate equal to those
with crowding of 32 animalslml, d3,/(a), since these animals in Frank’s
experiment appeared to be approximately half the volume of corresponding animals a t eightlml. For very small animals, m 5 s(a)/4, we
set the death rate at 2d3,, since no data are available. In Sinko and
Streifer (1969), death rates for intermediate masses were obtained by
linear interpolation. Smoother curves recently obtained by Flint ( 1972)
utilize quadratic interpolation with d,, d3,, and data for d, (four
animals/ml). His results are illustrated in Fig. 9, which shows two views
of the surface 9 ( a , m ) vs a, m.
3. Solution
Routine numerical methods described by Sinko (1969) were employed
to solve the equations. The initial age distribution was taken from
Frank’s survivorship data for 32 animals/ml. The initial mass was taken
as a triangular distribution between Richman’s s(a) and sl(a), which are,
respectively, the masses of individuals of age a fed with 25 000 cells/ml/
day and 100000 cells/ml/day. Furthermore, we assumed that each
animal produces two neonates during the first four days. In all cases,
after the first cycle, the initial conditions had virtually no effect on the
results.
Early in the course of the numerical computations we found that all
animals of a particular age a t a particular time had the same mass. This
does not imply that animals of a particular age a, at t, have the same
mass as animals of that age a, at t,. Nevertheless, a substantial simplification was possible in that Eqn (82) was replaced by the system
-
= -D(a, t)n(a, t )
an(a, t ) + an@, t )
at
aa
-
and
where
n(a, t ) =
~ ( a ,
m, t)dm
r
is Von Foerster’s age-specific density function and
The solution of Eqn (91b) determines m(a, t ) .