REALISTIC MODELS IN POPULATION ECOLOGY
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(6) The food supply at t’ is inversely proportional to the total populaThe birth function W determined by assumptions (3) to ( 6 ) has the
tion a t t’, N(t’).
form
where the age-dependent term expresses the fertility variation of
assumption (3), the 1/N term represents the inverse proportionality of
the food supply to N stated in assumption (6), and K is a constant
selected so that the total birth rate is realistic. Also S(m’- 10m) is the
Dirac 6-function1 which sets the mass of each neonate equal to exactly
10% the mass of its parent,
m = 0-lm’
Such functions will always appear in &f to relate neonate mass to parent
age and mass.
To evaluate the integral (30) using Eqn (31) for $9, one requires a
knowledge of ~ ( a ’ ,
m‘, t ) as well as N(t’) for t - ~ < t ’ < t . For this
example, we assume a particularly simple density function
m’< O.lm,
so that no individuals have masses under 0-lm,. The age and mass
distributions are illustrated in Fig. 5a and b, and the total population
at t is very nearly equal N,. We also assume N(t’) = N , for t - T < t’ < t.
When these expressions are substituted in Eqns (29) and (30), we obtain
da’dm’dt ( 3 3 )
where assumptions (1) and (2) have been employed in modifying the
integration limits. The results of evaluating the integral
rn < O.lm,
1 The 6-function and other mathematical details relating to
discussed in Appendix A.
are
(34)
this section are