REALISTIC MODELS I N POPULATION ECOLOGY
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Appendix A
CALCULATIONS F O R T H E BIRTH E X A M P L E
O F
SECTION I V B 3
The Dirac &function, 6 ( x ) , utilized in Eqn ( 3 1 ) , is defined by
0, x # O
I co, x = o
6 ( x ) =
such that with x 2 > x1
1, x , < 0 < x 2
S:'S(x)dx = { 0, 0 < x l , or x2 < 0
One can show that iff ( x ) is any function defined at x = 0,
This last result is employed in the evaluation of Eqn (33).
The function 6(m'- 10m) is zero for m' # 10m (or equivalently
m # 0.lm') and is infinite at m' = IOm or m = 0.lm'. The integral
over m' in Eqn ( 3 3 ) ,
is in the form (A3). Thus, I m is equal to the function multiplying
6(m'- 10m) in the integral evaluated at m' = 10m for m'> m, and is
zero for m' < m,,
m < O*lmm
which is the mass dependence in Eqn (34).
Peirce, 1929, formula 402) to obtain
The constant A in Eqn ( 3 4 ) is calculated by integrating over a' (see
The total population at t is obtained by integrating ~ ( a ' ,
m', t ) given by
Eqn ( 3 2 ) over a' and m', i.e.