A comparative study of models for predation and parasitism

(1)
assumption of a PoxssoN distribution,
~E [ {z (r, + Jr)'Aq s~ j!] e - ~'(' '+ dr)2X ,
j~l
(2) probability that all of these j points fall within a locus (ri, ri+dr), i.e.
[ {z (r~ +dr) 2_ r, ri~}/~r (ri +dr) 2],,.
With a similar calculation as for 7] in Appendix 1, we have
Thus,
Similarly,
probability that at least one point of A's falls within radius ri+dr; this is, under the
l. - wr~.X
~m ta~=2~rXe dr.
d r--*O
co PR - *rr2X
lira i=
J r--~0 '=
a~lFi=|o 2r, rXe dr
=l_e-~rR~X
oo PR ~ -wrO-X
lira ]E rq2~=l 2.'rr Xe dr
zIr~O i=1 dO
and integrating by parts,
(iii).
fR2~,r,e-,rrO~X, ar=Jo ['R-wr~
e ' dr-Re -'R~'x (iv).
Let O(t) be the normal probability function, i.e.
1 t - t 2/2
$(t) =. 7,-- f e dt,
V~Tr JO
and setting t equal to l/2zXr,
9 (1/~R)/V'X.
the integral in the right-hand side of eq. (iv) will be written as
Thus, from eqs. (ii), (iii), and (iv) and using symbol ~, we have
L.=[, )
89
Appendix 3. The proof of eq. (4i. 6)
Let Y be the density of particles distributed at random over an area A, each one of the
particles having a circle of area 6 around it. Then the density of the particles, Y", within the
total area covered with these circles, i.e. A', is
Y"-=-AY/A'
Suppose that a large number of points, i.e. U, with circles of, also, area ~ are placed to
cover the whole area of A independently of the distribution of the particles. Then, the number
U' of these points that include at least one of the particles will he, on the assumption of the
Poissos distribution,
U'= U(1-e -~r)
(vi),
and the proportion U/U ~ must be equal to the proportion A/A'. Thus from eqs. (v and vi), we
find
Y"=Y/(1-e-~Y).
Thus a circle of area c~ around each particle contains on the average 6Y~P number of particles.
However, since ~Y~ is, as defined in w 4i, the mean number of particles in each circle less
the one at the centre, we have
6Y'=6Y"--I
=~Y/(1-e -~r) -1.