130 Chapter 8
Box 8.1Model of the
Soay sheep population
on St Kilda.
Threshold effects on mortality can be well described by a sigmoid function:
where irefers to the age group (from 0 for newborns up to 2 for adults), Nis population density of
yearlings and adults, pmaxis maximum survival rate, and αand βare parameters determining the shape
of the sigmoid survival function. Clutton-Brock et al. (1997) estimated the parameters of the Ψfunc-
tion, from several years of data. These values are shown below:
By applying these sigmoid functions, we can mimic the threshold effect (Fig. 8.21).
Similar sigmoid functions can be fitted to age-specific fecundity rates of females:
By applying these density-dependent survival and fecundity rates to specific age classes, we can
estimate changes in abundance over time:
nt
nt
nt
nin in
nn
nn
jt jt
j
jt
i j
tjt
j
tjt
j
0
1
2
0
1
1
1
1
0
1
,
,
,
, ,
,
,
,, ,
,,
,,
+
+
+
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
=
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜⎜
∑ ∑∑
∑
∑
ΩΨ
Ψ
Ψ
⎞⎞
⎠
⎟⎟+
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟
nn,,tj, ∑ t⎟
j
2 Ψ^2
ββ
.
.
.
=
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
24 1
14 1
14 1
αα
.
.
.
=
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
0 00629
0 00589
0 00589
mmax
.
.
.
=
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
0 335
0 643
0 643
Ω(, )
()
iN mmax
N
i
= 1 +ααi ββi
β
.
.
.
=
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
15 3
946
893
α
.
.
.
=
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
0 00562
0 00484
0 00467
pmax
.
.
.
=
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
088
094
096
Ψ(, )
()
iN pmax
N
i
= 1 +αi βi