untitled

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