Environmental Engineering Reference
In-Depth Information
provide a reasonable approximation for this, although obtaining adequate data
from such depleted populations is likely to be a challenge. Captive individuals
might also be used, although this approach should be used with great caution.
Demography in captivity is unlikely to be equivalent to wild conditions, particu-
larly with regard to survivorship in the absence of natural mortality risks such as
predation.
The third option for estimating
r
max
, along with population carrying capacity
k
,
is to fit a density-dependent
population model
to a time-series of catch and abun-
dance data, extending the catch-effort method of abundance estimation intro-
duced in Section 2.3.5.1 (see Section 4.3.3 for details of how to do this). Estimates
of
r
max
made using this approach are most robust if at least some estimates of true
abundance are available along with catch data, but indirect indices of population
size can also be used. The method is most commonly applied to catch and effort
data, using catch per unit effort as the abundance index. The minimal data require-
ments are thus a time-series of total catch along with sub-samples of catch and the
effort required to produce it. It is preferable to monitor total effort as well as catch,
though, if it can be done without bias, since this minimises sampling error.
Box 2.8
Estimating intrinsic rate of increase from maximal survival and
productivity rates.
From age-specific estimates of maximal survival rate,
S
j
, calculate survivorship
(the probability that a newborn will survive to age
i
):
i
l
i
1
S
j
j
Given the maximal age-specific productivity rate,
P
i
(the number of young
females produced per year by a female aged
i
), and ages at first and last reproduc-
tion (respectively
a
and
b
),
r
max
can be estimated from:
b
l
i
P
i
e
r
max
i
1
i
a
This is a generalisation of Cole's (1954) equation, which has been widely used to
estimate
r
max
. However, unlike Cole's equation, it includes survival through the
term
l
i
, which makes it more appropriate for our purposes. There is no explicit
solution for
r
max
in this equation, but it can be solved in a spreadsheet by finding
the value of
r
max
that brings the sum to 1 (the solver add-in in Excel provides a
useful tool for achieving this). Slade
et al
. (1998) provides a range of ways in
which simplifying assumptions can make this task easier. The most basic case is
when reproduction starts in the first year and continues indefinitely, and survival
and productivity rates are constant with age. In this case,
r
max
is given by:
r
max
ln(
S
(1
P
))
