Environmental Engineering Reference
In-Depth Information
The risk in assuming logistic growth is that the factors regulating population
growth actually act in significantly
non-linear
ways, potentially leading to very dif-
ferent responses to harvesting. For example, it is thought that some species tend to
show little regulation at low population size, but strong regulation around carrying
capacity, while others tend to show the strongest regulation when their popula-
tions are small (Fowler 1981; Sibly
et al
. 2005), which may be due to differential
susceptibility to
competition
among different age or size classes (Owen-Smith
2006). In order to detect and quantify these complications, we require demo-
graphic rate estimates (either population growth rate or its constituent parameters,
survival and productivity), replicated in time or space at a wide range of different
abundances, allowing us to define the demographic response to abundance
(Box 2.15). Furthermore, it is highly desirable to separate demographic rates by
Box 2.15
Some useful density dependence functions.
In principle, the response of demographic rates to density might take any form.
In practice, a few relatively flexible functions have become commonly used, pre-
ferred for their wide applicability. Where population abundance,
N
, and growth
rate,
r
t
ln(
N
t
1
/
N
t
), measures are available, a theta-logistic model can be fitted
to the data:
N
t
K
r
t
r
max
1
This adds an additional parameter,
, to the logistic model, allowing non-linear
growth responses (Fowler 1981). When
1, regulation intensifies as the popu-
lation declines below
K
, when
1, regulation is most intense at around
K
, while
1, a simple linear model of
r
against
N
can be used, the intercept giving an estimate of
r
max
, and the intercept
divided by the slope estimating
K
. Otherwise, a non-linear fitting procedure is
required to estimate the parameters.
Where data are available on either productivity or survival rates (jointly denoted
1 gives standard logistic growth. When
), functions commonly used to define density responses are the Ricker function:
max
e
bN
or the Beverton-Holt function:
max
1
bN
In both cases
max
defines the maximum survival or productivity rate at low popu-
lation size, and
b
defines the rate of decline with increasing density. Where these
functions are used to relate recruitment to the abundance of breeders, the Ricker
function is more appropriate if the outcome is primarily affected by the density of
