Environmental Engineering Reference
In-Depth Information
increasing density. For example, under crowded conditions, food resources might
be depleted, predation might intensify, or disease might take a bigger toll, any of
which could cause increased mortality rates or reduced reproduction. Regardless
of the mechanism of density dependence, the net result is a declining per capita
population growth rate with increasing density.
The implications of density dependence for exploitation can be understood
by imagining what happens when a population first becomes subject to a regular
harvest. At its un-harvested equilibrium, births balance deaths and the net growth
of the population is zero. The first harvest reduces the population below its natural
equilibrium, internal competition eases and net growth becomes positive, resulting
in a partial recovery by the time the next round of harvest starts. As the population
falls, successive harvests become smaller while net growth increases. Finally, a new
equilibrium is reached at which the harvest is exactly balanced by growth, and we
have, in theory, a biologically sustainable harvest. Of course, a sustainable equilib-
rium is only achievable if the offtake does not exceed a certain limit, which we can
define as a reference point.
1.3.1.1 The logistic model
The logistic is the simplest model of density dependent population growth. Being
easy to analyse and understand, it has been used as the basis of much theory of
sustainable use. This theory forms important background for understanding how
models can be used to define sustainability benchmarks, and we therefore provide
a brief introduction here. For more complete coverage of harvesting theory, see
Clark (1990), Getz and Haight (1989) and Milner-Gulland and Mace (1998).
Despite its simplicity the logistic model can also be used to model specific systems,
and the theory can therefore be translated directly into practical applications in
some cases.
The logistic is characterised by a linear decline in per capita growth rate with
increasing population density (Figure 1.1(a)). At a very low population size this
rate is maximal, while at the other end of the scale equilibrium population size
(or carrying capacity) is defined by the point at which the growth rate falls to zero.
Carrying capacity is commonly denoted
K
, while the maximum per capita growth
rate is commonly denoted
r
max
, and these two parameters alone define logistic
growth. Net growth in the population is the product of population size and per
capita growth, and has a domed relationship with population size (Figure 1.1(b)).
At small population size, although per capita growth is high, the small size of the
population leads to little overall growth. Conversely, at large population size, low
per capita growth leads to low net growth, despite the large population. Maximum
net growth occurs at intermediate population size. The maximum sustainable
yield (MSY) is equal to the net growth at this maximum. Assuming that harvest is
directly proportional to harvester effort, we can also express equilibrium growth,
and hence yield, as a function of a consistently applied effort. In this case equilibrium
population size is linearly and inversely related to effort, and the yield-effort curve
is therefore also domed (Figure 1.1(c)).
