K, the PBR formula avoids this need by substituting an estimate of density for an
estimate of K. Essentially, this sets a target harvest rate, rather than a total harvest
quota, which, as discussed above, is a more conservative and stable strategy.
Thus, for application of the PBR formula, an estimate of Kis not required.
The logistic growth model assumes a quadratic relationship between Nt 1 and
Nt, and this results in a linear relationship between the equilibrium population
size and the harvest rate (Figure 13.1(a)). A more general form for the logistic
growth model includes another parameter, , to govern the shape of the density-
dependent relationship:
. (13.13)
When the shape parameter is greater than 1, the effects of density-dependence do
not occur until the density is high, such as might be the case when the density-
dependence is due to a limited number of territories (Gilpin et al. 1976). When
the shape parameter is less than 1, the effects of density-dependence are apparent
even at very low densities; this might be the case when resources are hetero-
geneous and the first individuals consume the highest quality resources (Gilpin
et al. 1976). The shape parameter affects the density at which yield is maximized—
when1,NK/2; when 1,NK/2 (Taylor and DeMaster 1993).
Thus, knowledge about can increase confidence in the sustainability of harvest.
The PBR formula assumes that 1. Estimation of requires measuring the
growth rate of a population for a range of densities, ideally in the absence of
harvest, and fitting those data to equation (13.13).
The models discussed above incorporate only negative density-dependence—the
reduction in survival and/or reproductive rates, hence growth rates, with increases
in density. But positive density-dependence at low density, known as an Allee effect
(e.g. Dennis 1989), can also occur. For example, Allee effects can occur when poten-
tial mates are too sparsely distributed to be able to locate each other. Allee effects can
create instability in harvested systems—if the population drops below a critical
density, an extinction vortex can result. That critical density, if it exists, is below
the equilibrium population size that produces maximum harvest. Since the PBR
method seeks to keep the population size above that optimal equilibrium size, it
guards against potential Allee effects. Nevertheless, if there are aspects of the behav-
ior or life-history of the organism that suggest a possible Allee effect, harvest strate-
gies should guard against inadvertently lowering the density below the critical level.
Equations (13.1) and (13.13) are phenomenological, in that they do not
ascribe a mechanism for density-dependence, but instead posit an empirical
relationship that captures the phenomenon. A more mechanistic treatment
of density-dependence would look at how individual life-history parameters
Nt 1 NtrmaxNt 1
Nt
K
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