Assessing current sustainability of use - Conservation and Sustainable Use - page 135

Box 4.1 Biological reference points from the logistic model.

The logistic model described in Section 1.3.1.1 allows us to define several possible
biological reference points for sustainability. Using the symbols defined in
Table 2.1 and Fig. 1.1, the absolute maximum sustainable yield (MSY) is given by:

r max K
4

MSY

Alternatively, MSY can be expressed as a proportion of the current population.
This maximum sustainable proportional yield is given by:

r max
2

c MSY

Managing to this reference point is safer because the absolute catch is allowed to
fall as the population decreases, but it requires current population size to be
known. If harvesting effort is known, but not population size, and it can be
assumed that effort and yield are directly proportional (see Section 2.3.5.1 for
potential problems with this assumption), then effort can be used as an index of
proportional harvest. Given a catchability coefficient, q (the proportion of the
population caught per unit effort), the maximum sustainable effort is given by:

r max
2 q

E MSY

While catch above MSY causes extinction, proportional harvest above c MSY can in
principle be sustained, albeit with lower yield and greater risk to the population.
A population is overexploited in this way if it is below half carrying capacity,
giving the reference point:

K
2

N MSY

Even a proportional catch can drive a population extinct if it exceeds a certain
threshold. These thresholds, the maximum proportional catch and maximum
effort beyond which extinction occurs, are given by:

c max r max

r max
q

E max

Strictly speaking, these equations work only when population production and har-
vest are both continuous —that is, they occur throughout the year at more-or-less
constant rates. In practice, the equations are still a reasonable approximation for
seasonal systems if r max is low (less than about 0.5), but for more productive sea-
sonal species, the reference points should be based on a discrete time model :

e r max 1

MSY K

e r max /2 ) 2

(1