Biology Reference
In-Depth Information
therefore of critical importance (Smith and Link [2005]). We would like
to determine the maximum level of harvesting that is sustainable over a
long period of time without driving a population to extinction. This is
known as the maximum sustainable yield (MSY) for the population. Our
next example illustrates how equilibrium states and long-term dynamic
behavior of a system are affected by harvesting.
4
When a population is left undisturbed, it maintains near equilibrium at a
level close to the carrying capacity K of the environment. The net per
capita growth rate is nearly zero, which means that the per capita birth
and death rates are nearly equal. Harvesting increases the mortality rate,
which, in turn, decreases the net per capita growth rate. Thus, excessive
harvesting can cause the mortality rate to exceed the maximum birth rate
and lead to extinction. Moderate harvesting, however, will only lower
the net per capita growth rate, causing the system to settle around a new
equilibrium level lower than K.
To illustrate this concept mathematically, assume that a population
grows according to the logistic model
dP
P and that
P
K
dt
¼
a 1
harvesting yield per time unit is proportional to the size of the
population. The harvesting will then decrease the rate of change for the
population by a factor of bP, where b
0 represents the harvesting effort.
The rate of change of the population size accounting for the harvesting
will then be
dP
>
P
P
K
dt
¼
a 1
bP
:
The new nonzero equilibrium state
, which corresponds to harvesting yield
b
a
for this model is P
¼
K 1
:
b
a
Y
ð
b
Þ¼
bP
¼
bK 1
This equilibrium state will be non-negative if
b/a<1; that is, if b<a. Therefore, if the harvesting effort b is less than the
inherent per capita growth rate a, the harvesting effort is sustainable.
Conversely, if b
>
a, the population will die out. The yield
achieves its maximum at b
a
2
(the reader
should verify this), which shows that the maximum sustainable yield in
this case is
b
a
Y
ð
b
Þ¼
bP
¼
bK 1
¼
2
Y
a
a
2
K 1
a
2a
aK
4
MSY
¼
Y
max
¼
¼
¼
:
In this example, we made the assumption that yield is proportional to
population size. This assumption is certainly justified when fishing or
hunting is involved. As our next exercise shows, in more controlled
environments, the harvesting rate may be independent from the
population size. In such cases, a model may have more than one nonzero
4. An expanded analysis of these models can be found in Hoppensteadt and
Peskin (2002).
