Biology Reference
In-Depth Information
P
n
:
P
n
K
P
n
þ
1
P
n
¼
r
ð
P
n
Þ
P
n
¼
a 1
(1-25)
The model from Eq. (1-25) is sometimes called the Verhulst model, after
the Belgian mathematician Pierre Verhulst (1804-1849), who first studied
it in 1846. The model given by Eq. (1-25) may also be obtained by
modifying the unlimited growth model from Eq. (1-1) to allow net per
capita growth rate to vary with population size. In Eq. (1-25), as in
Eq. (1-24), K is the carrying capacity for the population; a
0 is the
inherent per capita growth rate; and the arguments for choosing
>
will be the same as for the continuous logistic model
P
n
K
r
ð
P
n
Þ¼
a 1
developed in Section IV.
Although the Verhulst model has the same equilibrium states as the
continuous logistic model, it can exhibit radically different long-term
behavior. Recall that an equilibrium state P is one in which the quantity
in question remains constant over time. For discrete models such as
(1-25), these are the values at which the system exhibits no change
(i.e., P
n
¼
P, for all n
¼
0, 1, 2,
). Equivalently, these are the values for
...
which P
n
¼
P
n
1
for all values of n
¼
1, 2, 3,
.
...
E
XERCISE
1-11
Show that the equilibrium states for Verhulst model [Eq. (1-25)] are
P
¼
0 and P
¼
K.
In Section VI, we proved that for any value of a
0 and any nonzero
initial population size P(0), the logistic model (1-24) exhibits convergence
for t
>
K. For P(0) <K,the population
size P(t) is continuously increasing to K when t
!1
to its equilibrium state P
¼
!1
while if P(0)
>
K,
the population size P(t) is continuously decreasing to K when t
!1
(Figure 1-8). The Verhulst model offers cases of considerably more
complex long-term behavior—the system could converge to an
equilibrium state through oscillations, exhibit lack of convergence
because of periodic oscillatory behavior, or be driven to chaos.
P
n
K
so that x
n
is the fraction of the maximum
population the environment can sustain. With this notation, the Verhulst
model takes the equivalent form:
x
n
þ
1
To demonstrate this, let x
n
¼
x
n
¼
a
ð
1
x
n
Þ
x
n
;
(1-26)
and the carrying capacity of the model in Eq. (1-26) is equal to 1. Equation
(1-26) represents the nondimensional form of the Verhulst model from
Eq. (1-25). This is due to the fact that P
n
and K are measured in the same
units, so the quantity x
n
¼
P
n
K
is nondimensional. This representation has
