Biology Reference
In-Depth Information
P
(
t
)
P
2
K
P
1
0
t
FIGURE 1-8.
Behavior of trajectories beginning at different population sizes. Trajectories that begin at population
size values smaller than the carrying capacity K increase toward K, whereas those that begin at
population size values larger than K decrease toward K.
explicitly solve the many differential equations that we shall encounter
to understand how the solutions evolve.
E
XERCISE
1-7
What would happen if there were a time when P(t)
¼
K?
An equilibrium state is one in which the quantity in question remains
constant over time. These will be the values for which the derivative is
zero. In Eq. (1-19), they will be where P
¼
0 and P
¼
K.
VII. ANALYZING EQUILIBRIUM STATES
Suppose that we have an equation of the type:
dP
dt
¼
f
ð
P
Þ:
(1-20)
This shows that the rate of change depends only on the value of the
population and not on when that value is attained. The logistic model
from Eqs. (1-12) and (1-19) has this form. The values of P for which
0 define the equilibrium states, because then
dP
f (P)
In this
section, we shall show how to classify the equilibrium states as stable or
unstable, based on the sign of the function f (P) near each equilibrium
state.
¼
dt
¼
0
:
If f (P)
0, then the derivative is positive and P will increase. If f (P) < 0,
then the derivative is negative and P will decrease. A very helpful tool is
to graph y
>
f (P) versus P. From the graph, we can then easily decide
where P will increase or decrease. Suppose
dP
¼
dt
¼
ð
Þ;
f
P
and the graph of
f (P) is shown in Figure 1-9.
