Biology Reference
In-Depth Information
by Eq. (1-18). Notice that the inherent per capita growth rate
a corresponds to the special case of Eq. (1-15) when all of the available
resource is unbound. The expression for the carrying capacity K provides
insight to the dependence of this empirical parameter upon available
resources and rate of consumption. As should be expected, K grows with
F and declines as the consumption rate c increases.
VI. LONG-TERM BEHAVIOR AND EQUILIBRIUM STATES
In the continuous models from Eqs. (1-2) and (1-12), we found a solution
that gives P(t). Differential equations, however, are often impossible to
solve explicitly. In spite of this, we can still glean essential information
about the long-term behavior of the model from these equations. We
shall now examine techniques for this type of analysis with the logistic
equation.
We begin by recalling that there is dependence between the increase/
decrease behavior of a function and the sign of its derivative. Namely, if
the derivative is positive over a certain time interval, the function is
increasing, while a negative derivative indicates the function is
decreasing. When the derivative is zero, the function exhibits no change.
In the logistic model [Eq. (1-12)], the governing differential equation can
also be written as:
dP
dt
¼
a
K
:
a
0
ð
where a
0
¼
K
P
Þ
P
;
(1-19)
What does this differential equation tell us? The derivative is zero at two
values of P:whenP
K. When we graph P versus
t (population vs. time), these values divide the graph into two regions—
values of P larger than the carrying capacity K and values of P smaller
than K (see Figure 1-8). Suppose we begin a new culture with a very
small quantity of yeast. Because the population is small, P(t) <K, then
dP
dt
is positive, and P(t) will increase (see the curve labeled P
1
). This does not
give the complete information that the solution of the logistic curve
gave, but it gives valuable information for very little effort. Similarly,
if a huge amount of yeast was introduced (greater than the carrying
capacity, so P(t)
¼
0 and when P
¼
K), then the derivative is negative and the population
will diminish (see the curve labeled P
2
).
1
There is an underlying
lesson here that is very important: Namely, that we do not have to
>
1. Notice that none of the arguments determining the long-term behavior of P(t)
here depends on the actual value of the parameter a
>
0. We will need this
observation in Section VIII, where we discuss discrete analogues of the logistic
model (1-12). In contrast with the continuous logistic model, those may exhibit
radically different behavior, depending upon the value of a
>
0.
