Lecture Notes in Differential Equations - Bruce E. Shapiro

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Figure 7.2: Solutions of the logistic growth model, equation (7.23). All
nonzero initial populations tend towards the carrying capacity K as t → ∞.
y
K
1/r
t
Thresholding Model
If we switch the sign of equation (7.8) it becomes
dy
(1
dt = −ry − y )
T
(7.24)
where we still assume that r > 0 and T > 0. The analysis is illustrated
below. Instead of all populations approaching T , as it did in the logistic
model, all models diverge from T .
The number T is a threshold of the model. At the value y = T the behavior
of the solution changes. We expect unlimited growth if y 0 > T and that y
will decay towards zero if y 0 < T . This type of model describes a species in
which there is not sufficient procreation to overcome the death rate unless
the initial population is large enough. Otherwise the different members
don’t run into each other often enough to make new babies. Following the
same methods as previously we obtain the solution
y =
T y 0
y 0 + (T − y 0 )e rt (7.25)
We point out the difference between (7.25) and (7.23) - which is the sign of
r in the exponential in the denominator.
When y 0 < T then equation (7.25) predicts that y → 0 as t → ∞. It
would appear to tell us the same thing for T < y 0 but this is misleading.