Lecture Notes in Differential Equations - Bruce E. Shapiro

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Figure 7.4: Solutions of the threshold model given by (7.25). When y 0 < T
the population slow decays to zero; when y 0 > T the population increases
without bounds, increasing asymptotically to ∞ at a time t ∗ that depends
on y 0 and T (eq. (7.27)).
y
T
1/r
t
Suppose we have a logistic model, but let the rate of increase depend on
the population:
dy
(1
dt = r(y)y − y )
(7.28)
K
We want the rate of increase to have a threshold,
(
r(y) = −r 1 − y )
(7.29)
T
combining these two
dy
(1
dt = −ry − y ) (
1 − y )
T K
(7.30)
Now we have a growth rate with three zeros at y = 0, y = T , and y = K,
where we want 0 < T < K. If the population exceeds T then it approaches
K, while if it is less than T it diminishes to zero.