Lecture Notes in Differential Equations - Bruce E. Shapiro

58 LESSON 7. AUTONOMOUS ODES
Figure 7.3: Plot of the right hand side of the thresholding model (7.24).
Since f(y) < 0 when y < T , the population decreases; when y > T , the
population increases.
dy/dt
dy/dt < 0
-rT/4
T/2 T
dy/dt > 0
y
Suppose that y 0 is just a little bit larger than T . Then the population
because increasing because the right hand side of the equation is positive.
As time progresses the second term in the denominator, which is negative,
gets larger and larger. This means that the denominator is getting smaller.
This causes the population to grow even faster. The denominator becomes
zero at t = t ∗ given by
0 = y 0 + (T − y 0 )e rt∗ (7.26)
Solving for t ∗ t ∗ = 1 r ln y 0
y 0 − T
(7.27)
Since y > T the argument of the logarithm is positive, so the solution gives
a positive real value for t ∗ . At this point, the population is predicted to
reach ∞; in other words, t = t ∗ is a vertical asymptote of the solution.
Logistic Growth with a Threshold
The problem with the threshold model is that it blows up. More realistically,
when y 0 > T , we would expect the rate of growth to eventually
decrease as the population gets larger, perhaps eventually reaching some
carrying capacity. In other words, we want the population to behave like a
thresholding model for low populations, and like a logistic model for larger
populations.