Lecture Notes in Differential Equations - Bruce E. Shapiro

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Figure 7.1: A plot of the right-hand side of y ′ = f(y) for the logistic model
in equation 7.8. When f(y) > 0, we know that dy/dt > 0 and thus y will
increase, illustrated by a rightward pointing arrow. Similarly, y decreases
when f(y) < 0. All arrows point toward the steady state at y = K.
rK/4
dy/dt
dy/dt < 0
K/2 K
y
dy/dt > 0
When y < K, then dy/dx > 0, so the population will increase; this is
represented by the arrow pointing to the right. When y > K, dy/dt < 0,
so the population will decrease. So no matter what the starting value of
y (except for y = 0), the arrows always point towards y = K. This tells
us that the population will approach y = K as time progresses. Thus the
carrying capacity tells us the long-term (or steady state) population.
We can solve the logistic model explicitly. In general this is not something
we can do in mathematical modeling. hence the frequent use of simple
models like exponential or logistic growth. Rewrite (7.8) and separating
the variables:
∫
∫
dy
K
y(K − y) = r dt (7.9)
Using partial fractions,
1
y(K − y) = A y + B
K − y
(7.10)
Cross multiplying and equating the numerators,
1 = A(K − y) + By (7.11)
Substituting y = K gives B = 1/K; and substituting y = 0 gives A = 1/K.