Lecture Notes in Differential Equations - Bruce E. Shapiro

54 LESSON 7. AUTONOMOUS ODES
Hence
dy
= by (7.2)
dt
If we assume that people also die at a rate dy then
dy
dt
= by − dy = (b − d)y = ry (7.3)
where r = b − d. As we have seen, the solution of this equation is
y = y 0 e rt (7.4)
Hence if the birth rate exceeds the death rate, then r = b − d > 0 and the
population will increase without bounds. If the death rate exceeds the birth
rate then the population will eventually die if. Only if they are precisely
balanced will the population remain fixed.
Logistic Growth
Exponential population growth is certainly seen when resources (e.g., food
and water) are readily available and there is no competition; an example
is bacterial growth. However, eventually the population will become very
large and multiple members of the same population will be competing for
the same resources. The members who cannot get the resources will die off.
The rate at which members die is thus proportional to the population:
d = αy (7.5)
so that
dy
dt = ry − dy = by − αy2 = ry
(1 − α )
b y
for some constant α. It is customary to define
K = r α
(7.6)
(7.7)
which is called the carrying capacity of the population:
dy
(1
dt = ry − y )
K
(7.8)
Equation (7.8) is called the logistic growth model, or logistic differential
equation. We can analyze this differential equation (without solving it)
by looking at the right hand side, which tells us how fast the population
increases (or decreases) as a function of population. This is illustrated in
figure 7.1