The logistic differential equation dn dt

= kn 

1 −

n K



, n(0) = n0, is one of the most common equations used to model population This is the logistic equation, which is discussed in more detail in Section 4.1.1. growth and death.

a. Show that the solution of the logistic differential equation is n(t) =

K 1 + Ae−kt , where Hint: use a computer. Then you’re more likely to get the answer correct, and it’ll be a lot faster and easier.

A =

K − n0 n0

.

K is called the carrying capacity of the population. The population always tends to K as t → ∞ because the environment cannot support a larger population.

b. Show that if n0 > K the solution decreases to K, but if n0 < K the solution increases to K. Check this by plotting a few solutions.

c. How do the solutions change as k is varied? What is the biological interpretation of k?