The first model x
0
(t) = rx is not that realistic as it either led to a population explosion
or to extinction. This simple model was improved on by building into this differential
equation a way to prevent the runaway growth when r > 0. The death rate now
increases when x starts to become too large. This is reasonable as we start to use
up our resources when the population is too large. Verhulst proposed the following
differential equation as a mathematical model: