One very obvious situation in which this sort of modeling would apply is in the study of how populations shrink and grow. These may be populations of people, animals, bacteria, or the number of HIV infected cells in a patient's bloodstream. Malthus was an Anglican minister as well as a professor of history at Cambridge University in England. His idea, which seems fairly obvious over 150 years later, was that the rate at which a population grows is directly proportional to its current size . To justify Malthus' idea we might argue, quite simplistically, as follows.
Populations grow because people have babies.
The more people there are, the more babies they'll have. This means that the number of babies being born is a constant multiple of the number of people present in the population. So, the rate of population growth is directly proportional to the size of the population. You might want to point out that there's also a death rate that we haven't considered. However, this rate too will simply be a certain percentage of the population's size at any given time.