The logistic model shows how the size of a population grows within a constrained environment. As the growth rate increases the model provides perhaps the best known example of a transition to chaos.
This model shows the progress of growth that is limited by available “resources”, such as market size or environmental carrying capacity (K). Systems that grow in regular jumps (eg seasonal reproduction), behave very differently from those that grow continuously. When the growth rate (R) is low, both systems behave the same way. As you increase R note the appearance of cycles, period doubling, and eventually chaos.