Mostly for me, I am writing a little history of mathematical models. A timeline really.
1202, Fibonacci tried to model rabbits with the eponymous sequence.
1780, Malthus tries to model human populations as a deterministic birth death process, not valid as time goes to infinity of course, but he has
where the difference between birth and death rates.
1820, Verhulst makes this more realistic by adding a carrying capacity that depends on resources. Thus he uses
so that makes for and a decline in population, whereas below the carrying capacity, the population grows.
1925, Lotka writes his famous differential equations to describe predator-prey dynamical systems (1926 Volterra independently publishes… allegedly). Here I write them in terms of rabbits and foxes
1936ish, Kolmogorov generalizes these equations
1927, Kermack and McKendrick write the famous SIR model, describing an epidemic process in susceptible, infected and recovered individuals. Typically this model is a fractional model using and rates are written in terms of fractions