The application of mathematics to the prediction of population dynamics has challenged demographers for at least two hundred years. Various proponents have developed formulae for both the calculation of population growth as well as the potential limits to such growth. While these formulae on their own have not always been able to accurately predict human carrying capacity limits, in many cases they have contributed to the development of more complex carrying capacity models.

[i] As such, they have often been theoretic in nature, rather than having direct applicability to a particular landscape.

One of the earliest known equations relating to population dynamics was Thomas Malthus’ exponential growth theory of 1798. According to Malthus,[ii] “[p]opulation, when unchecked, increases in a geometric ratio,” while its means of subsistence, namely its food supply, increases only in a linear or arithmetic manner. The exponential growth formula is relatively simple and can be given as;

*P(t) = **P*_{o}* e*^{rt},

where *P(t)* is the population at a point in time, *P*_{o}_{ }is the initial population, e is the base of natural logarithms (2.718...), *r* is the growth rate and *t* is time. This formula generates a j-shaped curve with population reaching to infinity (figure 1a). However, according to Malthus, this infinite growth is inevitably halted by the inability of food production to keep up with the population’s exponential expansion (figure 1b).

*Figure 1a. (left): Malthusian exponential growth curve showing how the population increases infinitely. Figure 1b. (right): Malthus’ exponential population growth curve limited by the linearly increasing food supply. The assumed carrying capacity is the point at which the population projection intersects with the food supply projection. The carrying capacity is assumed in this instance because Malthus did not refer to it as carrying capacity.*