The oldest model describing how individual organisms survive the effects of competition for resources is the "logistic" equation, originally formulated by Verhulst (1838) but rediscovered by Pearl and Reed (1920). Lotka (1925) also discovered the logistic equation as a special case (when only one species is allowed to vary) of his "fundamental equations of kinetics of evolving systems". Lotka's "fundamental equations" can be stated as follows

where N_i is the density of the i^th population in a community composed of many species with densities N₁, N₂, N₃.....N_i..., P represents environmental influences (weather, soil moisture, etc.) and Q represents the genetic properties of the species. Under the condition that all but one species in the community have constant densities, then the "fundamental equations" reduce to

which can be expanded by Taylor's theorem to the polynomial form

From this, the simplest model that satisfies the conditions that populations cannot grow when their densities are zero (i.e., a₀ must be zero), grow exponentially when their densities are very small (i.e., a₁ > 0), and must decline when their densities are infinitely large (i.e., a₂ < 0) is

This equation is identical to Verhulst's logistic model for populations growing in a finite environment. Dividing through by N and defining the two parameters a = a₁ = the maximum per-capita rate of change of the species in a given environment and b = a₂ = the coefficient of intraspecific competition or, if you prefer, the "struggle for existence", then we obtain a logistic R-function

Carrying Capacity

A population is said to be in equilibrium when the rate of change is zero. Let the density of the population which gives a growth rate of zero be K, then

where K is often called the carrying capacity of the environment because it specifies the maximum population that can be sustained indefinitely in a particular environment.

We can now substitute 1 / K for a₂ / a₁ in the logistic R-function to obtain the more popular form for the logistic R-function with carrying capacity

where a = a₁.