A

model, or

approximation to the way in which the number of a certain type of

life form changes with

time, usually expressed in the form of a

differential equation.

The simplest population growth model is dP / dt = k*P, where P is the population, t is the time elapsed since some arbitrary start time, and k is some constant; that is, that the rate of growth of the population at a given time is proportional to its value at that time. Solving this differential equation gives the solution that the population grows exponentially, which is indeed true for most systems, at least in their early stages.

Obviously, this exponential growth cannot continue indefinitely, or the Earth would be completely overrun with life. Any closed system must have a limit to the population it can sustain. As the population approaches this limit, the rate of growth tapers off, causing the population to approach the limit asymptotically. Functions of this form, which start off by growing exponentially, but later slow down and approach an asymptotic limit are known as sigmoids, or sigmoidal curves.

Still more advanced population models take multiple species and predator/prey relationships into account, leading to systems of simultaneous differential equations.