Editing Theoretical ecology (section)

===Exponential growth===
{{Main|Exponential growth}}

The most basic way of modeling population dynamics is to assume that the rate of growth of a population depends only upon the population size at that time and the per capita growth rate of the organism. In other words, if the number of individuals in a population at a time t, is N(t), then the rate of population growth is given by:
:<math> \frac{dN(t)}{dt}=rN(t) </math>
where r is the per capita growth rate, or the intrinsic growth rate of the organism.  It can also be described as r = b-d, where b and d are the per capita time-invariant birth and death rates, respectively. This [[Linear differential equation#First order equation|first order]] [[linear differential equation]] can be solved to yield the solution
:<math> N(t) = N(0) \ e^{rt} </math>,
a trajectory known as [[Malthusian growth]], after [[Thomas Malthus]], who first described its dynamics in 1798. A population experiencing Malthusian growth follows an exponential curve, where N(0) is the initial population size. The population grows when r > 0, and declines when r < 0. The model is most applicable in cases where a few organisms have begun a colony and are rapidly growing without any limitations or restrictions impeding their growth (e.g. bacteria inoculated in rich media).