Exponential growth is known as the first principle of population dynamics, with the use of "principle" implying that it is a fundamental property of all population systems or, in fact, all self-reproducing entities (e.g., your compound interest savings account).

Exponential growth is best visualized by considering a single amoebae which, we assume, reproduces by division once every day:

Discrete exponential growth is characterized by changes over a fixed time interval, usually a year, month or day. For example, the equation

is a discrete growth equation because the number at a particular time (t), say June 1, 1997, is calculated from the number at a discrete unit of time earlier (t-1), say June 1, 1996. Hence, G is the annual growth rate. Remember that G = 1+ B -D is the finite per-capita rate of change, and B and D the per-capita birth and death rates, respectively (click here to refresh your memory?). This equation is also called a difference equation because growth is computed by the difference between population densities at two points in time. The equation can be solved for any length of time under the condition that G is constant; i.e.,

Continuous exponential growth is characterized by changes that occur instantaneously, or as the time between observations becomes very small. Continuous population growth is defined by the differential equation

where dN / dt is the rate of population change over an instant of time and R is the instantaneous per-capita rate of change. We can integrate this equation by employing the calculus; i.e., rearranging it so that only N appears on the left-hand side

and integrating over time, we obtain

with C the constant of integration. Letting C = ln N₀, we have

or

Note that R and G are equivalent through the logarithmic transformation

More information on exponential growth from Alexei Sharov?

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©1997 Alan A. Berryman