Equation of the Month

Equation of the Month

A blog run by the

Theoretical Population Ecology and Evolution Group,

Biology Dept.,

Lund University

The purpose of this blog is to emphasize the role of theory for our understanding of natural, biological systems. We do so by highlighting specific pieces of theory, usually expressed as mathematical 'equations', and describing their origin, interpretation and relevance.

Friday, March 11, 2011

Exponential growth

What it means
N is population density and t is time. This is the simplest model of population growth and assumes that the per capita growth rate, i.e., the difference between per capita birth and death rates, is a constant, r, often referred to as the intrinsic (per capita) growth rate. The solution of the differential equation above is

where N(0) is the population density at time zero. If r > 0 the population will grow to infinity, whereas if r < 0 it will decline towards zero.

Implications and importance
The 18th century reverend Thomas Malthus is often cited as the founder of the exponential growth model. This model is sometimes referred to as the exponential law (Turchin 2003); it certainly has similarities with the law of intertia in physics, and it is generally considered to be the first principle of population dynamics (e.g. Ginzburg 1986; Berryman 1999). Since it describes the dynamics of a population in a constant environment with no forces acting upon it, it efficiently serves as a starting point for more detailed models including e.g. structure (age, stage, space, etc), interactions and stochasticity. The stochastic version of the exponential growth model, which is a random walk on the log scale, is sometimes used in conservation biology for estimating extinction risks of populations at low density.

Further reading

Berryman, A. A. 1999. Principles of population dynamics and their applications. Stanley Thornes Publishers, Cheltenham, UK.

Ginzburg, L. R. 1986. The theory of population dynamics. I. Back to first principles. Journal of Theoretial Biology 122:385-399.

Turchin, P. 2003. Complex population dynamics. A theoretial/empirical synthesis. Princeton University Press, Princeton, NJ.


  1. Dear friends in Lund:
    Malthus' own words were:
    "Population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio. A slight acquaintance with numbers will shew the immensity of the first power in comparison of the second."
    I could not have translated his words into the formula, but fortunately Malthus' meme has evolved by itself...
    The book is free to read online at
    http://www.econlib.org/library/Malthus/malPop1.html#Chapter I

  2. Thank you for that contribution, Mikael!
    I'd also like to emphasize that exponential growth applies to structured populations as well, although it is not apparent from the standard formulation above. An age-, stage- or spatially structured population will, 'when unchecked', grow at an exponential rate (after some initial transients). The 'exponential law' is thus quite general.