# 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.

## Sunday, January 1, 2012

### Logistic growth  What it means
N is population density, r is the intrinsic rate of increase (i.e., the maximum per capita growth rate), K is the so-called carrying capacity (i.e., the maximum sustainable population) and t is time. A population following the logistic growth equation is regulated such that the per capita growth rate ((dN/dt)/N) declines linearly with density.
Hence, when there are very few individuals (N << K) the per capita growth rate is close to r, but it will decrease as the population density is increased. Eventually when the density has reached the carrying capacity (N=K) the growth rate equals zero and the population has reached a globally stable equilibrium (i.e., the population will end up at N=K independent on starting value except for N(0)=0). If the population density incidentally is larger than the carrying capacity (N > K), e.g., due to immigration, then the growth rate becomes negative until the population density has decreased down to the carrying capacity (N=K).

Where does it come from?
The logistic growth equation was originally formulated by Pierre-François Verhulst. The Verhulst equation was published after Verhulst had read Thomas Malthus' An Essay on the Principle of Population. Verhulst derived his logistic equation to describe the self-limiting growth of a biological population.

Importance
The logistic growths equation is a common model of single species population growth when there are limited resources. It means that the rate of increase is proportional to both the existing population and the amount of available resources, all else being equal. Due to the linear relationship between the per capita growth rate and population density the logistic growth model is the simplest model of population regulation. Actually, it is one of very few nonlinear differential equations in ecology having an exact solution. It is used to model single species populations of a great variety, e.g. in bacteria, yeast, fish, mammals and plants. The logistic growth model has also been extended in various ways and it can be an important building block when formulating multi-species models.

Anders Wikström

Literature
Case, T.J. 2000. An illustrated guide to theoretical ecology. Oxford University Press

Mangel, M.2006. The theoretical biologists toolbox. Cambridge University Press

Turchin, P. 2003. Complex population dynamics: a theoretical/empirical synthesis. Princeton University Press