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.

Thursday, May 3, 2012

Source-sink Dynamics

bj + ijdjej = (bide)j = 0
source: bj > dj and ej > ij
sink: bj < dj and ej < ij

What it means
bj, dj, ij and ej are the total number of births, deaths, immigrants and emigrants in habitat j. At equilibrium a source is a net exporter and a sink a net importer of individuals. The source-sink concept explains how populations can persist in poor habitats where extinction would be a fact if it wasn't for a net in-flow of individuals dispersing from high quality habitats.

Where does it come from? 
The source-sink concept is often credited H. Ronald Pulliam, who's seminal paper published in The American Naturalist have been cited almost 2,200 times since its publication in 1988. As always, there were other key contributions leading up to this influential paper. Until the 1960's ecological theory was non-spatial, but with the publication of several important papers and books by e.g. MacArthur, Wilson, Levins and not to mention habitat selection theory based on the ideal free distribution by Fretwell and Lucas (1970) there was an growing interest in population dynamics across heterogeneous landscapes. However, if the spatial distribution of a population conformed to the ideal free distribution, all individuals would have the same expected fitness (since none would benefit from dispersing). This was in contrast to the dispersal sink terminology suggested by Lidicker (1975) to describe that individuals may occupy poor habitats because they are excluded from better ones. Horn realised that this meant that local population size is not necessarily informative about the underlying habitat quality. The potential importance of source-sink dynamics may have led Holt (1985) to the first mathematical source-sink model three years before the publication of Pulliam (1988). Pulliams's paper was built on the bide (birth, immigration, death, emigration) structure which had been previously used by Cohen (1969). Thanks to the simple, yet instructive model analysis presented by Pulliam the message was available and could be understood by a wide range of scientists.

The source-sink concept has changed the way we think about niches (realized niches may be larger than fundamental niches), and the spatial dimension of population and community ecology (local patterns may no reflect local conditions). Identifying sources and sinks may however be challenging due to temporal variation in local conditions as well the existence of pseudo-sinks, where an overpopulated source (e.g. due to immigration) can have a negative growth rate. The source-sink concept has nevertheless vitalised population management by providing theory for spatial control and challenged optimal monitoring, exploitation and conservation in heterogeneous landscapes. Furthermore, source-sink dynamics provide a challenge for climate envelope models that try to predict future species ranges based on observed ranges. Today Pulliam's paper is cited many disciplines outside ecology, including law, medicine, toxicology and mathematics.

Niclas Jonzén

Liu, J., Hull, V., Morzillo, A.T. & Wiens, J.A (eds.). 2011. Sources, Sinks and Sustainability. Cambridge UP, Cambridge, UK. The source-sink literature is treated in detail by this volume and includes all papers cited above.
Pulliam, H.R. 1988. Sources, sinks and population regulation. Am. Nat. 132: 652-661.