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.

Monday, April 18, 2011

The Marginal Value Theorem

What it means:
The foraging in a patch (i) should be abandoned when the rate of energy acquisition in that patch (the left-hand side) equals the average intake rate including travelling time (the right-hand side, I*). E is energy gain and h is time spent in the patch. The assumptions is that all the animal is doing is searching for and handling food.

Implications and importance:

The theorem is based on the fact that resource acquisition often has diminishing returns and that it pays to leave an activity before the patch is depleted if there are alternative patches to exploit. Charnov (1976) and Parker & Stuart (1976) were the first to formalize this idea in evolutionary ecology.
One example solution to the Marginal Value Theorem (MVT) is

where hi* is the optimal patch residence time in patch i, si is the resource level in patch i, sa is the average resource level across patches, k is a parameter determining the initial slope of the gain function in a patch, and ts is the average travelling time between any two patches in the environment (Lundberg & Åström 1990). If patches are close to each other so little time is spent travelling, patch residence time decreases. The same is true if the average patch (i.e., the environment) is resource rich (high sa). The richer the focal patch (high si), the longer the patch residence time should be.
The MVT has made innumerable predictions for resource use in patchy environments (e.g., bees visiting flowers, browsers feeding on trees, mice exploiting seeds). Imagine yourself picking apples in an orchard or having one or many pork chops to eat when hungry. How many apples do you leave behind before changing trees if they are close and full of apples as opposed to the reverse?

Per Lundberg

Further reading:
Charnov, E. L. 1976. Optimal foraging, the marginal value theorem. Theor. Pop. Biol. 9: 129-136
Lundberg, P. & Åström, M. 1990. Functional response of optimally foraging herbivores. J. Theor. Biol. 144: 367-377.
Parker, G. A. & Stuart, R. A. 1976. Animal behaviour as a strategy optimizer: evolution of resource assessment strategies and optimal emigration thresholds. Am. Nat. 110: 1055-1076.
Stevens, D. W., Brown, J. S. & Ydenberg, R. C. (eds) 2007. Foraging. Chicago Univ. Press.