Measured Games · Paper 14 of 07 · Behavioural ecology

The ideal free distribution

After S. D. Fretwell and H. L. Lucas (1969); tested on mallards by D. G. C. Harper (1982) and on sticklebacks by M. Milinski (1979)

Abstract

Foragers distributing themselves across patches of differing quality should arrive at the allocation that equalises the return in every occupied patch, which puts the share of competitors in each patch equal to that patch's share of the total input. The equilibrium requires no communication, no leader and no memory of past distributions, only that each individual can tell whether it would do better somewhere else.

Keywordsinput matching · habitat selection · equal returns · undermatching · field test

1Setup

Setup

Two feeding patches deliver food at different rates. A flock of foragers can move freely between them and each individual takes whatever share of a patch's input its competition allows. Where should they settle?

Not all in the richer patch, because crowding there would drive the individual return below what the poorer patch offers. Not evenly, because the richer patch can support more. The distribution that no individual can improve on by moving is the one where the return is the same everywhere, and that condition turns out to fix the numbers exactly.

2The model

The model

# patch i delivers input at rate rᵢ, and holds nᵢ competitors intake per forager in patch i = ri / ni # nobody can gain by moving, so returns are equal everywhere rA / nA = rB / nB # which gives input matching ni / N = ri / Σr # the share of competitors in a patch equals that patch's share of the food

The prediction is unusually sharp for behavioural ecology. It contains no free parameters, is stated in observable quantities, and can be falsified by counting animals. It also predicts something stronger than the distribution itself: that the average intake should be the same in every occupied patch, which is a separate and independently checkable claim.

Figure 1Free movement between two patches

This figure needs JavaScript. The panels below are live simulations rather than images, so with scripting turned off they stay empty and the controls do nothing. Everything the figure demonstrates is also stated in the surrounding text: the model is given in full in section 2 and the result in section 3.

Two patches · foragers distribute themselves

Intake per forager in each patch · equal returns is the equilibrium condition

Observed share Predicted share Rich patch intake Poor patch intake

Charts are focusable. With a chart focused, use the left and right arrow keys to read values along the horizontal axis.

Input ratio, rich to poor

Foragers

Sampling rate

Travel cost

Predicted share—

Observed share—

Rich patch intake—

Poor patch intake—

Intake gap—

Round0

Figure 1. Foragers only ever compare their own intake with what they would get by moving, so the distribution is assembled from individual decisions and no forager knows the input rates. Swap the patches mid-run to watch the flock re-sort. Travel cost is the interesting control: raise it above zero and the distribution stops matching exactly and settles short of the prediction, which is the undermatching that field studies almost always report.

3Results

Results

Fretwell and Lucas (1969) set out the ideal free distribution as a theory of habitat selection: ideal because individuals assess patches correctly, free because nothing prevents them moving. Under those assumptions the equilibrium equalises fitness across occupied habitats, and with intake proportional to a patch's input rate that becomes input matching.

Two things make the result worth having. It is a genuine equilibrium prediction rather than an optimisation: no individual is doing anything clever, and the aggregate pattern is a by-product of each one refusing to stay somewhere worse. And it predicts a null result, equal intake everywhere, which is easier to test convincingly than a predicted difference.

The standard deviation from it is undermatching: too few animals in the rich patch and too many in the poor one. Travel costs, imperfect information, unequal competitive ability and perceptual limits all push in that direction, and the simulator reproduces it as soon as movement stops being free.

4Evidence

Evidence

Harper (1982) ran the test that is still cited first. Mallards on a pond were fed standardised pieces of bread by two throwers stationed apart, and the number of birds at each station settled in proportion to the rate of delivery there after an induction period on the order of a minute. The speed matters as much as the fit: the flock reached the distribution well before any individual could have sampled both stations thoroughly.

What made the study a landmark was not the distribution but the second measurement. Harper recorded the intake of individual identifiable birds as well as the numbers at each station, which nobody testing the ideal free distribution had managed before. That is what allows the equal-returns condition to be checked rather than assumed, and it is why a study of people throwing bread at ducks is still the reference experiment.

Milinski (1979) obtained the same result with three-spined sticklebacks in a tank with two feeding ends, and was able to show individual switching consistent with the aggregate pattern. Later work on the same system examined competitive differences between individuals, which the basic model assumes away.

Across the wider literature the qualitative prediction holds up well and the quantitative fit is usually imperfect in the undermatching direction. That pattern is consistent enough to be treated as a finding in its own right rather than as noise.

5Limitations

Limitations

The model assumes foragers are interchangeable. They are not: dominant individuals take more than their share of a patch, and once competitive ability varies the equilibrium is a distribution of phenotypes rather than of numbers, which can match the input ratio while the intake condition fails badly.

It also assumes movement is costless and information perfect. Harper's ducks came close because the patches were metres apart and visible; the assumption is much weaker across a landscape. Undermatching is so consistently observed that a fitted matching exponent is now often reported instead of a test against exact matching, which weakens the theory's falsifiability.

Finally, equal intake is the prediction only when intake is the currency that matters. Where patches differ in predation risk, the equilibrium equalises something else and a count of animals will not match the input ratio at all.

References

Fretwell, S. D., & Lucas, H. L. (1969). On territorial behavior and other factors influencing habitat distribution in birds. Acta Biotheoretica, 19(1), 16–36. https://doi.org/10.1007/BF01601953
Harper, D. G. C. (1982). Competitive foraging in mallards: Ideal free ducks. Animal Behaviour, 30(2), 575–584. https://doi.org/10.1016/S0003-3472(82)80071-7
Kennedy, M., & Gray, R. D. (1993). Can ecological theory predict the distribution of foraging animals? A critical analysis of experiments on the ideal free distribution. Oikos, 68(1), 158–166. https://doi.org/10.2307/3545322
Milinski, M. (1979). An evolutionarily stable feeding strategy in sticklebacks. Zeitschrift für Tierpsychologie, 51(1), 36–40. https://doi.org/10.1111/j.1439-0310.1979.tb00669.x
Parker, G. A., & Sutherland, W. J. (1986). Ideal free distributions when individuals differ in competitive ability: Phenotype-limited ideal free models. Animal Behaviour, 34(4), 1222–1242. https://doi.org/10.1016/S0003-3472(86)80182-8
Tregenza, T. (1995). Building on the ideal free distribution. Advances in Ecological Research, 26, 253–307. https://doi.org/10.1016/S0065-2504(08)60067-7

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