Use, misuse and extensions of “ideal gas” models of animal encounter

1. What they did

In 1860 the physicist James Clerk Maxwell worked out how often molecules in a gas should bump into each other, from how many there are, how fast they move and how big they are. Biologists have rediscovered this “ideal gas” idea again and again to predict how often animals, or groups of animals, should meet if they wander about at random and ignore each other. If animals meet much more or much less often than predicted, that hints that they are attracted to or avoiding each other.

The authors:

• Reviewed how the model has been used in biology: meetings between primate groups, males and females, predators and prey, sperm and eggs, and observers and the animals they count.
• Set out the basic formulae, and showed how to adjust them when animals vary in speed, travel mostly in one direction, live at uneven densities, or are confined to home ranges that only partly overlap.
• Used computer simulations to test what happens when animals do not travel in straight lines, and when they move apart after meeting.
• Corrected errors in eight areas of the published literature.
• Described three ways to test whether observed meetings differ from the prediction, something earlier studies rarely did.
• Reanalysed old field data on one group of grey-cheeked mangabeys in Kibale Forest, Uganda, first used by Waser (1976) to argue that mangabey groups avoid each other.

2. What they found

How robust the model is

• The average number of meetings predicted by the model holds even when animals wander along winding paths instead of straight lines.
• But winding paths make the numbers far more variable, and many meetings are re-meetings with the same animal shortly after parting. Fewer different individuals are met. Observers can easily miss the short gap between two meetings and count them as one.
• Small differences in speed between individuals make little difference. A predator moving at half the speed of its prey gains only about 6% more meetings than if it stayed still, which suggests advantages of a sit-and-wait tactic.
• When animals move apart on contact, the model underestimates meetings, but only at high densities (such as ants in a nest). With an encounter distance of 100 m, density would have to reach 7.6 per km² before meetings exceeded the prediction by 10%.
• Where home ranges only partly overlap, the predicted number of meetings drops considerably, by an amount that depends on how much the ranges overlap, how the shared areas are used, and how the ranges are arranged.

Errors in earlier papers

• Several studies (including Waser’s own 1984 and 1987 papers) overestimated how long meetings last, by a factor of about 1.57.
• An early formula for the distance at which spread-out groups detect each other was wrong and was copied into several theses.
• A model of infanticide in great apes gave much too low a risk, because it did not consider each male whose range overlaps the female’s separately. Using that study’s figures for Virunga gorillas, and imagining females were solitary, the corrected risk came out at 0.9999.
• Other corrections covered mating benefits of a larger home range, spiral swimming of sperm, jellyfish catching herring larvae, and seabird counts from ships and aircraft.

The mangabey reanalysis

• The group was followed for 1130.5 hours and moved 133.5 km. Only 4 meetings with other groups were seen within 200 m, and 11 within 500 m.
• The simplest model predicted 24.6 meetings within 200 m and 50.1 within 500 m. More realistic speeds raised the predictions to about 30 and 60.
• Allowing for partly overlapping home ranges roughly halved the predictions (12.5 to 13.5 within 200 m).
• If re-meetings within 1 hour are ignored, the simulated prediction within 200 m fell from 29.7 to 19.7; within 4 hours, to 14.1.
• The simulations gave very wide confidence limits (for example, 8 to 59 meetings within 200 m), which makes it hard to show a real difference statistically.
• Comparing the group’s paths with its own paths from other months predicted 16.0 meetings within 200 m, similar to the home range corrected model.

3. What they concluded

• When animals ignore each other, the ideal gas model predicts the average number of meetings well, but not the spread of those numbers or the number of different individuals met.
• The reanalysis tends to support the original conclusion that mangabey groups avoid each other. But three other things could help explain why so few meetings were seen: neighbouring home ranges overlap only partly, observers may miss re-meetings within an hour, and the confidence limits are wide.
• Because confidence limits are so wide, counting meetings may not be the best way to test whether animals avoid or are drawn to each other. Waser also used playbacks of neighbouring groups’ calls.
• The model remains a valuable null model and a useful, simpler stand-in for reality. Simulations built on the real movements of the species allow more realistic details to be added.
• Tracking technologies such as Global Positioning System (GPS) data loggers and satellite tracking now make it easier to measure animal speeds and home range overlap, and to record meetings directly when several animals are tracked at once.

6. Other technologies, skills or innovations

• Computer simulations of animals moving in short straight steps with random turns (correlated random walks), with step lengths and turning angles drawn from the mangabey group’s observed movements.
• A randomization test that uses the animals’ own real paths recorded at different times as a “no interaction” comparison, since animals on paths made at different times cannot have responded to each other.
• Formulae for observers counting animals from a moving ship, aircraft or along a path, including areas that are rectangular, elliptical, semicircular or keyhole shaped (a narrow strip of clear view ahead plus a circle around the observer).