Sibly (1983) Optimal group size unstable

Sibly RM (1983) Optimal group size is unstable. Animal Behaviour 31:947-948.

If the fitness advantage of group living depends on group size, then optimal group size can be defined as that size yielding maximum individual fitness. It is commonly believed that individuals choose to live in groups of optimal size with the consequence that actual groups are around optimal size. This expectation is false, because groups of around optimal size are unstable. The correct prediction is that groups are of greater than optimal size.

The basic idea is that there will be a benefit to a lone individual to join a group even after it has grown to suboptimal size.

Similar logic shows that groups of around optimal size would be unstable and would tend to increase in size. Consider the following example, in which the optimal group size is 20, and the fitness of being in a group is given by the function in the Figure.

Suppose there is one flock of size 18, two of 19, three of 20, two of 21 and one of 22. The most disadvantaged birds would be those in the flocks of 18 and 22. Suppose one member of the flock of 22 suspected he could do better elsewhere, he would join a flock of size 19, increasing its size to 20. If a member of the flock of 18 did likewise there would then be one flock of 17, none of 18 or 19, five of 20 and three of 21. The most disadvantaged birds would then be those in the small flock of 17: each recruit from this small flock would do best to join a flock of size 20, thereby increasing its size to 21. Soon there would be one flock of size 12 and eight of size 21. Further recruits from the small flock would join the flocks of size 21, increasing their size to 22. Soon there would be one flock of size four and eight of size 22. The four members of the small flock would join flocks of size 22 so that eventually there would be four flocks of size 22 and four of size 23. Average flock size would therefore increase by 1289~ and the number of flocks would decrease from nine to eight. This simple example demonstrates that flocks of around optimal size are unstable and will tend to increase in size.

There is a brief note at the end about additional complexities that the model ignores, such as individual differences in capabilities, each being able to make the best choice given the options, each making decisions one at a time, and that individuals are not forcibly expelled from groups.

NOTE: The main point of this paper was criticized in:
Giraldeau and Gillis (1985) Optimal group size can be stable: A reply to Sibly. Animal Behaviour 33:666-667.

It can be proven that group size will never fall below the optimal size (assuming a sufficient population size). However, the shape of the fitness function can allow for the stability of the optimally sized group. In Sibly's original analysis, fitness decreases rapidly as group size shrinks relative to optimal, but decreases more slowly when group size increases above optimal. If, instead, the reverse is true, such that fitness decreases very rapidly as group size increases, then the optimal group size can indeed be stable.

Thus, the degree to which actual group size will be above the optimal size depends on the shape of the fitness function, and how it decreases on either side of the optimal size.