J. theor. Biol. (1971) 31, 295-311 Geometry for the Selfish Herd W. D. HAMILTON Department of Zoology, Imperial College, London, S. W.7, England (Received 28 September 1970) This paper presents an antithesis to the view that gregarious behaviour is evolved through benefits to the population or species. Following Galton (1871) and Williams (1964) gregarious behaviour is considered as a form of cover-seeking in which each animal tries to reduce its chance of being caught by a predator. It is easy to see how pruning of marginal individuals can maintain centripetal instincts in already gregarious species; some evidence that marginal pruning actually occurs is summarized. Besides this, simply defined models are used to show that even in non-gregarious species selection is likely to favour individuals who stay close to others. Although not universal or unipotent, cover-seeking is a widespread and important element in animal aggregation, as the literature shows. Neglect of the idea has probably followed from a general disbelief that evolution can be dysgenic for a species. Nevertheless, selection theory provides no support for such disbelief in the case of species with outbreeding or un- subdivided populations. The model for two dimensions involves a complex problem in geo- metrical probability which has relevance also in metallurgy and com- munication science. Some empirical data on this, gathered from random number plots, is presented as of possible heuristic value. 1. A Model of Predation in One Dimension Imagine a circular lily pond. Imagine that the pond shelters a colony of frogs and a water-snake. The snake preys on the frogs but only does so at a certain time of day-up to this time it sleeps on the bottom of the pond. Shortly before the snake is due to wake up all the frogs climb out onto the rim of the pond. This is because the snake prefers to catch frogs in the water. If it can’t find any, however, it rears its head out of the water and surveys the disconsolate line sitting on the rim-it is supposed that fear of terrestial predators prevents the frogs from going back from the rim-the snake surveys this line and snatches the nearest one. 295296 W. D. HAMILTON Now suppose that the frogs are given opportunity to move about on the rim before the snake appears, and suppose that initially they are dispersed in some rather random way. Knowing that the snake is about to appear, will all the frogs be content with their initial positions? No; each will have a better chance of not being nearest to the snake if he is situated in a narrow gap between two others. One can imagine that a frog that happens to have climbed out into a wide open space will want to improve his position. The part of the pond’s perimeter on which the snake could appear and find a certain frog to be nearest to him may be termed that frog’s “domain of danger” : its length is half that of the gap between the neighbours on either side. The diagram below shows the best move for one particular frog and how his domain of danger is diminished by it: E%z k!%? -aLzi&x...? xaz But usually neighbours will be moving as well and one can imagine a confused toing-and-froing in which the desirable narrow gaps are as elusive as the croquet hoops in Alice’s game in Wonderland. From the positions of the above diagram, assuming the outside frogs to be in gaps larger than any others shown, the following moves may be expected: What will be the result of this communal exercise ? Devious and unfair as usual, natural justice does not, in general, equalize the risks of these selfish frogs by spacing them out. On the contrary, with any reasonable assumptions about the exact jumping behaviour, they quickly collect in heaps. Except in the case of three frogs who start spaced out in an acute- angled triangle I know of no rule of jumping that can prevent them aggregating. Some occupy protected central positions from the start; some are protected only initially in groups destined to dissolve; some, on the margins of groups, commute wildly from one heap to another and yet continue to bear most of the risk. Figure 1 shows the result of a computerGEOMETRY FOR THE SELFISH HERD 297 IO” segments of pool margin (degrees) 0 90 180 270 360 I I 2’3’33”“““““““““““““1”’ 6133 4 I35 I345294 56354 122 43 2 522 8 32 61 17 247 II? 75255 32 44 3 4 5 6 7 8 29 E IO c’ II .ii, 12 % 13 a 14 15 16 17 I8 El 612 8 31 8 6 3 3 9 7 9 2 9 7 9 9 a i 9 9 5 9 II! 4 a i2 3 7 II 2 6 14 6 14 5 1s 3 16 10 19 9 49 II a 411 10 a 512 a 8 514 7 8 516 5 9 418 3 IO 32c I II 2 22 12 I 22 I3 22 13 22 I3 22 13 22 12 22 II 22 9 22 7 22 94 75 22 44 103 86 a2 45 121 96 3 45 13 95 5 36 15 33 6 36 17 9 I 6 5 5 I3 e 6 74 20 6 6 92 ?.1 4 7 92 24 3 7 92 26 7 8 82 28 I 9 72 31, IO 62 3; 9 62 3 : a 72 3s 6 I32 37 5 9 I FIG. 1. Gregarious behaviour of 100 frogs is shown in terms of the numbers found successively within 10” segments on the margin of the pool. The initial scatter (position 1) is random. Frogs jump simultaneously giving the series of positions shown. They pass neighbours’ positions by one-third of the width of the gap. For further explanation, see text simulation experiment in which 100 frogs are initially spaced randomly round the pool. In each “round” of jumping a frog stays put only if the “gap” it occupies is smaller than both neighbouring gaps; otherwise it jumps into the smaller of these gaps, passing the neighbour’s position by one-third of the gap-length. Note that at the termination of the experiment only the largest group is growing rapidly. The idea of this round pond and its