* Author to whom correspondence should be addressed. E-mail: j.beecham@mluri.sari.ac.uk. J. theor. Biol. (1999) 198, 533}548 Article No. jtbi.1999.0930, available online at http://www.idealibrary.com on Animal Group Forces Resulting from Predator Avoidance and Competition Minimization J. A. BEECHAM* AND K. D. FARNSWORTH Macaulay ¸and ;se Research Institute, Aberdeen AB15 8QH, ;K (Received on 23 November 1998, Accepted in revised form on 5 March 1999) A new model to explain animal spacing, based on a trade-o! between foraging e$ciency and predation risk, is derived from biological principles. The model is able to explain not only the general tendency for animal groups to form, but some of the attributes of real groups. These include the independence of mean animal spacing from group population, the observed variation of animal spacing with resource availability and also with the probability of predation, and the decline in group stability with group size. The appearance of &&neutral zones'' within which animals are not motivated to adjust their relative positions is also explained. The model assumes that animals try to minimize a cost potential combining the loss of intake rate due to foraging interference and the risk from exposure to predators. The cost potential describes a hypothetical "eld giving rise to apparent attractive and repulsive forces between animals. Biologically based functions are given for the decline in interference cost and increase in the cost of predation risk with increasing animal separation. Predation risk is calculated from the probabilities of predator attack and predator detection as they vary with distance. Using example functions for these probabilities and foraging interference, we calcu- late the minimum cost potential for regular lattice arrangements of animals before generalizing to "nite-sized groups and random arrangements of animals, showing optimal geometries in each case and describing how potentials vary with animal spacing.  1999 Academic Press Introduction For many animals the presence of, and distance to other animals is important in determining their spatial distribution through apparently social forces (e.g. Breder, 1954; Arnold & Dud- zinski, 1978; Taylor, 1981; Warburton & Lazarus, 1991; Gueron et al., 1996). Warburton & Lazarus (1991) who investigated several models of social distance regulation in the presence of a simple random searching behaviour, recognized that many animals form groups under the in#uence of social forces so that &&an animal's movements might be in#uenced by a few or many other individuals in the group''. The resulting patterns of movement and group structure were claimed to have functional implications. The spacing and geometry of social animals appears to result from a trade-o! between the need to avoid predators and the need to optimize accessibility to food or other resources (Rayor & Uetz, 1990). This trade-o! is illustrated by a tendency for animals to be less aggregated when resources are scarce (Dudzinski et al., 1969) and for apparently hungry animals to move to the front or outside of a group where resources are more plentiful but predation risks are higher (Romey, 1995; Krause, 1993). As a result, behav- ioural ecologists have recognized the need to 0022}5193/99/013533#16 $30.00/0  1999 Academic Press