## Optimal Group Size

Despite the great variation in types of animal groups, the biology of flock behavior has important commonalities across all species. An example is the determinants of group size, which can be seen as a function of the costs and benefits of being part of a group.

A group consisting of N individuals can increase its size without recruitment of individuals from outside as long as the group's per capita growth rate r in eqn  is positive:

r is likely to be a function of group size, reflecting the difference between benefits and costs of adding further individuals to the group. Hence, r may be modeled as a difference between two functions:

where f (N) denotes the benefit function and g(N) the cost function. Both functions can either be independent of N or increase with N. Since groups cannot be infinitely large, there will exist a value for N satisfying the condition that r(N) = 0 for N = Nmax and r(N) < 0 for N> Nmax. Benefits of increasing group size are likely to level off with N, while costs are likely to accelerate as N becomes large as indicated in Figure 4a. Nmax corresponds to the value of N when costs and benefits balance.

When group size reaches its maximum size, r will be 0, indicating that the fitness of group members is low. By reducing group size from this point, the fitness of each  Figure 4 Group formation is predicted to occur when the relative growth rate (r) increases with group size. Depending on how the benefits (B) and costs (C) of group behavior change with group size (N), three different cases can be identified: (a) Benefits increase more steeply than costs when group size is small, and vice versa when group size is large, leading to an optimal group size (Nopt) toward which the group is predicted to converge. If a group does not split up when N > Nopt, it will continue to grow until N = Nmax. (b) Solitary behavior (i.e., N = 1) is predicted if costs increase more than benefits as group size increases. (c) A group must be largerthan Nmin in orderto persist and grow, whereas it is goes toward to extinction if N falls below Nmin because the relative growth rate is negative (the so-called Allee effect).

Figure 4 Group formation is predicted to occur when the relative growth rate (r) increases with group size. Depending on how the benefits (B) and costs (C) of group behavior change with group size (N), three different cases can be identified: (a) Benefits increase more steeply than costs when group size is small, and vice versa when group size is large, leading to an optimal group size (Nopt) toward which the group is predicted to converge. If a group does not split up when N > Nopt, it will continue to grow until N = Nmax. (b) Solitary behavior (i.e., N = 1) is predicted if costs increase more than benefits as group size increases. (c) A group must be largerthan Nmin in orderto persist and grow, whereas it is goes toward to extinction if N falls below Nmin because the relative growth rate is negative (the so-called Allee effect).

individual will increase. The optimal group size (denoted Nopt) is reached when r is at maximum (Figure 4b). If Nopt is close to 1, individuals will gain by living alone, for instance, in territories, and their spatial distribution will tend to be regular or random. In contrast, species with high values of Nopt will be patchily distributed. This will apply to species that can adjust their group size in accordance to net benefits, for instance, in animals where individuals can join (if N< Nopt) or leave groups (if N> Nopt). If groups are not formed by such behavioral mechanisms, as with plants, group size will continue to grow until N = Nmax. A special case of eqn  is when benefits and costs balance at two different group sizes, as shown in Figure 4c. When Nis below Nmin, the group will go extinct because the smaller the group the more negative r becomes (the so-called Allee effect) (see Allee Effects). The only way the extinction of such a group can be avoided is by recruiting individuals from outside. 