Mechanistic Models of Conflict between Ant Colonies and Their Consequences for Territory Scaling

Territory size in social insects depends on the rules by which border conflicts are resolved. We present three mechanistic mathematical models of conflict, inspired by the behavior of the pavement ant Tetramorium immigrans, to predict the advantage of larger colonies in pairwise contests and the resulting scaling of territory size with worker force. The models track the number of ants in the nest traveling to and from the boundary or engaged at the boundary. Ants at the boundary base their recruitment response on the relative numbers of ants from the two colonies. With two colonies, our central result is that the larger colony gains a territory disproportionately larger than the ratio of worker forces would indicate. This disproportionate territory control determines the scaling relation of territory size with worker force in a population. In two dimensions, if territory size were proportional to worker force, the slope of the scaling relation between log territory size and log worker force would be 1.0. With disproportionate territories, this slope is larger and can be explicitly approximated in terms of model parameters, and it is steepest when colonies are packed close to each other, when ants run quickly, or when colonies are small. A steeper slope exaggerates the advantage of larger colonies, creating a positive feedback that could amplify the inequality of the worker force distribution.