The kin facilitation hypothesis for red grouse population cycles: territorial dynamics of the family cluster

The kin facilitation hypothesis attributes red grouse population cycles to the formation of spatial assemblages comprising related territory holders, and to a reduction in aggressiveness within such family clusters. This well-documented process is hypothesised to increase annual recruitment into the territorial population and the resulting increase in density leads to increased aggressiveness within the population. After some years, the cumulative increase in density and aggressiveness is assumed to lead to the breakdown of the family clusters, inflation of territorial requirements and consequent population decline. The territorial dynamics of a single family cluster constitute an important and little investigated part of the hypothesis. We develop a simple deterministic model to examine the effects of crowding and family size on the formation of a single family cluster. Analysis of two versions of the model, one containing no territory sharing between neighbouring relatives and one containing a territory-sharing response function developed elsewhere, indicates that a continuous increase in crowding has a discontinuous effect on the ability of the cluster to form. In the case of the model containing territory sharing this change is irreversible as, due to the occurrence of multiple equilibria, solutions are sensitive to initial conditions. We discuss the implications of this result for the kin facilitation hypothesis.