Socially Induced Red Grouse Population Cycles Need Abrupt Transitions Between Tolerance and Aggression

Recent field experiments tested the hypothesis that variations in the aggressiveness of territorial cocks drive Red Grouse (Lagopus lagopus scoticus) population cycles. The implications of these results were previously explored with parametrically flexible models that made specific assumptions about the functional form of direct density dependence and the form and timing of delayed density dependence. Although these models were characterized by apparently different stability conditions, they pointed at the same conclusion: that the occurrence of population cycles under this hypothesis relies on the strength of the interaction between, density and aggressiveness around the system's equilibrium. To investigate if this important result is valid more generally, we develop a minimally specified model by lifting most of the assumptions on direct and delayed density dependence. Stability analysis of this functionally flexible model confirms that unstable dynamics are indeed more likely if small perturbations from equilibrium density have a strong impact on aggressiveness, and it unifies the stability conditions previously derived for the more specific models under a single, general condition. Further, we derive global, necessary, and sufficient conditions for instability and express them in terms of proportional changes in aggressiveness and density. For the first time since the inception of the hypothesis, the necessary condition quantifies the minimum strength of the intrinsic mechanism that would be required to cause unstable dynamics. We predict that unstable population dynamics are possible if proportional perturbations from equilibrium density are at least matched by proportional changes in aggressiveness. Existing field data indicate that the necessary condition for intrinsic cycles is satisfied in Red Grouse populations. In contrast, the sufficient condition is considerably more strict, implying that intrinsic instability is not an inevitable feature of the system. We conclude that the model is consistent with the demographic patterns of cyclic population fluctuations in Red Grouse and other birds of the grouse family.