Jason Matthiopoulos, John M. Halley, and Robert Moss. 2005. Socially induced Red Grouse population cycles need abrupt transitions between tolerance and aggression. Ecology 86:1883–1893.

Appendix C. Perturbation analysis of the necessary and sufficient conditions for cycles.

In the model, aggressiveness and density are related by the general function

.

(C.1)

As the stability conditions are given in terms of the parameter c, which is the partial derivative  calculated at the system’s equilibrium (), we are interested in using the slope of the function A at equilibrium. We fix the value of aggressiveness at  and examine the function

.

(C.2)

Let  be the proportional change in aggressiveness resulting from a (small) proportional perturbation  of the variable  from the equilibrium value . These unitless quantities are defined as

	(C.3)
and
.	(C.4)

We now recast the necessary and sufficient conditions for unstable dynamics in terms of these two quantities. For small values of  we consider the line that is tangent to the graph of  at the point (). This is given by the expression

.

(C.5)

Using Eq. C.5 we can approximate the quantity  as follows

.

(C.6)

From Eq. C.3 we have  so

.

(C.7)

Placing this result into the conditions in Eqs. 22 and 23

	(C.8)
.	(C.9)

These can be simplified further by noticing that

.

(C.10)

This gives

	(C.11)
.	(C.12)