Appendix 8. Index of attractor width.

The steady-state solution of equation A.1.1, with integer q > 2 and disturbance variance set to zero, is a polynomial in P with one or three positive real roots. An index of size of the low-P attractor, comparable to resilience as defined by Holling (1973), was calculated as follows. At any point in time, given It and Mt, the roots are calculated. If there are three real roots, the size of the low-P attractor is the distance from the low-P root to the unstable intermediate root. If there is one real root, it is either on the low-P limb of the phase space, or on the high-P limb (Fig. 1). If the root is on the high-P limb, then the size of the low-P attractor is zero. If the root is on the low-P limb, then the size of the low-P attractor is infinite. In this case, we set the attractor size to a relatively large value (2.0 in Figs. 8-11) so that results could be plotted.