APPENDIX

Return times and resilience

It is important to distinguish between behavior near a stable equilibrium and behavior near the boundary of a domain of attraction, which is an unstable equilibrium or separatrix. As discussed in Section 2, the long return times associated with a loss of resilience are caused by slow dynamics near the unstable  equilibrium, not by slow dynamics near the stable  equilibrium point. Unfortunately, there are two conflicting definitions of resilience and consequent confusion about the connection between resilience and return times.

Pimm (1991:13) defines resilience as "how fast a variable that has been displaced from equilibrium returns to it. Resilience could be estimated by a return time, the amount of time taken for the displacement to decay to some specified fraction of its initial value.'' Pimm (1991: 33) describes return to equilibrium by the equation

where is the population density at time , is the initial population density, and is the equilibrium density. The differential equation for that corresponds to this formula is

A similar model with discrete time could be given instead, but that would not alter the following argument. If we measure displacement from by , then satisfies

which is equivalent to Eq. (A.1) if is replaced by . Strictly speaking, Pimm's definition depends upon this simplicity, because the amount of time required for to decay to some specified fraction of its initial value is only constant if the model (A.1) is used. In fact, if the initial displacement is and the fraction is , then (A.1) implies that

from which we conclude that the return time is given by

The remarkable feature is that the magnitude does not appear in this formula. This is a feature of this model only, as we shall see below. In more general circumstances, such a result can be expected to hold only in the limit as . Such results are called "local." As pointed out in Section 2, a common error is to extrapolate local results to global ones. In the present context, it amounts to replacing a complicated function by a linear approximation. Such approximations are certainly easy to work with, but they may miss essential features of the dynamics. In fact, failure to recognize the distinction between local stability and global stability can lead to unwarrented optimism about the likely consequences of interventions in natural systems. If we think that stability to small perturbations necessarily implies stability to large perturbations, then precautions are never required.

In order to distinguish behavior near the equilibrium at from behavior near an unstable equilibrium, we must use a model with more parameters than (A.2). We set

This particular form leads to an especially simple equation for the return time: the time to reach a position starting at is given by

and the form for was chosen so that

as can be verified algebraically. In view of (A.7) and (A.8),

Now, if we replace by , (A.9) becomes

where

If the last two terms in (A.10) are omitted, this result is identical to Pimm's assumption (A.1). Our more complicated dynamical assumption (A.6) is the analogue of Pimm's assumption if there are three equilibria. Under what conditions does (A.10) imply large return times? The first term, which corresponds to Pimm's model, implies a long return time if the ratio is small or if is small. In Pimm's discussion, is a parameter that describes a probe or observation of the system. Ordinarily, is fixed, and the return time provides an estimate for .

The second term in (A.10) implies a long return time if is small or is small. Our previous discussion was concerned with a possibly variable and disturbances that might take the system near an unstable equilibrium. That corresponds to near , or near . In such a case, will be large even if the parameter is large. That is, return times may be long, even for systems that show very rapid return when close to the stable equilibrium. According to this point of view, long return times may be diagnostic for a small or for disturbances that are large enough to take the system near an unstable equilibrium. They may also correspond to weak repulsion from the unstable equilibrium, i. e., small . If a disturbance takes the system beyond the unstable equilibrium, there is no return at all.

In summary, according to Pimm (1991) and according to us, long return times may be diagnostic for a loss of resilience, but the meanings of the terms are quite different in the two cases. Pimm is concerned with behavior near a stable equilibrium. In that case, a long return time for a given displacement from the equilibrium indicates a small coefficient or, equivalently, a small derivative of . We are concerned with behavior of a system with two or three equilibria, one of which is stable. Resilience describes the tendency of the system to return to its stable equilibrium. A long return time is due to disturbances that bring the system near an unstable equilibrium, or possibly to a weak repulsion from an unstable equilibrium.