Alternatives to resilience for measuring the responses of ecological systems to perturbations

Ecology, April, 1997 by Michael G. Neubert, Hal Caswell

INTRODUCTION

Ecological systems are subject to continual perturbation. Their responses to perturbation are characterized qualitatively by stability (does the system return to its original state after perturbation, or does it not?), and quantitatively by resilience, or its reciprocal return time, which measure how rapidly a stable system returns to its original state after a perturbation (Holling 1973, Webster et al. 1975, Beddington et al. 1976, Harrison 1979, DeAngelis 1980, 1992, Pimm 1982, 1984, 1991). Theoretical and experimental ecologists have studied how resilience is affected by ecosystem characteristics including energy flow (O'Neill 1976, DeAngelis 1980), nutrient loads and nutrient cycling (Harwell et al. 1977, DeAngelis 1980, DeAngelis et al. 1989a, b, Steinman et al. 1991, Cottingham and Carpenter 1994, Loreau 1994), environmental stochasticity (Ives 1995), life history strategies (Leps et al. 1982), food chain length (Pimm and Lawton 1977, Vincent and Anderson 1979, DeAngelis et al. 1989a, Steinman et al. 1991, Carpenter et al. 1992, Cottingham and Carpenter 1994), food web connectance (Pimm 1979, Leps et al. 1982) and connectivity (Armstrong 1982), herbivory (Lee and Inman 1975), and omnivory (Pimm and Lawton 1978, Pimm 1979).

A number of indices have been suggested for measuring resilience in model ecosystems (Patten and Witkamp 1967, Jordan et al. 1972, Pimm and Lawton 1977, Harte 1979, DeAngelis 1980). The most frequently used and easily calculated index is based on the eigenvalues of the system near its equilibrium. Consider a linear system

dx / dt = Ax x(0) = [x.sub.0] (1)

which may represent either an intrinsically linear system (such as certain compartment models) or the linearization of a nonlinear system near an equilibrium point. Eq. 1 has the unique solution

x(t) = [e.sup.At][x.sub.0]. (2)

When the eigenvalues of A are all negative, then [e.sup.At] [approaches] 0 as t [approaches] [infinity] and the equilibrium solution [x.sup.*] = 0 is asymptotically stable. In fact, for almost all initial conditions,

[Mathematical Expression Omitted] (3)

where [[Lambda].sub.1](A) is the eigenvalue of A with largest real part and [w.sub.1] is the corresponding eigenvector. Thus x is approximately proportional to [w.sub.1] and decays like [e.sup.[[Lambda].sub.1](A)t] for t large.

Because, asymptotically, the magnitude of x decreases by the factor 1/e in a time interval of length -1/Re([[Lambda].sub.1](A)), Pimm and Lawton (1977) used -1/Re([[Lambda].sub.1](A)) as a measure of return time. Thus resilience, defined as

resilience [equivalence] -Re([[Lambda].sub.1](A)) (4)

is an asymptotic approximation of the decay rate of perturbations to the linear system (Eq. 1). The larger the resilience, the faster perturbations eventually decay. Expression 4, or the equivalent version for a discrete-time system, is widely used (e.g., Beddington et al. 1976, Pimm and Lawton 1977, 1978, Harwell and Ragsdale 1979, Pimm 1979, 1982, 1984, Vincent and Anderson 1979, DeAngelis 1980, Harwell et al. 1981, Armstrong 1982, DeAngelis et al. 1989a, b, Carpenter et al. 1992, Nakajima 1992, Cottingham and Carpenter 1994, Loreau 1994).

Resilience, measured in terms of the dominant eigenvalue of A (Expression 4), is an asymptotic property, giving the rate of decay of perturbations as t [approaches] [infinity]. Short-term transient behavior, immediately after the perturbation, is ignored. The question arises whether asymptotic behavior adequately characterizes the response to perturbations. Because of the short duration of many ecological experiments, transients may dominate the observed responses to perturbations. In addition, transient responses may be at least as important as asymptotic responses. Managers charged with ecosystem restoration, for example, are likely to be interested in both the short-term and long-term effects of their manipulations, particularly if the short-term effects can be large (National Research Council 1992). It is therefore the goal of this article to find mathematically simple measures of transient responses that complement resilience as a measure of the response to perturbation.

Our analysis shows that even in stable, resilient systems, transient behavior can be dramatic, long lasting, and counterintuitive. Although a perturbation eventually decays, its size can grow rapidly at first, and the growth can continue for times on the order of the return time. This transient growth is not the result of nonlinearity, although nonlinearity can enhance the effect. It is a generic characteristic of linear or linearized equations with asymptotically stable equilibria and should therefore be common in ecosystem models.

Here is an example. Consider the solution to the system (Eq. 1) when A is either

[Mathematical Expression Omitted] or [Mathematical Expression Omitted]. (5)

Measuring the size of the solution by the Euclidean norm, i.e.,

[Mathematical Expression Omitted] (6)

we can examine the dynamics under each matrix following a perturbation for which [[[x.sub.d]]] = 1. Since [A.sub.1] and [A.sub.2] have the same eigenvalues ([[Lambda].sub.1] = -1, [[Lambda].sub.2] = -2), they have the same resilience, and [[[x(t)]]] asymptotically approaches zero at the same rate for each. However, as shown in Fig. 1a, the solutions initially behave differently. One solution exhibits transient growth, temporarily moving farther away from the equilibrium, despite the negative eigenvalues. Fig. lb shows that the same phenomenon can occur when the eigenvalues are complex by comparing