## Introduction

Analyses of population and community dynamic models have tended to focus on asymptotic (long-term) model behaviour (Hastings 2004). Recognizing that natural dynamics tend to fluctuate through time, a historical debate (Kendall *et al*. 1999; Turchin 1999) has focused on the relative contribution of endogenous effectors of complicated dynamics (May 1974) and exogenous, stochastic causes of population (Bjornstad & Grenfell 2001) or community (Ives *et al*. 2003; Boyce, Haridas & Lee 2006) fluctuations. Recently, transients have gained theoretical and empirical attention as a third mechanism of complicated dynamics (Hastings 2001, 2004). Transient dynamics are now implicated in promoting species invasions and epidemics (Drake 2005), food web complexity (Chen & Cohen 2001; Caswell & Neubert 2005; Noonburg & Abrams 2005), the amplitude of population cycles (Costantino *et al*. 1998) and species co-existence (Nelson, McCauley & Wrona 2005). A better understanding of transient dynamics should allow their exploitation in pest management, harvesting and conservation strategies.

When natural systems are subject to environmental change and stochastic disturbances, their structure and dynamics may never settle to behaviour predicted by model asymptotics. Instead, populations and communities can show transient amplification, decrease or oscillation, with exogenous perturbations or disturbances resetting the process (Neubert & Caswell 1997; Neubert, Klanjscek & Caswell 2004). Transients can dominate the dynamics of systems that are described by ‘non-normal’ projection matrices. Non-normality is now well-established as a mechanism for strange system behaviour that defies explanation using eigenvalues (Trefethen *et al*. 1993; Trefethen & Embree 2005). Ecological projection matrix models are often extremely non-normal.

This paper introduces the Kreiss bound, an index of transient amplification of ecological dynamics based on stage-structured matrix models (Kreiss 1962), and develops a simple formula showing how this index depends on model parameters. Throughout, the simplicity and utility of the Kreiss bound is compared with two prevailing methods, reactivity and the amplification envelope (Neubert & Caswell 1997), however, we also refer to a wider array of potentially useful indices of transient dynamics (Table 1).

Indicator | What is it? | Use | Parametric | Precision | Comments |
---|---|---|---|---|---|

λ(A) | Max eigenvalue of A | Yes | Poor | Useful only if A is normal | |

Transient bound P_{t} | Upper-bounds future system magnitude | Yes | Good | ||

Asymptotic amplification ρ_{∞}(A) | Yes | Good | Excellent for t >> 1 | ||

Asymptotic amplifier | Often achieves ρ_{t} | Yes | Good | Asymptotic amplification, t→∞ | |

D, reactivity | Yes | Good for t << ∞ | |||

C, reactivity µ(A) | Yes | Good for t << ∞ | e^{µt} useful when combined with P_{t} | ||

Amplification envelope ρ_{t} | No | Excellent | Computational, no meaningful parameterization | ||

Maximum amplification ρ_{max} | max_{t0}ρ_{t} | No | Excellent | Computational, no meaningful parameterization | |

Kreiss bound | Lower bounds ρ_{max} (D: θ= 1; C: θ= 0) | Yes | Good | Compromise |