A framework for studying transient dynamics of population projection matrix models

Empirical models are central to effective conservation and population management, and should be predictive of real?world dynamics. Available modelling methods are diverse, but analysis usually focuses on long?term dynamics that are unable to describe the complicated short?term time series that can arise even from simple models following ecological disturbances or perturbations. Recent interest in such transient dynamics has led to diverse methodologies for their quantification in density?independent, time?invariant population projection matrix (PPM) models, but the fragmented nature of this literature has stifled the widespread analysis of transients. We review the literature on transient analyses of linear PPM models and synthesise a coherent framework. We promote the use of standardised indices, and categorise indices according to their focus on either convergence times or transient population density, and on either transient bounds or case?specific transient dynamics. We use a large database of empirical PPM models to explore relationships between indices of transient dynamics. This analysis promotes the use of population inertia as a simple, versatile and informative predictor of transient population density, but criticises the utility of established indices of convergence times. Our findings should guide further development of analyses of transient population dynamics using PPMs or other empirical modelling techniques.</p

Beyond sensitivity: nonlinear perturbation analysis of transient dynamics

1. Perturbation analyses of population models are integral to population management: such analyses evaluate how changes in vital rates of members of the population translate to changes in population dynamics. Sensitivity and elasticity analyses of long?term (asymptotic) growth are popular, but limited: they ignore short?term (transient) dynamics and provide a linear approximation to nonlinear perturbation curves.2. Population inertia measures how much larger or smaller a non?stable population becomes compared with an equivalent stable population, as a result of transient dynamics. We present formulae for the transfer function of population inertia, which describes nonlinear perturbation curves of transient population dynamics. The method comfortably fits into wider frameworks for analytical study of transient dynamics, and for perturbation analyses that use the transfer function approach.3. We use case studies to illustrate how the transfer function of population inertia may be used in population management. These show that strategies based solely on asymptotic perturbation analyses can cause undesirable transient dynamics and/or fail to exploit desirable transient dynamics. This highlights the importance of considering both transient and asymptotic population dynamics in population management.4. Our case studies also show a tendency towards marked nonlinearity in transient perturbation curves. We extend our method to measure sensitivity of population inertia and show that it often fails to capture dynamics resulting from perturbations typical of management scenarios.</p

A framework for studying transient dynamics of population projection matrix models

Empirical models are central to effective conservation and population management, and should be predictive of real?world dynamics. Available modelling methods are diverse, but analysis usually focuses on long?term dynamics that are unable to describe the complicated short?term time series that can arise even from simple models following ecological disturbances or perturbations. Recent interest in such transient dynamics has led to diverse methodologies for their quantification in density?independent, time?invariant population projection matrix (PPM) models, but the fragmented nature of this literature has stifled the widespread analysis of transients. We review the literature on transient analyses of linear PPM models and synthesise a coherent framework. We promote the use of standardised indices, and categorise indices according to their focus on either convergence times or transient population density, and on either transient bounds or case?specific transient dynamics. We use a large database of empirical PPM models to explore relationships between indices of transient dynamics. This analysis promotes the use of population inertia as a simple, versatile and informative predictor of transient population density, but criticises the utility of established indices of convergence times. Our findings should guide further development of analyses of transient population dynamics using PPMs or other empirical modelling techniques.</p

Beyond sensitivity: nonlinear perturbation analysis of transient dynamics

1. Perturbation analyses of population models are integral to population management: such analyses evaluate how changes in vital rates of members of the population translate to changes in population dynamics. Sensitivity and elasticity analyses of long?term (asymptotic) growth are popular, but limited: they ignore short?term (transient) dynamics and provide a linear approximation to nonlinear perturbation curves.2. Population inertia measures how much larger or smaller a non?stable population becomes compared with an equivalent stable population, as a result of transient dynamics. We present formulae for the transfer function of population inertia, which describes nonlinear perturbation curves of transient population dynamics. The method comfortably fits into wider frameworks for analytical study of transient dynamics, and for perturbation analyses that use the transfer function approach.3. We use case studies to illustrate how the transfer function of population inertia may be used in population management. These show that strategies based solely on asymptotic perturbation analyses can cause undesirable transient dynamics and/or fail to exploit desirable transient dynamics. This highlights the importance of considering both transient and asymptotic population dynamics in population management.4. Our case studies also show a tendency towards marked nonlinearity in transient perturbation curves. We extend our method to measure sensitivity of population inertia and show that it often fails to capture dynamics resulting from perturbations typical of management scenarios.</p