Transient dynamics and nonlinear fitness: A matrix approach to pulse and press perturbation

Disturbances can occur as short-lived pulses (e.g., storms) or sustained presses (e.g., chronic drought). Much work in ecology has developed methods to help predict how natural populations respond to disturbances, but analyses of pulse and press disturbances have been largely disconnected. We present a matrix framework that links press and pulse perturbations within the same analytical approach, showing how transient nonlinearities and demography shape fitness. We find that transient responses to pulse perturbations accumulate to determine the long-term response to press disturbances. For structured population models, this cumulative change is given by a new transient response matrix (TRM). Strikingly, the TRM also yields the second derivatives of the population growth rate with respect to matrix elements. Thus, there is an intimate but unexpected relationship between nonlinear selection pressures on demographic rates, and the transient dynamics of populations. This relationship yields a strong correlation between TRM and generation time across 439 unique plant and animal species (2690 population models). We also show that the TRM is directly related to Cohen's cumulative distance measure for populations converging to stability. Our framework provides ecologists with a general tool to predict population responses to diverse environmental changes.