• amplification;
  • attenuation;
  • reactivity;
  • stage-structure;
  • projection matrix;
  • transient;
  • asymptotic;
  • population dynamics;
  • Kreiss bound


  • 1
    Not all members of natural populations contribute equally to population growth or decline. Populations that are disturbed away from stable stage structure will amplify (i.e. get bigger than expected) and/or attenuate (i.e. get smaller than expected) in the short term.
  • 2
    We provide mathematical bounds for the magnitude of this amplification and attenuation, both in terms of absolute population change and population change relative to the long-term rate of population increase.
  • 3
    Our results correct an important error in an earlier analysis of transient population amplification, and provide new transient bounds for the analysis of population attenuation.
  • 4
    Synthesis and applications. Bounds on transient amplification and attenuation help population managers to gauge ‘worst case’ and ‘best case’ scenarios for the response of stage-structured populations to disturbance and management strategies. Such bounds help to create an envelope of possible future population scenarios around the mean, long-term predictions made by eigenvalues and eigenvectors of projection matrix models. Transient amplification, caused by stage structures biased towards reactive life stages, may be exploited by conservation managers wishing to boost population densities in the short term and may be avoided in pest species by stage-specific control strategies. Similarly, transient attenuation should be avoided by conservation managers and exploited by pest managers.