Transient dynamics often differ drastically from the asymptotic dynamics of systems. In this paper we analyze transient dynamics in birth-jump metapopulations where dispersal occurs immediately after birth (e.g., via larval dispersal). We address the choice of appropriate norms as well as the effect of stage structure on transient dynamics. We advocate the use of the $\ell_1$ norm, because of its biological interpretation, and extend the transient metrics of reactivity and attenuation to birth-jump metapopulations in this norm. By way of examples we compare this norm to the more commonly used $\ell_2$ norm. Our focus is the case where transient dynamics are very different than asymptotic dynamics. We provide simple examples of metapopulations where this is the case and also show how increasing the number of habitat patches can increase this difference. We then connect the reactivity and attenuation of metapopulations to the source-sink classification of habitat patches and demonstrate how to meaningfully measure reactivity when metapopulations are stage-structured, with a focus on marine metapopulations. Our paper makes three primary contributions. First, it provides guidance to readers as to the appropriate norm and scalings for studying transients in birth-jump metapopulations. Second, it provides three examples of transient behavior in metapopulations involving slow-fast systems, crawl-bys, and high dimensionality. Third, it connects the concepts of reactivity and attenuation to the source-sink classification of habitat patches more commonly found in marine metapopulations.


  1. transient dynamics
  2. metapopulation
  3. reactivity
  4. source-sink dynamics
  5. marine systems

MSC codes

  1. 92D25
  2. 92D40
  3. 34A30
  4. 34C11

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Information & Authors


Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 1287 - 1321
ISSN (online): 1536-0040


Submitted: 18 March 2021
Accepted: 21 December 2021
Published online: 1 June 2022


  1. transient dynamics
  2. metapopulation
  3. reactivity
  4. source-sink dynamics
  5. marine systems

MSC codes

  1. 92D25
  2. 92D40
  3. 34A30
  4. 34C11



Funding Information

Canada Research Chairs https://doi.org/10.13039/501100001804

Funding Information

Natural Sciences and Engineering Research Council of Canada https://doi.org/10.13039/501100000038

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