A framework for studying transients in marine metapopulations

Transient dynamics can often differ drastically from the asymptotic dynamics of systems. In this paper we provide a unifying framework for analysing transient dynamics in marine metapopulations, from the choice of norms to the addition of stage structure. We use the l1 norm, because of its biological interpretation, to extend the transient metrics of reactivity and attenuation to marine metapopulations, and use examples to compare these metrics under the more commonly used l2 norm. We then connect the reactivity and attenuation of marine metapopulations to the source-sink distribution of habitat patches and demonstrate how to meaningfully measure reactivity when metapopulations are stage-structured.

[1]  Will F. Figueira,et al.  Defining patch contribution in source-sink metapopulations: the importance of including dispersal and its relevance to marine systems , 2009, Population Ecology.

[2]  Claire B Paris-Limouzy,et al.  Scaling of Connectivity in Marine Populations , 2006, Science.

[3]  J. Roughgarden,et al.  Recruitment dynamics in complex life cycles. , 1988, Science.

[4]  A. Hastings.  Transients: the key to long-term ecological understanding? , 2004, Trends in ecology & evolution.

[5]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[6]  Paul R. Armsworth,et al.  RECRUITMENT LIMITATION, POPULATION REGULATION, AND LARVAL CONNECTIVITY IN REEF FISH METAPOPULATIONS , 2002 .

[7]  Ilkka Hanski,et al.  Metapopulation dynamics and conservation: A spatially explicit model applied to butterflies , 1994 .

[8]  Yu Jin,et al.  R0 Analysis of a Benthic-Drift Model for a Stream Population , 2016, SIAM J. Appl. Dyn. Syst..

[9]  Yu Jin,et al.  R0 Analysis of a Spatiotemporal Model for a Stream Population , 2012, SIAM J. Appl. Dyn. Syst..

[10]  A. Hastings.  Transient dynamics and persistence of ecological systems , 2001 .

[11]  R. F. Morris.  The Dynamics of Epidemic Spruce Budworm Populations , 1963 .

[12]  S. Townley,et al.  Erratum et addendum: transient amplification and attenuation in stage-structured population dynamics , 2008 .

[13]  Horst R. Thieme,et al.  Mathematics in Population Biology , 2003 .

[14]  Lothar Reichel,et al.  Tridiagonal Toeplitz matrices: properties and novel applications , 2013, Numer. Linear Algebra Appl..

[15]  J. Arino,et al.  A multi-city epidemic model , 2003 .

[16]  M. Lewis,et al.  A Next-Generation Approach to Calculate Source–Sink Dynamics in Marine Metapopulations , 2020, Bulletin of Mathematical Biology.

[17]  R. Steneck,et al.  Larval retention and connectivity among populations of corals and reef fishes: history, advances and challenges , 2009, Coral Reefs.

[18]  H. Caswell,et al.  Transient dynamics and pattern formation: reactivity is necessary for Turing instabilities. , 2002, Mathematical biosciences.

[19]  A. Rinaldo,et al.  A generalized definition of reactivity for ecological systems and the problem of transient species dynamics , 2017 .

[20]  M. Lewis,et al.  Homing fidelity and reproductive rate for migratory populations , 2015, Theoretical Ecology.

[21]  S. Yoshizawa,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 1962, Proceedings of the IRE.

[22]  B. Husband,et al.  A METAPOPULATION PERSPECTIVE IN PLANT POPULATION BIOLOGY , 1996 .

[23]  David W. Williams,et al.  SPATIAL SYNCHRONY OF SPRUCE BUDWORM OUTBREAKS IN EASTERN NORTH AMERICA , 2000 .

[24]  G. de Vries,et al.  Birth-jump processes and application to forest fire spotting , 2015, Journal of biological dynamics.

[25]  Martin Krkošek,et al.  An R0 theory for source–sink dynamics with application to Dreissena competition , 2010, Theoretical Ecology.

[26]  H. Caswell,et al.  ALTERNATIVES TO RESILIENCE FOR MEASURING THE RESPONSES OF ECOLOGICAL SYSTEMS TO PERTURBATIONS , 1997 .

[27]  F. Lutscher,et al.  Reactivity of communities at equilibrium and periodic orbits. , 2020, Journal of theoretical biology.

[28]  Stuart Townley,et al.  A framework for studying transient dynamics of population projection matrix models. , 2011, Ecology letters.

[29]  S. Townley,et al.  Boom or bust? A comparative analysis of transient population dynamics in plants , 2010 .

[30]  Bill Broyles.  Sources , 1976, Journal of the Southwest.

[31]  Karen C. Abbott,et al.  Long transients in ecology: Theory and applications. , 2019, Physics of life reviews.

[32]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[33]  S. Townley,et al.  Predicting transient amplification in perturbed ecological systems , 2007 .

[34]  A. Hastings,et al.  Stochastic Dynamics and Deterministic Skeletons: Population Behavior of Dungeness Crab , 1997 .

[35]  A. Hastings,et al.  Persistence of Transients in Spatially Structured Ecological Models , 1994, Science.

[36]  R. Levins.  Some Demographic and Genetic Consequences of Environmental Heterogeneity for Biological Control , 1969 .

[37]  Ying-Cheng Lai,et al.  Transient phenomena in ecology , 2018, Science.

[38]  M. Tsatsomeros.  Matrix Equalities and Inequalities , 2006 .

[39]  A. Carroll,et al.  The biology and epidemiology of the mountain pine beetle in lodgepole pine forests. , 2006 .

[40]  A. Hastings,et al.  The influence of spatially and temporally varying oceanographic conditions on meroplanktonic metapopulations , 1994 .

[41]  J. Cushing,et al.  The many guises of R0 (a didactic note). , 2016, Journal of theoretical biology.

[42]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[43]  H. Caswell,et al.  Reactivity and transient dynamics of discrete-time ecological systems , 2005 .

[44]  Robin E. Snyder.  What makes ecological systems reactive? , 2010, Theoretical population biology.

[45]  Hal Caswell,et al.  Reactivity and transient dynamics of predator–prey and food web models , 2004 .

[46]  R. Cowen,et al.  Larval dispersal and marine population connectivity. , 2009, Annual review of marine science.

[47]  A L Lloyd,et al.  Spatial heterogeneity in epidemic models. , 1996, Journal of theoretical biology.

[48]  H. Pulliam,et al.  Sources, Sinks, and Population Regulation , 1988, The American Naturalist.

[49]  C. S. Holling,et al.  Qualitative Analysis of Insect Outbreak Systems: The Spruce Budworm and Forest , 1978 .

[50]  K. Black,et al.  Connected networks of sea lice populations: dynamics and implications for control , 2015 .

[51]  H. Caswell,et al.  Sensitivity Analysis of Reactive Ecological Dynamics , 2008, Bulletin of mathematical biology.

[52]  Horst R. Thieme,et al.  Spectral Bound and Reproduction Number for Infinite-Dimensional Population Structure and Time Heterogeneity , 2009, SIAM J. Appl. Math..