Reactivity, Attenuation, and Transients in Metapopulations
Abstract
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.
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References
1.
T. P. Adams, R. Proud, and K. D. Black, Connected networks of sea lice populations: Dynamics and implications for control, Aquac. Environ. Interact., 6 (2015), pp. 273--284, https://doi.org/10.3354/aei00133.
2.
J. Arino and P. van den Driessche, A multi-city epidemic model, Math. Popul. Stud., 10 (2003), pp. 175--193, https://doi.org/10.1080/08898480306720.
3.
P. R. Armsworth, Recruitment limitation, population regulation, and larval connectivity in reef fish metapopulations, Ecology, 83 (2002), pp. 1092--1104, https://doi.org/10.1890/0012-9658(2002)083[1092:RLPRAL]2.0.CO;2.
4.
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics Appl. Math. SIAM, Philadelphia, 1994.
5.
M. Bode, L. Bode, and P. R. Armsworth, Larval dispersal reveals regional sources and sinks in the Great Barrier Reef, Mar. Ecol. Prog. Ser., 308 (2006), pp. 17--25.
6.
L. W. Botsford, C. L. Moloney, A. Hastings, J. L. Largier, T. M. Powell, K. Higgins, and J. F. Quinn, The influence of spatially and temporally varying oceanographic conditions on meroplanktonic metapopulations, Deep. Res. Part II, 41 (1994), pp. 107--145, https://doi.org/10.1016/0967-0645(94)90064-7.
7.
J. E. Byers and J. M. Pringle, Going against the flow: Retention, range limits and invasions in advective environments, Mar. Ecol. Prog. Ser., 313 (2006), pp. 27--41, https://doi.org/10.3354/meps313027.
8.
H. Caswell, Matrix Population Models, Vol. 1, Sinauert, Sunderland, MA, 2000.
9.
H. Caswell and M. G. Neubert, Reactivity and transient dynamics of discrete-time ecological systems, J. Difference Equ. Appl., 11 (2005), pp. 295--310, https://doi.org/10.1080/10236190412331335382.
10.
R. K. Cowen, C. B. Paris, and A. Srinivasan, Scaling of connectivity in marine populations, Science, 311 (2006), pp. 522--527, https://doi.org/10.1126/science.1122039.
11.
R. K. Cowen and S. Sponaugle, Larval dispersal and marine population connectivity, Ann. Rev. Mar. Sci., 1 (2009), pp. 443--466, https://doi.org/10.1146/annurev.marine.010908.163757.
12.
J. M. Cushing and O. Diekmann, The many guises of $R_0$ (a didactic note), J. Theoret. Biol., 404 (2016), pp. 295--302, https://doi.org/10.1016/j.jtbi.2016.06.017.
13.
W. F. Figueira, Connectivity or demography: Defining sources and sinks in coral reef fish metapopulations, Ecol. Model., 220 (2009), pp. 1126--1137, https://doi.org/10.1016/j.ecolmodel.2009.01.021.
14.
W. F. Figueira and L. B. Crowder, Defining patch contribution in source-sink metapopulations: The importance of including dispersal and its relevance to marine systems, Popul. Ecol., 48 (2006), pp. 215--224, https://doi.org/10.1007/s10144-006-0265-0.
15.
R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), pp. 445--466, https://doi.org/10.1016/S0006-3495(61)86902-6.
16.
M. Gyllenberg, A. Hastings, and I. Hanski, Structured metapopulation models, in Metapopulation Biology, I. Hanski and M. E. Gilpin, eds., Academic Press, San Diego, 1997, pp. 93--122, https://doi.org/10.1016/B978-012323445-2/50008-0.
17.
I. Hanski and C. D. Thomas, Metapopulation dynamics and conservation: A spatially explicit model applied to butterflies, Biol. Conserv., 68 (1994), pp. 167--180, https://doi.org/10.1016/0006-3207(94)90348-4.
18.
P. D. Harrington and M. A. Lewis, A next-generation approach to calculate source--sink dynamics in marine metapopulations, Bull. Math. Biol., 82 (2020), pp. 1--44, https://doi.org/10.1007/s11538-019-00674-1.
19.
A. Hastings, Transient dynamics and persistence of ecological systems, Ecol. Lett., 4 (2001), pp. 215--220, https://doi.org/10.1046/j.1461-0248.2001.00220.x.
20.
A. Hastings, Transients: The key to long-term ecological understanding?, Trends Ecol. Evol., 19 (2004), pp. 39--45, https://doi.org/10.1016/j.tree.2003.09.007.
21.
A. Hastings, K. C. Abbott, K. Cuddington, T. Francis, G. Gellner, Y. C. Lai, A. Morozov, S. Petrovskii, K. Scranton, and M. L. Zeeman, Transient phenomena in ecology, Science, 361 (2018), https://doi.org/10.1126/science.aat6412.
22.
A. Hastings and K. Higgins, Persistence of transients in spatially structured ecological models, Science, 263 (1994), pp. 1133--1136, https://doi.org/10.1126/science.263.5150.1133.
23.
K. Higgins, A. Hastings, J. N. Sarvela, and L. W. Botsford, Stochastic dynamics and deterministic skeletons: Population behavior of Dungeness crab, Science, 276 (1997), pp. 1431--1434, https://doi.org/10.1126/science.276.5317.1431.
24.
T. Hillen, B. Greese, J. Martin, and G. de Vries, Birth-jump processes and application to forest fire spotting, J. Biol. Dyn., 9 (2015), pp. 104--127, https://doi.org/10.1080/17513758.2014.950184.
25.
Q. Huang, Y. Jin, and M. A. Lewis, $R_0$ analysis of a benthic-drift model for a stream population, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 287--321, https://doi.org/10.1137/15M1014486.
26.
Q. Huang and M. A. Lewis, Homing fidelity and reproductive rate for migratory populations, Theor. Ecol., 8 (2015), pp. 187--205, https://doi.org/10.1007/s12080-014-0243-7.
27.
B. C. Husband and S. C. H. Barrett, A metapopulation perspective in plant population biology, J. Ecol., 84 (1996), pp. 461--469, https://doi.org/10.2307/2261207.
28.
G. Jones, G. Almany, G. Russ, P. Sale, R. Steneck, M. van Oppen, and B. Willis, Larval retention and connectivity among populations of corals and reef fishes: History, advances and challenges, Coral Reefs, 28 (2009), pp. 307--325, https://doi.org/10.1007/s00338-009-0469-9.
29.
M. Krkošek and M. A. Lewis, An $R_0$ theory for source-sink dynamics with application to Dreissena competition, Theor. Ecol., 3 (2010), pp. 25--43, https://doi.org/10.1007/s12080-009-0051-7.
30.
R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control, Bull. Entomol. Soc. Amer., 15 (1969), pp. 237--250, https://doi.org/10.1093/besa/15.3.237.
31.
A. L. Lloyd and R. M. May, Spatial heterogeneity in epidemic models, J. Theoret. Biol., 179 (1996), pp. 1--11, https://doi.org/10.1006/jtbi.1996.0042.
32.
D. Ludwig, D. D. Jones, and C. S. Holling, Qualitative analysis of insect outbreak systems: The spruce budworm and forest, J. Anim. Ecol., 47 (1978), pp. 315--332, https://doi.org/10.2307/3939.
33.
F. Lutscher and X. Wang, Reactivity of communities at equilibrium and periodic orbits, J. Theoret. Biol., 493 (2020), 110240, https://doi.org/10.1016/j.jtbi.2020.110240.
34.
L. Mari, R. Casagrandi, A. Rinaldo, and M. Gatto, A generalized definition of reactivity for ecological systems and the problem of transient species dynamics, Methods Ecol. Evol., 8 (2017), pp. 1574--1584, https://doi.org/10.1111/2041-210X.12805.
35.
H. W. Mckenzie, Y. Jin, J. Jacobsen, and M. A. Lewis, $R_0$ analysis of a spatiotemporal model for a stream population, SIAM J. Appl. Dyn. Syst., 11 (2012), pp. 567--596, https://doi.org/10.1137/100802189.
36.
A. Morozov, K. Abbott, K. Cuddington, T. Francis, G. Gellner, A. Hastings, Y. C. Lai, S. Petrovskii, K. Scranton, and M. L. Zeeman, Long transients in ecology: Theory and applications, Phys. Life Rev., 32 (2020), pp. 1--40, https://doi.org/10.1016/j.plrev.2019.09.004.
37.
R. F. Morris, The dynamics of epidemic spruce budworm populations, Mem. Entomol. Soc. Canad., 95 (1963), pp. 1--12, https://doi.org/10.4039/entm9531fv.
38.
J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), pp. 2061--2070, https://doi.org/10.1109/JRPROC.1962.288235.
39.
M. G. Neubert and H. Caswell, Alternatives to resilience for measuring the responses of ecological systems to perturbations, Ecology, 78 (1997), pp. 653--665, https://doi.org/10.1890/0012-9658(1997)078[0653:ATRFMT]2.0.CO;2.
40.
M. G. Neubert, H. Caswell, and J. D. Murray, Transient dynamics and pattern formation: Reactivity is necessary for Turing instabilities, Math. Biosci., 175 (2002), pp. 1--11, https://doi.org/10.1016/S0025-5564(01)00087-6.
41.
M. G. Neubert, T. Klanjscek, and H. Caswell, Reactivity and transient dynamics of predator-prey and food web models, Ecol. Model., 179 (2004), pp. 29--38, https://doi.org/10.1016/j.ecolmodel.2004.05.001.
42.
S. Noschese, L. Pasquini, and L. Reichel, Tridiagonal Toeplitz matrices: Properties and novel applications, Numer. Linear Algebra Appl., 20 (2013), pp. 302--326, https://doi.org/10.1002/nla.1811.
43.
H. R. Pulliam, Sources, sinks and population regulation, Amer. Nat., 132 (1988), pp. 652--661, https://doi.org/10.1086/284880.
44.
J. Roughgarden, S. Gaines, and H. Possingham, Recruitment dynamics in complex life cycles, Science, 241 (1988), pp. 1460--1466, https://doi.org/10.1126/science.11538249.
45.
L. Safranyik and A. L. Carroll, The biology and epidemiology of the mountain pine beetle in lodgepole pine forests, in The Mountain Pine Beetle: A Synthesis of Biology, Management, and Impacts of Lodgepole Pine, Canadian Forest Service, Victoria, Canada, 2007, pp. 3--66.
46.
R. E. Snyder, What makes ecological systems reactive?, Theor. Popul. Biol., 77 (2010), pp. 243--249, https://doi.org/10.1016/j.tpb.2010.03.004.
47.
I. Stott, M. Franco, D. Carslake, S. Townley, and D. Hodgson, Boom or bust? A comparative analysis of transient population dynamics in plants, J. Ecol., 98 (2010), pp. 302--311, https://doi.org/10.1111/j.1365-2745.2009.01632.x.
48.
I. Stott, S. Townley, and D. J. Hodgson, A framework for studying transient dynamics of population projection matrix models, Ecol. Lett., 14 (2011), pp. 959--970, https://doi.org/10.1111/j.1461-0248.2011.01659.x.
49.
S. J. Theuerkauf, B. J. Puckett, and D. B. Eggleston, Metapopulation dynamics of oysters: Sources, sinks, and implications for conservation and restoration, Ecosphere, 12 (2021), https://doi.org/10.1002/ecs2.3573.
50.
H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, 2003.
51.
H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), pp. 188--211, https://doi.org/10.1137/080732870.
52.
S. Townley, D. Carslake, O. Kellie-Smith, D. Mccarthy, and D. Hodgson, Predicting transient amplification in perturbed ecological systems, J. Appl. Ecol., 44 (2007), pp. 1243--1251, https://doi.org/10.1111/j.1365-2664.2007.01333.x.
53.
S. Townley and D. J. Hodgson, Erratum et addendum: Transient amplification and attenuation in stage-structured population dynamics, J. Appl. Ecol., 45 (2008), pp. 1836--1839, https://doi.org/10.1111/j.1365-2664.2008.01562.x.
54.
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), pp. 29--48, https://doi.org/10.1016/S0025-5564(02)00108-6.
55.
A. Verdy and H. Caswell, Sensitivity analysis of reactive ecological dynamics, Bull. Math. Biol., 70 (2008), pp. 1634--1659, https://doi.org/10.1007/s11538-008-9312-7.
56.
X. Wang, M. Efendiev, and F. Lutscher, How spatial heterogeneity affects transient behavior in reaction--diffusion systems for ecological interactions?, Bull. Math. Biol., 81 (2019), pp. 3889--3917, https://doi.org/10.1007/s11538-019-00659-0.
57.
D. W. Williams and A. M. Liebhold, Spatial synchrony of spruce budworm outbreaks in eastern North America, Ecology, 81 (2000), pp. 2753--2766, https://doi.org/10.1890/0012-9658(2000)081[2753:SSOSBO]2.0.CO;2.
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Published In
SIAM Journal on Applied Dynamical Systems
Pages: 1287 - 1321
DOI: 10.1137/21M140451X
ISSN (online): 1536-0040
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© 2022, Society for Industrial and Applied Mathematics.
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Submitted: 18 March 2021
Accepted: 21 December 2021
Published online: 1 June 2022
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Canada Research Chairs https://doi.org/10.13039/501100001804
Natural Sciences and Engineering Research Council of Canada https://doi.org/10.13039/501100000038
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