Sudden Shifts in Ecological Systems: Intermittency and Transients in the Coupled Ricker Population Model

Abstract

Many real ecological systems show sudden changes in behavior, phenomena sometimes categorized as regime shifts in the literature. The relative importance of exogenous versus endogenous forces producing regime shifts is an important question. These forces’ role in generating variability over time in ecological systems has been explored using tools from dynamical systems. We use similar ideas to look at transients in simple ecological models as a way of understanding regime shifts. Based in part on the theory of crises, we carefully analyze a simple two patch spatial model and begin to understand from a mathematical point of view what produces transient behavior in ecological systems. In particular, since the tools are essentially qualitative, we are able to suggest that transient behavior should be ubiquitous in systems with overcompensatory local dynamics, and thus should be typical of many ecological systems.

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References

  1. Amritkar, R.E., Gade, P.M., 1993. Wavelength doubling bifurcations in coupled map lattices. Phys. Rev. Lett. 70(22), 3408–3411.

    Article  Google Scholar 

  2. Amritkar, R.E., Gade, P.M., Gangal, A.D., Nandkumaran, V.M., 1991. Stability of periodic-orbits of coupled-map lattices. Phys. Rev. A 44(6), R3407–R3410.

    Article  MathSciNet  Google Scholar 

  3. Anteneodo, C., Pinto, S.E.D., Batista, A.M., Viana, R.L., 2003. Analytical results for coupled-map lattices with long-range interactions. Phys. Rev. E 68(4), 045202.

    Article  Google Scholar 

  4. Astakhov, V.V., Anishchenko, V.S., Shabunin, A.V., 1995. Controlling spatiotemporal chaos in a chain of the coupled logistic maps. IEEE Trans. Circuits Syst. I-Fundam. Theory Appl. 42(6), 352–357.

    Article  Google Scholar 

  5. Atkinson, K.E., 1978. An Introduction to Numerical Analysis. Wiley, New York.

    MATH  Google Scholar 

  6. Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.M., 1978. All Lyapunov characteristic numbers are effectively computable. C. R. Hebd. Seances Acad. Sci. Ser. A 286(9), 431–433.

    MATH  MathSciNet  Google Scholar 

  7. Bjornstad, O.N., Grenfell, B.T., 2001. Noisy clockwork: time series analysis of population fluctuations in animals. Science 293(5530), 638–643.

    Article  Google Scholar 

  8. Carpenter, S.R., Brock, W.A., 2006. Rising variance: a leading indicator of ecological transition. Ecol. Lett. 9(3), 308–315.

    Google Scholar 

  9. De Monte, S., d’Ovidio, F., Chate, H., Mosekilde, E., 2004. Noise-induced macroscopic bifurcations in globally coupled chaotic units. Phys. Rev. Lett. 92(25), 254101.

    Article  Google Scholar 

  10. Eckmann, J.-P., Ruelle, D., 1985. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656.

    Article  MathSciNet  Google Scholar 

  11. Feigenbaum, M.J., 1978. Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25–52.

    MATH  Article  MathSciNet  Google Scholar 

  12. Gade, P.M., Amritkar, R.E., 1993. Spatially periodic-orbits in coupled-map lattices. Phys. Rev. E 47(1), 143–154.

    Article  MathSciNet  Google Scholar 

  13. Grebogi, C., Ott, E., Yorke, J.A., 1982. Chaotic attractors in crisis. Phys. Rev. Lett. 48(22), 1507–1510.

    Article  MathSciNet  Google Scholar 

  14. Grebogi, C., Ott, E., Yorke, J.A., 1983. Crises, sudden changes in chaotic attractors, and transient chaos. Physica D 7(1–3), 181–200.

    Article  MathSciNet  Google Scholar 

  15. Grebogi, C., Ott, E., Yorke, J.A., 1986. Critical exponent of chaotic transients in nonlinear dynamic-systems. Phys. Rev. Lett. 57(11), 1284–1287.

    Article  MathSciNet  Google Scholar 

  16. Grebogi, C., Ott, E., Romeiras, F., Yorke, J.A., 1987. Critical exponents for crisis-induced intermittency. Phys. Rev. A 36(11), 5365–5380.

    Article  MathSciNet  Google Scholar 

  17. Guckenheimer, J., Holmes, P., 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences, vol. 42. Springer, New York.

    MATH  Google Scholar 

  18. Gyllenberg, M., Söderbacka, G., Ericsson, S., 1993. Does migration stabilize local population dynamics? Analysis of a discrete metapopulation model. Math. Biosci. 118, 25–49.

    MATH  Article  MathSciNet  Google Scholar 

  19. Hastings, A., 1982. Dynamics of a single species in a spatially varying environment: the stabilizing role of high dispersal rates. J. Math. Biol. 16, 49–55.

    MATH  MathSciNet  Google Scholar 

  20. Hastings, A., 1995. A metapopulation model with population jumps of varying sizes. Math. Biosci. 128, 285–298.

    MATH  Article  Google Scholar 

  21. Hastings, A., 2004. Transients: the key to long-term ecological understanding? Trends Ecol. Evol. 19, 39–45.

    Article  Google Scholar 

  22. Hastings, A., Higgins, K., 1994. Persistence of transients in spatially structured ecological models. Science 263, 1133–1136.

    Article  Google Scholar 

  23. Hastings, A., Hom, C.L., Ellner, S., Turchin, P., Godfray, H.C.J., 1993. Chaos in ecology—is mother-nature a strange attractor? Annu. Rev. Ecol. Syst. 24, 1–33.

    Google Scholar 

  24. Hsu, G.H., Ott, E., Grebogi, C., 1988. Strange saddles and the dimension of their manifolds. Phys. Lett. A 127, 199–204.

    Article  MathSciNet  Google Scholar 

  25. Janaki, T.M., Rangarajan, G., Habib, S., Ryne, R.D., 1999. Computation of the Lyapunov spectrum for continuous-time dynamical systems and discrete maps. Phys. Rev. E 60, 6614–6626.

    MATH  Article  MathSciNet  Google Scholar 

  26. Kaneko, K., 1992a. Supertransients, spatiotemporal intermittency and stability of fully developed spatiotemporal chaos. Phys. Lett. A 149(2–3), 105–112.

    Google Scholar 

  27. Kaneko, K., 1992b. Overview of coupled map lattices. Chaos 2, 279–282.

    MATH  Article  MathSciNet  Google Scholar 

  28. Kaneko, K., 1993. Chaotic traveling waves in a coupled map lattice. Physica D 68(3–4), 299–317.

    MATH  Article  MathSciNet  Google Scholar 

  29. Katok, A., Hasselblatt, B., 1995. Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and its Applications, vol 54. Cambridge University Press, New York.

    MATH  Google Scholar 

  30. Kendall, B.E., Fox, G.A., 1998. Spatial structure, environmental heterogeneity, and population dynamics: analysis of the coupled logistics map. Theor. Popul. Biol. 54, 11–37.

    MATH  Article  Google Scholar 

  31. Konishi, T., Kaneko, K., 1992. Clustered motion in symplectic coupled map systems. J. Phys. A 25, 6283–6296.

    MATH  Article  MathSciNet  Google Scholar 

  32. Kuznetsov, Y.A., 1998. Elements of Applied Bifurcation Theory. Applied Mathematical Sciences, vol. 112. Springer, New York.

    MATH  Google Scholar 

  33. Labra, F.A., Lagos, N.A., Marquet, P.A., 2003. Dispersal and transient dynamics in metapopulations. Ecol. Lett. 6, 197–204.

    Article  Google Scholar 

  34. Lai, Y.C., 1995. Persistence of supertransients of spatiotemporal chaotic dynamical systems in noisy environment. Phys. Lett. A 200, 418–422.

    Article  Google Scholar 

  35. Lloyd, A.L., 1995. The coupled logistic map: a simple model for the effects of spatial heterogeneity on population dynamics. J. Theor. Biol. 173, 217–230.

    Article  Google Scholar 

  36. Ludwig, D., Jones, D.D., Holling, C.S., 1978. Qualitative-analysis of insect outbreak systems—spruce budworm and forest. J. Animal Ecol. 47(1), 315–332.

    Article  Google Scholar 

  37. Manrubia, S.C., Mikhailov, A.S., 2000. Very long transients in globally coupled maps. Europhys. Lett. 50, 580–586.

    Article  Google Scholar 

  38. May, R.M., 1973. Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton.

    Google Scholar 

  39. Morita, S., 1996. Bifurcations in globally coupled chaotic maps. Phys. Lett. A 211(5), 258–264.

    MATH  Article  MathSciNet  Google Scholar 

  40. Parekh, N., Parthasarathy, S., Sinha, S., 1998. Global and local control of spatiotemporal chaos in coupled map lattices. Phys. Rev. Lett. 81, 1401–1404.

    Article  Google Scholar 

  41. Press, W., Teukolsky, S., Vetterling, W., Flannery, B., 2002. Numerical Reicipes in C++: The Art of Scientific Computing. Cambridge University Press, Cambridge.

    Google Scholar 

  42. Ricker, W., 1954. Stock and recruitment. J. Fish. Res. Board Can. 11, 559–663.

    Google Scholar 

  43. Robinson, C., 1995. Dynamical Systems. CRC Press, Boca Raton.

    MATH  Google Scholar 

  44. Saravia, L.A., Ruxton, G.D., Coviella, C.E., 2000. The importance of transients’ dynamics in spatially extended populations. Proc. Roy. Soc. Lond. 267, 1781–1786.

    Article  Google Scholar 

  45. Scheffer, M., van Nes, E.H., 2004. Mechanisms for marine regime shifts: can we use lakes as microcosms for oceans? Prog. Oceanogr. 60(2–4), 303–319.

    Article  Google Scholar 

  46. Silva, J.A.L., De Castro, M.L., Justo, D.A.R., 2001. Stability in a metapopulation model with density-dependent dispersal. Bull. Math. Biol. 63, 485–505.

    Article  Google Scholar 

  47. Wysham, D.B., Meiss, J.D., 2006. Iterative techniques for computing the linearized manifolds of quasiperiodic tori. Chaos 16(2), 023129.

    Article  MathSciNet  Google Scholar 

  48. Zhu, K.E., Chen, T.L., Bian, G.X., 2003. Controlling spatiotemporal chaos in coupled map lattices to periodic orbits. Commun. Theor. Phys. 40(5), 527–532.

    MathSciNet  Google Scholar 

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Correspondence to Derin B. Wysham.

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This work has been supported by NSF Grant EF-0434266.

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Wysham, D.B., Hastings, A. Sudden Shifts in Ecological Systems: Intermittency and Transients in the Coupled Ricker Population Model. Bull. Math. Biol. 70, 1013–1031 (2008). https://doi.org/10.1007/s11538-007-9288-8

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Keywords

  • Population dynamics
  • Transients
  • Intermittency
  • Coupled-map lattices
  • Crises