Skip to main content
SpringerLink
Log in

Transient Responses to Spatial Perturbations in Advective Systems

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

We study the transient dynamics, following a spatially-extended perturbation of models describing populations residing in advective media such as streams and rivers. Our analyses emphasize metrics that are independent of initial perturbations—resilience, reactivity, and the amplification envelope—and relate them to component spatial wavelengths of the perturbation using spatial Fourier transforms of the state variables. This approach offers a powerful way of understanding the influence of spatial scale on the initial dynamics of a population following a spatially variable environmental perturbation, an important property in determining the ecological implications of transient dynamics in advective systems. We find that asymptotically stable systems may exhibit transient amplification of perturbations (i.e., have positive reactivity) for some spatial wavelengths and not others. Furthermore, the degree and duration of amplification varies strongly with spatial wavelength. For two single-population models, there is a relationship between transient dynamics and the response length that characterizes the steady state response to spatial perturbations: a long response length implies that peak amplification of perturbations is small and occurs fast. This relationship holds less generally in a specialist consumer-resource model, likely due to the model’s tendency for flow-induced instabilities at an alternative characteristic spatial scale.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Anderson, K.E., Nisbet, R.M., Diehl, S., Cooper, S.D., 2005. Scaling population responses to spatial environmental variability in advection-dominated systems. Ecol. Lett. 8, 933–43.

    Article  Google Scholar 

  • Anderson, K.E., Nisbet, R.M., Diehl, S., 2006. Spatial scaling of consumer-resource interactions in advection-dominated systems. Am. Nat. 168, 358–72.

    Article  Google Scholar 

  • Botton, S., van Heusden, M., Parsons, J.R., Smidt, H., van Straalen, N., 2006. Resilience of microbial systems towards disturbances. Crit. Rev. Microbiol. 32, 101–12.

    Article  Google Scholar 

  • Cooper, S.D., Diehl, S., Kratz, K., Sarnelle, O., 1998. Implications of scale for patterns and processes in stream ecology. Aust. J. Ecol. 23, 27–0.

    Article  Google Scholar 

  • Diehl, S., Cooper, S.D., Kratz, K.W., Nisbet, R.M., Roll, S.K., Wiseman, S.W., Jenkins, T.M. Jr., 2000. Effects of multiple, predator-induced behaviors on short-term producer-grazer dynamics in open systems. Am. Nat. 156, 293–13.

    Article  Google Scholar 

  • Diehl, S., Anderson, K.E., Nisbet, R.M., 2008. Population responses of drifting stream invertebrates to spatial environmental variability: new theoretical developments. In: Lancaster, J., Briers, R.A. (Eds.), Aquatic Insects: Challenges to Populations. CABI Publishing, invited chapter, in press.

  • Eisenman, I., 2005. Non-normal effects on salt-finger growth. J. Phys. Oceanogr. 35, 616–27.

    Article  Google Scholar 

  • Elliott, J.M., 1971. The distances traveled by drifting invertebrates in a Lake District stream. Oecologia 6, 350–79.

    Article  Google Scholar 

  • Englund, G., Cooper, S.D., Sarnelle, O., 2001. Application of a model of scale dependence to quantify scale domains in open predation experiments. Oikos 92, 501–14.

    Article  Google Scholar 

  • Fisher, S.G., Grimm, N.B., Marti, E., Holmes, R.M., Jones, J.B. Jr., 1998. Material spiraling in stream corridors: A telescoping ecosystem model. Ecosystems 1, 19–4.

    Article  Google Scholar 

  • Gaylord, B., Gaines, S.D., 2000. Temperature or transport? Range limits in marine species mediated solely by flow. Am. Nat. 155, 769–89.

    Article  Google Scholar 

  • Gurney, W.S.C., Nisbet, R.M., 1998. Ecological Dynamics. Oxford University Press, New York.

    Google Scholar 

  • Holling, C.S., 1973. Resilience and stability of ecological systems. Annu. Rev. Ecol. Syst. 4, 1–3.

    Article  Google Scholar 

  • Humborg, C., Conley, D.J., Rahm, L., Wulff, F., Cociasu, A., Ittekkot, V., 2000. Silicon retention in river basins: far-reaching effects on biogeochemistry and aquatic food webs in coastal marine environments. Ambio 29, 45–0.

    Google Scholar 

  • Ives, A.R., Dennis, B., Cottingham, K.L., Carpenter, S.R., 2003. Estimating community stability and ecological interactions from time-series data. Ecol. Monogr. 73, 301–30.

    Article  Google Scholar 

  • Kim, L., Moehlis, J., 2006. Transient growth for streak-streamwise vortex interactions. Phys. Lett. A 358, 431–37.

    Article  MATH  Google Scholar 

  • Kot, M., 2001. Elements of Mathematical Ecology. Cambridge University Press, New York.

    Google Scholar 

  • Levine, J.M., 2003. A patch modeling approach to the community-level consequences of directional dispersal. Ecology 84, 1215–224.

    Article  Google Scholar 

  • Lutscher, F., Pachepsky, E., Lewis, M.A., 2005. The effect of dispersal patterns on stream populations. SIAM J. Appl. Math. 65, 1305–327.

    Article  MATH  MathSciNet  Google Scholar 

  • Lutscher, F., Lewis, M.A., McCauley, E., 2006. Effects of heterogeneity on spread and persistence in rivers. Bull. Math. Biol. 68, 2129–160.

    Article  MathSciNet  Google Scholar 

  • Malchow, H., 1995. Flow- and locomotion-induced pattern formation in nonlinear population dynamics. Ecol. Model. 82, 257–64.

    Article  Google Scholar 

  • Malchow, H., 2000. Motional instabilities in prey-predator systems. J. Theor. Biol. 204, 639–47.

    Article  Google Scholar 

  • McGillem, C.D., Cooper, G.R., 1991. Continuous and Discrete Signal and Systems Analysis, 3rd edn. Saunders College Publishing, Philadelphia.

    Google Scholar 

  • McLay, C., 1970. A theory concerning the distance traveled by animals entering the drift of a stream. J. Fish. Res. Board Can. 27, 359–70.

    Google Scholar 

  • Melbourne, B.A., Chesson, P., 2005. Scaling up population dynamics: integrating theory and data. Oecologia 145, 179–87.

    Article  Google Scholar 

  • Mulholland, P.J., Rosemond, A.D., 1992. Periphyton response to longitudinal nutrient depletion in a woodland stream: evidence of upstream downstream linkage. J. North Am. Benthol. Soc. 11, 405–19.

    Article  Google Scholar 

  • Murdoch, W.W., Kendall, B.E., Nisbet, R.M., Briggs, C.J., McCauley, E., Bolser, R., 2002. Single-species models for many species food webs. Nature 417, 541–43.

    Article  Google Scholar 

  • Murray, J.D., 2003. Mathematical Biology, 3rd edn. Springer, New York.

    MATH  Google Scholar 

  • Neubert, M.G., Caswell, H., 1997. Alternatives to resilience for measuring the responses of ecological systems to perturbations. Ecology 78, 653–65.

    Google Scholar 

  • Neubert, M.G., Caswell, H., Murray, J.D., 2002. Transient dynamics and pattern formation: reactivity is necessary for Turing instabilities. Math. Biosci. 175, 1–1.

    Article  MATH  MathSciNet  Google Scholar 

  • Neubert, M.G., Klanjscek, T., Caswell, H., 2004. Reactivity and transient dynamics of predator-prey and food web models. Ecol. Model. 179, 29–8.

    Article  Google Scholar 

  • Nisbet, R.M., Gurney, W.S.C., 2003. Modelling Fluctuating Populations. Blackburn Press, Caldwell.

    Google Scholar 

  • Nisbet, R.M., Diehl, S., Wilson, W.G., Cooper, S.D., Donalson, D.D., Kratz, K.W., 1997. Primary productivity gradients and short-term population dynamics in open systems. Ecol. Monogr. 67, 535–53.

    Article  Google Scholar 

  • Nisbet, R.M., Anderson, K.E., McCauley, E., Lewis, M.A., 2007. Response of equilibrium states to spatial environmental heterogeneity in advective systems. Math. Biosci. Eng. 4, 1–3.

    MATH  MathSciNet  Google Scholar 

  • Pachepsky, E., Lutscher, F., Nisbet, R.M., Lewis, M.A., 2005. Persistence, spread and the drift paradox. Theor. Popul. Biol. 67, 61–3.

    Article  MATH  Google Scholar 

  • Poff, N.L., Ward, J.V., 1989. Implications of streamflow variability and predictability for lotic community structure: a regional analysis of streamflow patterns. Can. J. Fish. Aquat. Sci. 46, 1805–818.

    Article  Google Scholar 

  • Rovinsky, A.B., Adiwidjaja, H., Yakhnin, V.Z., Menzinger, M., 1997. Patchiness and enhancement of productivity in plankton ecosystems due to the differential advection of predator and prey. Oikos 78, 101–06.

    Article  Google Scholar 

  • Shanks, A.L., Eckert, G.L., 2005. Population persistence of California current fishes and benthic crustaceans: a marine drift paradox. Ecol. Monogr. 75, 505–24.

    Article  Google Scholar 

  • The Mathworks, Inc., 2006. Matlab, version 7.1. Natick, Massachusetts, USA.

  • Thorp, J.H., Thoms, M.C., Delong, M.D., 2006. The riverine ecosystem synthesis: biocomplexity in river networks across space and time. River Res. Appl. 22, 123–47.

    Article  Google Scholar 

  • Townsend, C.R., 1989. The patch dynamics concept of stream community ecology. J. North Am. Benthol. Soc. 8, 36–0.

    Article  Google Scholar 

  • Vannote, R.L., Minshall, G.W., Cummins, K.W., Sedell, J.R., Cushing, C.E., 1980. The river continuum concept. Can. J. Fish. Aquat. Sci. 37, 130–37.

    Article  Google Scholar 

  • Waters, T.F., 1965. Interpretation of invertebrate drift in streams. Ecology 46, 327–34.

    Article  Google Scholar 

  • Waters, T.F., 1972. The drift of stream insects. Ann. Rev. Entomol. 17, 253–72.

    Article  Google Scholar 

  • Williams, D.D., Williams, N.E., 1993. The upstream/downstream movement paradox of lotic invertebrates: quantitative evidence from a Welsh mountain stream. Freshw. Biol. 30, 199–18.

    Article  Google Scholar 

  • Wolfram Research, Inc., 2005. Mathematica, version 5.2. Champaign, Illinois, USA.

  • Woodward, G., Hildrew, A.G., 2002. Food web structure in riverine landscapes. Freshw. Biol. 47, 777–98.

    Article  Google Scholar 

Download references

Author information

Author notes

  1. Kurt E. Anderson

    Present address: Department of Biology, University of California, Riverside, CA, 92521, USA

Authors and Affiliations

  1. Department of Ecology, Evolution, and Marine Biology, University of California, Santa Barbara, CA, 93106-9610, USA

    Kurt E. Anderson & Roger M. Nisbet

  2. Department of Biological Sciences, University of Calgary, Calgary, AB, T2N 1N4, Canada

    Kurt E. Anderson & Edward McCauley

Authors
  1. Kurt E. Anderson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roger M. Nisbet
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edward McCauley
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kurt E. Anderson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Anderson, K.E., Nisbet, R.M. & McCauley, E. Transient Responses to Spatial Perturbations in Advective Systems. Bull. Math. Biol. 70, 1480–1502 (2008). https://doi.org/10.1007/s11538-008-9309-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-008-9309-2

Keywords

Search

Navigation