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.
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.
Anderson, K.E., Nisbet, R.M., Diehl, S., 2006. Spatial scaling of consumer-resource interactions in advection-dominated systems. Am. Nat. 168, 358–72.
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.
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.
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.
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.
Elliott, J.M., 1971. The distances traveled by drifting invertebrates in a Lake District stream. Oecologia 6, 350–79.
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.
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.
Gaylord, B., Gaines, S.D., 2000. Temperature or transport? Range limits in marine species mediated solely by flow. Am. Nat. 155, 769–89.
Gurney, W.S.C., Nisbet, R.M., 1998. Ecological Dynamics. Oxford University Press, New York.
Holling, C.S., 1973. Resilience and stability of ecological systems. Annu. Rev. Ecol. Syst. 4, 1–3.
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.
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.
Kim, L., Moehlis, J., 2006. Transient growth for streak-streamwise vortex interactions. Phys. Lett. A 358, 431–37.
Kot, M., 2001. Elements of Mathematical Ecology. Cambridge University Press, New York.
Levine, J.M., 2003. A patch modeling approach to the community-level consequences of directional dispersal. Ecology 84, 1215–224.
Lutscher, F., Pachepsky, E., Lewis, M.A., 2005. The effect of dispersal patterns on stream populations. SIAM J. Appl. Math. 65, 1305–327.
Lutscher, F., Lewis, M.A., McCauley, E., 2006. Effects of heterogeneity on spread and persistence in rivers. Bull. Math. Biol. 68, 2129–160.
Malchow, H., 1995. Flow- and locomotion-induced pattern formation in nonlinear population dynamics. Ecol. Model. 82, 257–64.
Malchow, H., 2000. Motional instabilities in prey-predator systems. J. Theor. Biol. 204, 639–47.
McGillem, C.D., Cooper, G.R., 1991. Continuous and Discrete Signal and Systems Analysis, 3rd edn. Saunders College Publishing, Philadelphia.
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.
Melbourne, B.A., Chesson, P., 2005. Scaling up population dynamics: integrating theory and data. Oecologia 145, 179–87.
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.
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.
Murray, J.D., 2003. Mathematical Biology, 3rd edn. Springer, New York.
Neubert, M.G., Caswell, H., 1997. Alternatives to resilience for measuring the responses of ecological systems to perturbations. Ecology 78, 653–65.
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.
Neubert, M.G., Klanjscek, T., Caswell, H., 2004. Reactivity and transient dynamics of predator-prey and food web models. Ecol. Model. 179, 29–8.
Nisbet, R.M., Gurney, W.S.C., 2003. Modelling Fluctuating Populations. Blackburn Press, Caldwell.
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.
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.
Pachepsky, E., Lutscher, F., Nisbet, R.M., Lewis, M.A., 2005. Persistence, spread and the drift paradox. Theor. Popul. Biol. 67, 61–3.
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.
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.
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.
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.
Townsend, C.R., 1989. The patch dynamics concept of stream community ecology. J. North Am. Benthol. Soc. 8, 36–0.
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.
Waters, T.F., 1965. Interpretation of invertebrate drift in streams. Ecology 46, 327–34.
Waters, T.F., 1972. The drift of stream insects. Ann. Rev. Entomol. 17, 253–72.
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.
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.
Author information
Authors and Affiliations
Corresponding author
Rights 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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-008-9309-2