×

A three-species model explaining cyclic dominance of Pacific salmon. (English) Zbl 1405.92286

Summary: The four-year oscillations of the number of spawning sockeye salmon (Oncorhynchus nerka) that return to their native stream within the Fraser River basin in Canada are a striking example of population oscillations. The period of the oscillation corresponds to the dominant generation time of these fish. Various – not fully convincing – explanations for these oscillations have been proposed, including stochastic influences, depensatory fishing, or genetic effects. Here, we show that the oscillations can be explained as an attractor of the population dynamics, resulting from a strong resonance near a Neimark Sacker bifurcation. This explains not only the long-term persistence of these oscillations, but also reproduces correctly the empirical sequence of salmon abundance within one period of the oscillations. Furthermore, it explains the observation that these oscillations occur only in sockeye stocks originating from large oligotrophic lakes, and that they are usually not observed in salmon species that have a longer generation time.

MSC:

92D40 Ecology
92D25 Population dynamics (general)
34C23 Bifurcation theory for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Beddington, J.R., Mutual interference between parasites or predators and its effects on searching efficiency, J. anim. ecol., 44, 331-340, (1975)
[2] Blasius, B.; Huppert, A.; Stone, L., Complex dynamics and phase synchronization in spatially extended ecological systems, Nature, 399, 354-359, (1999)
[3] Brown, J.H.; Gillooly, J.F.; Allen, A.P.; Savage, V.M.; West, G.B., Toward a metabolic theory of ecology, Ecology, 85, 1771-1789, (2004)
[4] Elton, C.; Nicholson, M., The ten-year cycle in numbers of lynx in Canada, J. anim. ecol., 11, 215-244, (1942)
[5] Fair, L.F., Critical elements of kvichak river sockeye salmon management, Alaska fish. res. bull., 10, 95-103, (2003)
[6] Gende, S.M.; Edwards, R.T.; Willson, M.F.; Wipfli, M.S., Pacific salmon in aquatic and terrestrial ecosystems, Bioscience, 52, 917-928, (2002)
[7] Godfray, H.J.C.; Hassell, M.P., Discrete and continuous insect populations in tropical environments, J. anim. ecol., 58, 153-174, (1989)
[8] Guill, C., Reichhardt, B., Drossel, B., Just, W. Coexistence of two periodic attractors: the degenerate Neimark Sacker bifurcation as a generic mechanism, in press, Phys. Rev. E, doi:10.1103/PhysRevE.00.001900; Guill, C., Reichhardt, B., Drossel, B., Just, W. Coexistence of two periodic attractors: the degenerate Neimark Sacker bifurcation as a generic mechanism, in press, Phys. Rev. E, doi:10.1103/PhysRevE.00.001900
[9] Gurney, W.S.C.; Nisbet, R.M., Fluctuation periodicity, generation separation, and the expression of larval competition, Theor. popul. biol., 28, 150-180, (1985) · Zbl 0568.92018
[10] Hanski, I.; Turchin, P.; Korpimäki, E.; Henttonen, H., Population oscillations of boreal rodents—regulation by mustelid predators leads to chaos, Nature, 364, 232-235, (1993)
[11] Healey, M., The adaptive significance of age and size at maturity in female sockeye salmon (oncorhynchus nerka), (), 110-117
[12] Hume, J.M.B.; Shortreed, K.S.; Morton, K.F., Juvenile sockeye rearing capacities of three lakes in the fraser river system, Can. J. fish. aquat. sci., 53, 719-733, (1996)
[13] Hume, J., Shortreed, K., Whitehouse, T., 2005. Sockeye fry, smolt, and nursery lake monitoring of Quesnel and Shuswap lakes in 2004. Available at: \( \langle\) http://www.unbc.ca/qrrc/historical_research.html \(\rangle \); Hume, J., Shortreed, K., Whitehouse, T., 2005. Sockeye fry, smolt, and nursery lake monitoring of Quesnel and Shuswap lakes in 2004. Available at: \( \langle\) http://www.unbc.ca/qrrc/historical_research.html \(\rangle \)
[14] Kuznetsov, Y.A., Elements of applied bifurcation theory, (2004), Springer Verlag New York · Zbl 1082.37002
[15] Larkin, P.A., Simulation studies of the Adams river sockeye salmon, oncorhynchus nerka, J. fish. res. board can., 18, 1493-1502, (1971)
[16] Levy, D.A.; Wood, C.C., Review of proposed mechanisms for sockeye salmon population cycles in the fraser river, Bull. math. biol., 54, 241-261, (1992)
[17] Ludwig, D.; Jones, D.D.; Holling, C.S., Qualitative analysis of insect outbreak systems—spruce budworm and forest, J. anim. ecol., 47, 315-332, (1978)
[18] Marsden, A.D.; Martell, S.J.D.; Sumaila, U.R., Retrospective bioeconomic analysis of fraser river sockeye salmon fishery management, Fish. res., 95, 32-41, (2009)
[19] Martell, S.J.D.; Walters, C.J.; Hilborn, R., Retrospective analysis of harvest management performance for Bristol bay and fraser river sockeye salmon, Can. J. fish. aquat. sci., 65, 409-424, (2008)
[20] Murdoch, W.W.; Kendall, B.E.; Nisbet, R.M.; Briggs, C.J.; McCauley, E.; Boiser, R., Single-species models for many-species food webs, Nature, 417, 541-543, (2002)
[21] Myers, R.A.; Mertz, G.; Bridson, J.M.; Bradford, M.J., Simple dynamics underlie sockeye salmon (oncorhynchus nerka) cycles, Can. J. fish. aquat. sci., 55, 2355-2364, (1998)
[22] Odum, E.P., Fundamentals of ecology, (1953), W.B. Saunders Philadelphia
[23] Pauley, G.B., Risher, R., Thomas, G.L., 1989. Species profiles: life histories and environmental requirements of coastal fishes and invertebrates (Pacific Northwest)—Sockeye salmon. US Department of the Interior, Fish and Wildlife Service, Biological Report 82(11.116).; Pauley, G.B., Risher, R., Thomas, G.L., 1989. Species profiles: life histories and environmental requirements of coastal fishes and invertebrates (Pacific Northwest)—Sockeye salmon. US Department of the Interior, Fish and Wildlife Service, Biological Report 82(11.116).
[24] Persson, L.; Leonardsson, K.; de Roos, A.M.; Gyllenberg, M.; Christensen, B., Ontogenetic scaling of foraging rates and the dynamics of a size-structured consumer – resource model, Theor. popul. biol., 54, 270-293, (1998) · Zbl 0954.92032
[25] Ricker, W.E., Cycle dominance among the fraser sockeye, Ecology, 31, 6-26, (1950)
[26] Ricker, W.E., Cycles of abundance among fraser river sockeye salmon (oncorhynchus nerka), Can. J. fish. aquat. sci., 54, 950-968, (1997)
[27] Rogers, L.A.; Schindler, D.E., Asynchrony in population dynamics of sockeye salmon in southwest alaska, Oikos, 117, 1578-1586, (2008)
[28] de Roos, A.M.; Metz, J.A.J.; Evers, E.; Leipoldt, A., A size dependent predator – prey interaction who pursues whom?, J. math. biol., 28, 609-643, (1990) · Zbl 0718.92026
[29] Rounsefell, G.A.; Kelez, G.B., The salmon and salmon fisheries of swiftsure bank, puget sound, and the fraser river, Bull. bur. fish., 49, 692-823, (1938)
[30] Royama, T., Population dynamics of the spruce budworm choristoneura-fumiferana, Ecol. mon., 54, 429-462, (1984)
[31] Skalski, G.T.; Gilliam, J.F., Functional responses with predator interference: viable alternatives to the Holling type II model, Ecology, 82, 3083-3092, (2001)
[32] Townsend, C.R., Population cycles in freshwater fish, J. fish biol., 35, Supplement A, 125-131, (1989)
[33] Walters, C.J.; Christensen, V., Adding realism to foraging arena predictions of trophic flows in ecosim ecosystem models: shared foraging arenas and bout feeding, Ecol. mod., 209, 342-350, (2007)
[34] Walters, C.J., Staley, M.J.,1987. Evidence against the existence of cyclic dominance in Fraser River sockeye salmon (Oncorhynchus nerkaOncorhynchus nerka; Walters, C.J., Staley, M.J.,1987. Evidence against the existence of cyclic dominance in Fraser River sockeye salmon (Oncorhynchus nerkaOncorhynchus nerka
[35] Walters, C.J.; Woodey, J.C., Genetic models for cyclic dominance in sockeye salmon (oncorhynchus nerka), Can. J. fish. aquat. sci., 49, 281-292, (1992)
[36] West, F.W., Fair, L.F., 2006. Abundance, age, sex, and size statistics for pacific salmon in Bristol Bay, 2003. Fishery Data Series No. 06-47, Alaska Department of Fish and Game, Division of Sport Fish and Commercial Fisheries, Anchorage.; West, F.W., Fair, L.F., 2006. Abundance, age, sex, and size statistics for pacific salmon in Bristol Bay, 2003. Fishery Data Series No. 06-47, Alaska Department of Fish and Game, Division of Sport Fish and Commercial Fisheries, Anchorage.
[37] West, F.W., Fair, L.F., Baker, T., Morstad, S., Weiland, K., Sands, T., Westings, C., 2009. Abundance, age, sex, and size statistics for pacific salmon in Bristol Bay, 2004. Fishery Data Series No. 09-51, Alaska Department of Fish and Game, Division of Sport Fish and Commercial Fisheries, Anchorage.; West, F.W., Fair, L.F., Baker, T., Morstad, S., Weiland, K., Sands, T., Westings, C., 2009. Abundance, age, sex, and size statistics for pacific salmon in Bristol Bay, 2004. Fishery Data Series No. 09-51, Alaska Department of Fish and Game, Division of Sport Fish and Commercial Fisheries, Anchorage.
[38] Yodzis, P.; Innes, S., Body size and consumer – resource dynamics, Am. nat., 139, 1151-1175, (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.