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Long-period endogenous oscillations in fish population size: Mathematical modeling

  • Complex Systems Biophysics
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Abstract

We present a mathematical model of an aquatic community, where the size-and-age structure of hydrobiont populations is taken into account and the corresponding trophic interactions between zooplankton, peaceful fish, and predatory fish are described. We show that interactions between separate components of the aquatic community can give rise to long-period oscillations in fish population size. The period of these oscillations is on the order of decades. With this model we also show that an increase in the zooplankton growth rate may entail a sequence of bifurcations in the fish population dynamics: steady states → regular oscillations → quasicycles → dynamic chaos.

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References

  1. E. A. Kriksunov and A. E. Bobyrev, in Problems of Regulation in Biological Systems (Inst. Komp’yut. Issl., Moscow-Izhevsk, 2007), p. 453 [in Russian].

    Google Scholar 

  2. V. Kaitala, E. Ranta, and J. Lindström, J. Animal Ecol. 65, 249 (1996).

    Article  Google Scholar 

  3. N. C. Stenseth, O. N. Bjørnstad, W. Falck, et al., Proc. Royal Soc. London, Ser. B 266, 1645 (1999).

    Article  Google Scholar 

  4. D. H. Cushing, Population Production and Regulation in the Sea: A Fisheries Perspective (Cambridge University Press, Cambridge, 1995).

    Google Scholar 

  5. J.-M. Fromentin, R. M. Myers, O. Bjørnstad, et al., Ecology 82, 567 (2001).

    Article  Google Scholar 

  6. R. M. Cook, A. Sinclair, and G. Stefansson, Nature 385, 521 (1997).

    Article  ADS  Google Scholar 

  7. R. R. Dickson and K. Brander, Fisheries Oceanography 2, 124 (1993).

    Article  Google Scholar 

  8. G. Ottersen and S. Sundby, Fisheries Oceanography 4, 278 (1995).

    Article  Google Scholar 

  9. L. Persson, A. M. De Roos, D. Claessen, et al., Proc. Natl. Acad. Sci. USA 100, 4035 (2003).

    Article  ADS  Google Scholar 

  10. A. E. Bobyrev and E. A. Kriksunov, Mathematical Modeling of Fish Population Dynamics with Variable Replenishment Tempo (Nauka, Moscow, 1996) [in Russian].

    Google Scholar 

  11. M. Kot, Elements of Mathematical Ecology (Cambridge University Press, Cambridge, 2001).

    Book  Google Scholar 

  12. G. Yu. Riznichenko and A. B. Rubin, Biophysical Dynamics of Production Processes (Inst. Komp’yut. Issl., Moscow-Izhevsk, 2004) [in Russian].

    Google Scholar 

  13. A. M. Gilyarov, Dynamics of the Numbers of Freshwater Planktonic Crustaceans (Nauka, Moscow, 1987) [in Russian].

    Google Scholar 

  14. C. S. Holling, Canadian Entomologist 91, 293 (1959).

    Article  Google Scholar 

  15. Z. Zakęś, M. Szkudlarek, M. Woźniak, et al., Electr. J. Polish Agric. Univ. Fisheries 6(1), # 4 (2003).

    Google Scholar 

  16. A. Kowalska, Z. Zakęś, and K. Demska-Zakeś, Electr. J. Polish Agric. Univ. Fisheries 9(2), # 5 (2006).

    Google Scholar 

  17. S. Hulsmann, K. Rinke, and W. M. Mooij, Oikos 110, 43 (2005).

    Article  Google Scholar 

  18. K. Rinke and J. Vijverberg, Ecol. Modelling 186, 326 (2005).

    Article  Google Scholar 

  19. O. P. Dubovskaya, Zh. Obshch. Biol. 70, 168 (2009).

    Google Scholar 

  20. K. Rinke, S. Hulsmann, and W. M. Mooij, Oikos 117, 273 (2008).

    Article  Google Scholar 

  21. E. A. Kriksunov, A. E. Bobyrev, V. A. Burmenskii, et al., Balance Model of the Biotic Community of Syamozero (KNTs RAN, Petrozavodsk, 2005) [in Russian].

    Google Scholar 

  22. L. F. Antipova and N. Ya. Kontsevaya, Sb. Nauch. Tr. GosNIORKh 209, 102 (1983).

    Google Scholar 

  23. A. P. Shirkova, Gidrobiol. Issl. 4, 213 (1966).

    Google Scholar 

  24. O. P. Sterligova, V. N. Pavlov, N. V. Il’mast, et al., Ecosystem of Syamozero (biological regime and use) (KNTs RAN, Petrozavodsk, 2002) [in Russian].

    Google Scholar 

  25. L. A. Kuderskii, S. A. Vetkasov, and V. N. Kartsev, Sb. Nauch. Tr. GosNIORKh 237, 31 (1985).

    Google Scholar 

  26. A. M. Gilyarov, Population Ecology (Izd MGU, Moscow, 1990) [in Russian].

    Google Scholar 

  27. L. N. Mitrofanova, Tr. Pskov. Otd. GosNIORKh 2, 94 (1976).

    Google Scholar 

  28. R. Kortet, A. Vainikka, M. J. Rantala, et al., J. Evolut. Biol. 17, 1337 (2004).

    Article  Google Scholar 

  29. M. M. Meshkov, Gidrobiol. Issl. 4, 249 (1966).

    Google Scholar 

  30. V. D. Spanovskaya and V. A. Grigorash, in Food Demands and Energy Balance in Fish (Naukova Dumka, Kiev, 1973), p. 119 [in Russian].

    Google Scholar 

  31. L. F. Antipova, Sb. Nauch. Tr. GosNIORKh 156, 65 (1980).

    Google Scholar 

  32. I. A. Stolbunov, Biol. Vnutr. Vod, No. 3, 73 (2006).

    Google Scholar 

  33. L. Urho, Folia Zoologica 51, 161 (2002).

    Google Scholar 

  34. L. Persson, A. M. De Roos, and P. Byström, J. Anim. Ecology 76, 94 (2007).

    Article  Google Scholar 

  35. G. L. Mel’nichuk, in Food Demands and Energy Balance in Fish (Naukova Dumka, Kiev, 1973), p. 50 [in Russian].

    Google Scholar 

  36. Yu. Virbitskas, A. Gyarulaitis, D. Misyusene, and D. Sinyavichene, Zander Biology and Fishing in Reservoirs of Lithuania (Mintis, Vilnius, 1974) [in Russian].

    Google Scholar 

  37. T. R. Marshall, J. Fisheries Res. Board of Canada 34, 1515 (1977).

    Article  Google Scholar 

  38. R. S. Zaripova and V. A. Kozlov, Sb. Nauch. Tr. Gos-NIORKh 239, 74 (1985).

    Google Scholar 

  39. K. A. Snezhina, Tr. AtlantNIRO 46, 89 (1971).

    Google Scholar 

  40. O. I. Kudrinskaya, in Food Demands and Energy Balance in Fish (Naukova Dumka, Kiev, 1973), p. 5 [in Russian].

    Google Scholar 

  41. B. B. Collette and P. Banarescu, J. Fisheries Res. Board of Canada 34, 1450 (1977).

    Article  Google Scholar 

  42. O. A. Popova and L. A. Sytina, J. Fisheries Res. Board of Canada 34, 1559 (1977).

    Article  Google Scholar 

  43. P. Frankiewicz, K. Dabrowski, and M. Zalewski, Annal. Zoologici Fennici 33, 321 (1996).

    Google Scholar 

  44. G. V. Fedorova and K. S. Drozzhina, Sb. Nauch. Tr. GosNIORKh 157, 64 (1980).

    Google Scholar 

  45. E. Kh. Pikhu and E. R. Pikhu, Izv. GosNIORKh 83, 136 (1974).

    Google Scholar 

  46. V. A. Kozlov, Sb. Nauch. Tr. GosNIORKh 156, 77 (1980).

    Google Scholar 

  47. J. Lappalainen, M. Olin, and M. Vinni, Annal. Zoologici Fennici 43, 35 (2006).

    Google Scholar 

  48. J. Willemsen, J. Fisheries Res. Board of Canada 34, 1710 (1977).

    Article  Google Scholar 

  49. M. Gielen, C. Rougeot, Y. Neus, et al., in Proc. 3rd Intern. Percid Fish Symp. (University of Wisconsin, Sea Grant Institute, 2004), p. 29.

    Google Scholar 

  50. J.-P. Eckmann, S. O. Kamphorst, and D. Ruelle, Europhys. Lett. 4, 973 (1987).

    Article  ADS  Google Scholar 

  51. A. E. Bobyrev, V. A. Burmenskii, E. A. Kriksunov, et al., Biophysics 57, 115 (2012).

    Article  Google Scholar 

  52. E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 2002).

    Book  MATH  Google Scholar 

  53. H. Kantz and T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, 1997).

    MATH  Google Scholar 

  54. R. M. Nisbet and W. S. Gurney, Nature 263, 319 (1976).

    Article  ADS  Google Scholar 

  55. G. V. Nikol’skii, Zool. Zh. 29, 16 (1950).

    Google Scholar 

  56. G. V. Nikol’skii, Theory of Fish Flock Dynamics (Pishchevaya Prom., Moscow, 1974) [in Russian].

    Google Scholar 

  57. Methods for Assessment of Fish Production in Fresh Waters Ed. by W. E. Ricker, (F.A. Davis, Philadelphia, 1968).

    Google Scholar 

  58. E. A. Kriksunov and M. A. Snetkov, in Theory of Forming the Numbers and Rational Use of Food Fish Flocks (Nauka, Moscow, 1985), c. 46 [in Russian].

    Google Scholar 

  59. V. D. Fedorov an T. G. Gil’manov, Ecology (Izd. MGU, Moscow, 1980) [in Russian].

    Google Scholar 

  60. A. B. Medvinsky and A. V. Rusakov, Chaos, Solitons and Fractals 44, 390 (2011).

    Article  MathSciNet  ADS  Google Scholar 

  61. R. M. Nisbet and W. S. C. Gurney, Modelling Fluctuating Populations (Wiley, New York, 1982).

    MATH  Google Scholar 

  62. M. Pascual and P. Mazzega, Theor. Popul. Biol. 64, 385 (2003).

    Article  MATH  Google Scholar 

  63. A. B. Medvinsky, A. V. Rusakov, and N. I. Nurieva, Ecol. Complexity (in press).

  64. A. B. Medvinsky, I. A. Tikhonova, R. R. Aliev, Phys. Rev. E 64, 021915 (2001).

    Article  ADS  Google Scholar 

  65. A. B. Medvinsky, S. V. Petrovskii, I. A. Tikhonova, et al., Uspekhi Fiz. Nauk 172, 31 (2002).

    Article  Google Scholar 

  66. Zh. A. Chernyaev, Doctoral Dissertation in Biology (Moscow, 1990).

    Google Scholar 

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Authors and Affiliations

  1. Severtsov Institute of Ecology and Evolution, Russian Academy of Sciences, Moscow, 119071, Russia

    A. E. Bobyrev

  2. Biological Faculty, Moscow State University, Moscow, 119991, Russia

    V. A. Burmensky & E. A. Kriksunov

  3. Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Moscow, Russia

    A. B. Medvinsky, N. I. Nurieva & A. V. Rusakov

  4. Pushchino State Institute of Natural Sciences, Pushchino, Moscow Region, 142290, Russia

    A. B. Medvinsky & N. I. Nurieva

Authors
  1. A. E. Bobyrev
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  2. V. A. Burmensky
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  3. E. A. Kriksunov
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  4. A. B. Medvinsky
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  5. N. I. Nurieva
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  6. A. V. Rusakov
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Correspondence to A. V. Rusakov.

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Original Russian Text © A.E. Bobyrev, V.A. Burmensky, E.A. Kriksunov, A.B. Medvinsky, N.I. Nurieva, A.V. Rusakov, 2013, published in Biofizika, 2013, Vol. 58, No. 2, pp. 334–348.

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Bobyrev, A.E., Burmensky, V.A., Kriksunov, E.A. et al. Long-period endogenous oscillations in fish population size: Mathematical modeling. BIOPHYSICS 58, 245–257 (2013). https://doi.org/10.1134/S000635091302005X

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  • DOI: https://doi.org/10.1134/S000635091302005X

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