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On synchronization in semelparous populations

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Abstract.

Synchronization, i.e., convergence towards a dynamical state where the whole population is in one age class, is a characteristic feature of some population models with semelparity. We prove some rigorous results on this, for a simple class of nonlinear one- population models with age structure and semelparity: (i) the survival probabilities are assumed constant, and (ii) only the last age class is reproducing (semelparity), with fecundity decreasing with total population. For this model we prove: (a) The synchronized, or Single Year Class (SYC), dynamical state is always attracting. (b) The coexistence equilibrium is often unstable; we state and prove simple results on this. (c) We describe dynamical states with some, but not all, age classes populated, which we call Multiple Year Class (MYC) patterns, and we prove results extending (a) and (b) into these patterns.

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References

  1. Behncke, H.: Periodical cicadas. J. Math Biol. 40, 413–431 (2000)

    Google Scholar 

  2. Bergh, M.O., Getz, W.M.: Stability of discrete age- structured and aggregated delay-difference population models. J. Math. Biol. 26, 551–581 (1988)

    MathSciNet  MATH  Google Scholar 

  3. Bulmer, M.G.: Periodical insects. Am. Naturalist 111, 1099–1117 (1977)

    Google Scholar 

  4. Caswell, H.: Matrix population models. Sinauer Associates, Inc. Publishers, Sunderland, Massachusetts, 2001

  5. Davydova, N.V., Diekmann, O., van Gils, S.A.: Year class coexistence or competitive exclusion for strict biennials? J. Math. Biol. 46, 95–131 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Davydova, N.V., Diekmann, O., van Gils, S.A.: On circulant populations. I. The algebra of semelparity. Subm. to J. Math. Biol., 2003

  7. Devaney, R.L.: An introduction to Chaotic Dynamical Systems. Addison–Wesley, 1989

  8. Hassell, M.P.: Density dependence in single-species populations. J. Animal Ecology 44, 283–296 (1974)

    Google Scholar 

  9. Hoppensteadt, F.C., Keller, J.B.: Synchronization of periodical cicada emergences. Science 194, 335–337 (1976)

    CAS  PubMed  Google Scholar 

  10. Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. Cambridge University Press, 1995

  11. Levin, S.A.: Age-structure and stability in multiple-age spawning populations. In: Renewable Resource Managements, T.L. Vincent, J.M. Skowronski (eds.), Springer Verlag, Heidelberg, 1981

  12. Levin, S.A., Goodyear, P.H.: Analysis of an age-structured fishery model. J. Math. Biol. 9, 245–274 (1980)

    MathSciNet  MATH  Google Scholar 

  13. Meyer, C.: Matrix Analysis and Applied Linear Algebra. SIAM, 2000

  14. Muir, T.: A treatise on the theory of determinants. Dover, New York, 1960

  15. Murray, J.D.: Mathematical Biology. 2. ed. Springer- Verlag, 1993

  16. Neubert, M.G., Caswell, H.: Density dependent vital rates and their population and dynamic consequences. J. Math. Biol. 41, 103–121 (2000)

    Google Scholar 

  17. Ricker, W.E.: Stock and recruitment. J. Fisheries Res. Board Canada 11, 559–623 (1954)

    Google Scholar 

  18. Silva, J.A., Hallam, T.G.: Effects of delay, truncation and density dependence in reproduction schedules on stability of nonlinear Leslie matrix models. J. Math. Biol. 31, 367–395 (1993)

    MathSciNet  MATH  Google Scholar 

  19. Solberg, T.: Rare Phenomena in an Age-Structured Population Model. Master thesis, University of Tromsø, Norway, 1998

  20. Thunberg, H.: Periodicity versus Chaos in One-Dimensional Dynamics. SIAM Rev. 43, 3–30 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tuljapurkar, S., Boe, C., Wachter, K.W.: Nonlinear Feedback Dynamics in Fisheries: Analysis of the Deriso–Schnute Model. Can. J. Fish. Aquat. Sci. 51, 1462–1472 (1994)

    Google Scholar 

  22. Wikan, A.: Dynamic consequences of reproductive delay in Leslie matrix models with nonlinear survival probabilities. Math. Biosci. 146, 37–62 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wikan, A., Mjølhus, E.: Overcompensatory recruitment and generation delay in discrete age-structured population models. J. Math. Biol. 35, 195–239 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Science, Department of Mathematics and Statistics, University of Tromsø, N-9037, Tromsø, Norway

    E. Mjølhus

  2. Harstad College, Havnegata 5, N-9480, Harstad, Norway

    A. Wikan

  3. Norwegian Institute of Defense, 25, N-2027, Kjeller, Norway

    T. Solberg

Authors
  1. E. Mjølhus
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  2. A. Wikan
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  3. T. Solberg
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Correspondence to E. Mjølhus.

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Acknowledgement Boris Kruglikov contributed the “nonlinear” part of the formulation as well as the proof of Theorem 1. The authors are grateful for critical and constructive comments by N. Davydova and O. Diekmann. E.M. is also grateful for discussions with Marius Overholt concerning problems of proving Theorem 2.

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Mjølhus, E., Wikan, A. & Solberg, T. On synchronization in semelparous populations. J. Math. Biol. 50, 1–21 (2005). https://doi.org/10.1007/s00285-004-0275-5

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  • DOI: https://doi.org/10.1007/s00285-004-0275-5

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