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Journal of Applied Mathematics
Volume 2017 (2017), Article ID 8934295, 14 pages
https://doi.org/10.1155/2017/8934295
Research Article

An Analysis of a Semelparous Population Model with Density-Dependent Fecundity and Density-Dependent Survival Probabilities

School of Business and Economics, The Arctic University of Norway, Campus Harstad, Havnegata 5, 9480 Harstad, Norway

Correspondence should be addressed to Arild Wikan

Received 12 June 2017; Revised 23 August 2017; Accepted 24 September 2017; Published 17 December 2017

Academic Editor: Urmila Diwekar

Copyright © 2017 Arild Wikan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Guckenheimer, G. Oster, and A. Ipaktchi, “The dynamics of density dependent population models,” Journal of Mathematical Biology, vol. 4, no. 2, pp. 101–147, 1977. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. S. A. Levin and C. P. Goodyear, “Analysis of an age-structured fishery model,” Journal of Mathematical Biology, vol. 9, no. 3, pp. 245–274, 1980. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. J. A. Silva and T. G. Hallam, “Effects of delay, truncations and density dependence in reproduction schedules on stability of nonlinear Leslie matrix models,” Journal of Mathematical Biology, vol. 31, no. 4, pp. 367–395, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. A. Wikan and E. Mj\o lhus, “Overcompensatory recruitment and generation delay in discrete age-structured population models,” Journal of Mathematical Biology, vol. 35, no. 2, pp. 195–239, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  5. A. Wikan and A. Eide, “An analysis of a nonlinear stage-structured cannibalism model with application to the northeast arctic cod stock,” Bulletin of Mathematical Biology, vol. 66, no. 6, pp. 1685–1704, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. I. Ugarcovici and H. Weiss, “Chaotic dynamics of a nonlinear density dependent population model,” Nonlinearity, vol. 17, no. 5, pp. 1689–1711, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  7. W. Govaerts and R. K. Ghaziani, “Numerical bifurcation analysis of a nonlinear stage structured cannibalism population model,” Journal of Difference Equations and Applications, vol. 12, no. 10, pp. 1069–1085, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. J. M. Cushing, “Nonlinear matrix models and population dynamics,” Natural Resource Modeling, vol. 2, no. 4, pp. 539–580, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J. M. Cushing, “A strong ergodic theorem for some nonlinear matrix models for the dynamics of structured populations,” Natural Resource Modeling, vol. 3, no. 3, pp. 331–357, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  10. K. M. Crowe, “A nonlinear ergodic theorem for discrete systems,” Journal of Mathematical Biology, vol. 32, no. 3, pp. 179–191, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. R. Kon, Y. Saito, and Y. Takeuchi, “Permanence of single-species stage-structured models,” Journal of Mathematical Biology, vol. 48, no. 5, pp. 515–528, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. D. L. Deangelis, L. J. Svoboda, S. W. Christensen, and D. S. Vaughan, “Stability and return times of Leslie matrices with density-dependent survival: applications to fish populations,” Ecological Modelling, vol. 8, no. C, pp. 149–163, 1980. View at Publisher · View at Google Scholar · View at Scopus
  13. R. A. Desharnais and L. Liu, “Stable demographic limit cycles in laboratory populations of tribolium castaneum,” Journal of Animal Ecology, vol. 56, no. 3, pp. 885–906, 1987. View at Publisher · View at Google Scholar · View at Scopus
  14. T. V. Burkey and N. C. Stenseth, “Population dynamics of territorial species in seasonal and patchy environments,” Oikos, vol. 69, no. 1, pp. 47–53, 1994. View at Publisher · View at Google Scholar · View at Scopus
  15. A. Wikan, “Four-periodicity in Leslie matrix models with density dependent survival probabilities,” Theoretical Population Biology, vol. 53, no. 2, pp. 85–97, 1998. View at Publisher · View at Google Scholar · View at Scopus
  16. A. Wikan, “Age or stage structure? a comparison of dynamic outcomes from discrete age- and stage-structured population models,” Bulletin of Mathematical Biology, vol. 74, no. 6, pp. 1354–1378, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  17. M. G. Bulmer, “Periodical insects,” The American Naturalist, vol. 111, no. 982, pp. 1099–1117, 1977. View at Publisher · View at Google Scholar
  18. F. C. Hoppensteadt and J. B. Keller, “Synchronization of periodical cicada emergences,” Science, vol. 194, no. 4262, pp. 335–337, 1976. View at Publisher · View at Google Scholar · View at Scopus
  19. H. Behncke, “Periodical cicadas,” Journal of Mathematical Biology, vol. 40, no. 5, pp. 413–431, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. N. V. Davydova, O. Diekmann, and S. A. van Gils, “Year class coexistence or competitive exclusion for strict biennials?” Journal of Mathematical Biology, vol. 46, no. 2, pp. 95–131, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. E. Mj\o lhus, A. Wikan, and T. Solberg, “On synchronization in semelparous populations,” Journal of Mathematical Biology, vol. 50, no. 1, pp. 1–21, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. R. Kon, “Nonexistence of synchronous orbits and class coexistence in matrix population models,” SIAM Journal on Applied Mathematics, vol. 66, no. 2, pp. 616–626, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. J. M. Cushing, “Nonlinear semelparous Leslie models,” Mathematical Biosciences and Engineering, vol. 3, no. 1, pp. 17–36, 2006. View at Publisher · View at Google Scholar · View at Scopus
  24. J. M. Cushing, “Three stage semelparous Leslie models,” Journal of Mathematical Biology, vol. 59, no. 1, pp. 75–104, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. J. M. Cushing and S. M. Henson, “Stable bifurcations in semelparous Leslie models,” Journal of Biological Dynamics, vol. 6, no. suppl. 2, pp. 80–102, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. J. M. Cushing, “On the fundamental bifurcation theorem for semelparous Leslie models,” in Dynamics, Games And Science, J. P. Bourguignon, R. Jeltsch, A. Pinto, and M. Viana, Eds., vol. 1 of CIM Ser. Math. Sci., pp. 215–251, Springer, Cham, Berlin, Germany, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  27. J. D. Murray, Mathematical Biology: Spatial Models And Biomedical Applications, vol. 18, Springer, Berlin, Germany, 3rd edition, 2003.
  28. Y. H. Wan, “Computation of the stability condition for the Hopf bifurcation of diffeomorphisms on ,” SIAM Journal on Applied Mathematics, vol. 34, no. 1, pp. 167–175, 1978. View at Publisher · View at Google Scholar · View at MathSciNet
  29. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer, NY, USA, 2002. View at MathSciNet
  30. J. Jost, Dynamical Systems, Universitext, Springer-Verlag, Berlin, Germany, 2005. View at MathSciNet
  31. H. N. Agiza, E. M. ELabbasy, H. EL-Metwally, and A. A. Elsadany, “Chaotic dynamics of a discrete prey-predator model with holling type II,” Nonlinear Analysis: Real World Applications, vol. 10, no. 1, pp. 116–129, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  32. M. G. Neubert and H. Caswell, “Density-dependent vital rates and their population dynamic consequences,” Journal of Mathematical Biology, vol. 41, no. 2, pp. 103–121, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus