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A juvenile-adult model with periodic vital rates

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Abstract

A global branch of positive cycles is shown to exist for a general discrete time, juvenile-adult model with periodically varying coefficients. The branch bifurcates from the extinction state at a critical value of the mean, inherent fertility rate. In comparison to the autonomous system with the same mean fertility rate, the critical bifurcation value can either increase or decrease with the introduction of periodicities. Thus, periodic oscillations in vital parameter can be either advantageous or deleterious. A determining factor is the phase relationship among the oscillations in the inherent fertility and survival rates.

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References

  1. Caswell, H .: Matrix Population Models: Construction, Analysis and Interpretation, 2nd edn. Sunderland, Massachusetts: Sinauer Associates, Inc. Publishers, 2001

  2. Coleman, B.D.: On the growth of populations with narrow spread in reproductive age. I. General theory and examples. J. Math. Biol. 6, 1–19 (1978)

    Google Scholar 

  3. Costantino, R.F., Cushing, J.M., Dennis, B., Desharnais, R.A., Henson, S.M.: Resonant population cycles in temporally fluctuating habitats. Bull. Math. Biol. 60 (2), 247–275 (1998)

    Google Scholar 

  4. Cushing, J.M.: Periodic time-dependent predator prey systems. SIAM J. Appl. Math. 23, 972–979 (1977)

    Google Scholar 

  5. Cushing, J.M.: Periodic Kolomogov systems. SIAM J. Appl. Math. 13, 811–827 (1982)

    Google Scholar 

  6. Cushing, J.M.: Periodic two-predator, one-prey interactions and the time sharing of a resource niche. SIAM J. Appl. Math. 44, 392–410 (1984)

    Google Scholar 

  7. Cushing, J.M.: Periodic Lotka-Volterra competition equations. J. Math. Biol. 24, 381–403 (1986)

    Google Scholar 

  8. Cushing, J.M.: Oscillatory population growth in periodic environments. Theor. Popu. Biol. 30, 289–308 (1987)

    Google Scholar 

  9. Cushing, J.M.: An Introduction to Structured Population Dynamics. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 71. Philadelphia: SIAM, 1998

  10. Cushing, J.M.: Periodically forced nonlinear systems of difference equations. J. Diff. Equ. and Appl. 3, 547–561 (1998)

    Google Scholar 

  11. Cushing, J.M., Henson, S.M.: Global dynamics of some periodically forced, monotone difference equations. J. Diff. Equ. Appl. 7, 859–872 (2001)

    Google Scholar 

  12. deMottoni, P., Schiaffino, A.: Competition systems with periodic coefficients: a geometrical approach. J. Math. Biol. 11, 319–335 (1982)

    Google Scholar 

  13. Elaydi, S.N.: An Introduction to Difference Equations. Springer-Verlag, New York, 1996

  14. Elaydi, S., Sacker, R.J.: Global stability of periodic orbits of non-autonomous difference equations and population biology. J. Diff. Equ. 208, 258–273 (2005)

    Google Scholar 

  15. Elaydi, S., Sacker, R.J.: Non-autonomous Beverton-Holt equations and the Cushing-Henson conjectures. J. Diff. Equ. Appl. 11, 337–346 (2005)

    Google Scholar 

  16. Elaydi, S., Sacker, R.J.: Global stability of periodic orbits of non-autonomous difference equations in population biology and the Cushing/Henson conjectures. In: Proceedings of the 8th International Conference of Difference Equations and Applications, pp. 113-126, Chapman & Hall/CRC, Boca Raton, FL, 2005

  17. Franke, J.E., Yakubu, A.-A.: Multiple attractors via CUSP bifurcation in periodically varying environments. J. Diff. Equ. Appl. 11, 365–377 (2005)

    Google Scholar 

  18. Franke, J.E., Yakubu, A.-A.: Attenuant cycles in periodically forced discrete-time age-structured population models, J. Math. Analy. Appl. 316, 69–86 (2006)

    Google Scholar 

  19. Franke, J.E., Yakubu, A.-A.: Population models with periodic recruitment functions and survival rates, J. Diff. Equ. Appl. 11 (14), 1169–1184 (2005)

    Google Scholar 

  20. Franke, J.E., Yakubu, A.-A.: Globally attracting attenuant versus resonant cycles in periodic compensatory Leslie models, to appear

  21. Franke, J.E., Yakubu, A.-A.: Signature function for predicting resonant and attenuant population cycles, to appear in Bull. Math. Biol.

  22. Henson, S.M.: Existence and stability of nontrivial periodic solutions of periodically forced discrete dynamical systems. J. Diff. Equ. and Appl. 2, 315–331 (1996)

    Google Scholar 

  23. Henson, S.M.: The effect of periodicity in maps. J. Diff. Equ. Appl. 5, 31–56 (1999)

    Google Scholar 

  24. Henson, S.M.: Multiple attractors and resonance in periodically forced population models. Physica D 140, 33–49 (2000)

    Google Scholar 

  25. Henson, S.M., Cushing, J.M.: The effect of periodic habitat fluctuations on a nonlinear insect population model. J. Math. Biol. 36, 201–226 (1997)

    Google Scholar 

  26. Jillson, D.A.: Insect populations respond to fluctuating environments. Nature 288, 699–700 (1980)

    Google Scholar 

  27. Kielhöfer, H.: Bifurcation Theory: An Introduction with Applications to PDEs. Applied Mathematical Sciences 156, Springer, New York, 2004

  28. Kocic, V.L.: A note on the non-autonomous Beverton-Holt model, J. Diff. Equ. Appl. 11 (4–5), 415–422 (2005)

    Google Scholar 

  29. Kon, R.: A note on attenuant cycles of population models with periodic carrying capacity. J. Diff. Equ. Appl. 10 (8), 791–793 (2004)

    Google Scholar 

  30. Kon, R.: Attenuant cycles of population models with periodic carrying capacity. J. Diff. Equ. Appl. 11, 423–430 (2005)

    Google Scholar 

  31. May, R.M.: Stability and Complexity in Model Ecosystems. Princeton Landmarks in Biology, Princeton University Press, Princeton, New Jersey, 2001, p. 123

  32. Namba, T.: Competitive co-existence in a seasonally fluctuating environment. J. Theo. Biol. 111, 369–386 (1984)

    Google Scholar 

  33. Nisbet, R.M., Gurney, W.S.C.: Population dynamics in a periodically varying environment. J. Theo. Biol. 56, 459–475 (1976)

    Google Scholar 

  34. Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Func. Anal. 7 (3), 487-513b (1971)

    Google Scholar 

  35. Rosenblat, S.: Population models in a periodically fluctuating environment. J. Math. Biol. 9, 23–36 (1980)

    Google Scholar 

  36. Slegrade, J.F., Roberds, J.H.: On the structure of attractors for discrete, periodically forced systems with applications to population models. Physica D 158, 69–82 (2001)

    Google Scholar 

  37. Smith, H.L.: Competitive coexistence in an oscillating chemostst. SIAM J. Appl. Math. 40, 498–522 (1981)

    Google Scholar 

  38. Smith, H.L., Waltman, P.: The Theory of the Chemostat: Dynamics of Microbial Competition. Cambridge Studies in Mathematical Biology, Cambridge University Press, Cambridge, 1995

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Authors and Affiliations

  1. Department of Mathematics, Interdisciplinary Program in Applied Mathematics, University of Arizona, 617 N Santa Rita, Tucson, AZ, 85721, USA

    J. M. Cushing

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  1. J. M. Cushing
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Research supported by NSF grant DMS-0414212.

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Cushing, J. A juvenile-adult model with periodic vital rates. J. Math. Biol. 53, 520–539 (2006). https://doi.org/10.1007/s00285-006-0382-6

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  • DOI: https://doi.org/10.1007/s00285-006-0382-6

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