Skip to main content

Two-parameter bifurcations in LPA model

Abstract

The structured population LPA model is studied. The model describes flour beetle (Tribolium) population dynamics of four stage populations: eggs, larvae, pupae and adults with cannibalism between these stages. We concentrate on the case of non-zero cannibalistic rates of adults on eggs and adults on pupae and no cannibalism of larvae on eggs, but the results can be numerically continued to non-zero cannibalism of larvae on eggs. In this article two-parameter bifurcations in LPA model are analysed. Various stable and unstable invariant sets are found, different types of hysteresis are presented and abrupt changes in dynamics are simulated to explain the complicated way the system behaves near two-parameter bifurcation manifolds. The connections between strong 1:2 resonance and Chenciner bifurcations are presented as well as their very significant consequences to the dynamics of the Tribolium population. The hysteresis phenomena described is a generic phenomenon nearby the Chenciner bifurcation or the cusp bifurcation of the loop.

This is a preview of subscription content, access via your institution.

Fig. 1

Source Elaydi (2005)

Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Notes

  1. In the case where \(c_{EL} \ne 0\) the flip bifurcation manifold depends on cannibalism rates.

  2. Non-degeneracy and transversality conditions should be involved for each bifurcation for correct analysis. It can be done analytically using the reduction to the center manifold. We used program MactontM (see e.g. Kuznetsov 2013) to get numerical results, because we do not continue with any further analysis of codimension 3 bifurcations.

  3. The Chenciner bifurcation is an analogy of continuous Bautin (or generalized Hopf) bifurcation.

  4. Fold bifurcation of a fixed point gives rise to two fixed points with opposite stability, fold bifurcation of a cycle gives rise to two cycles with opposite stability, for more details see Kuznetsov (1998).

  5. Topologically equivalent diagram is obtained for small \(c_{EL}>0\).

  6. The bifurcation diagram for the Chenciner bifurcation is presented and explained in total in the Sect. 4.1. Comparing Fig. 5 and \(\textcircled {{2}}\) we obtain the correspondence of domain II. In Fig. 5 to type \(\textcircled {{1}}\) in Fig. 2 and correspondence between domain III. in Fig. 5 and type \(\textcircled {{3}}\) in Fig. 2.

References

  • Chenciner A (1987) Bifurcations de points fixes elliptiques. III. Orbites périodiques de “petites” périodes et élimination résonnante des couples de courbes invariantes, Inst. Hautes Études Sci. Publ Math 66:5–91

    Article  MATH  Google Scholar 

  • Costantino RF, Desharnais RA, Cushing JM, Dennis B (1997) Chaotic dynamics in an insect population. Science 275(5298):389–391

    Article  MATH  Google Scholar 

  • Cushing JM (1998) An introduction to structured population dynamics. Society for Industrial and Applied Mathematics, Philadelphia

    Book  MATH  Google Scholar 

  • Cushing JM (2003) Chaos in ecology: experimental nonlinear dynamics. Academic Press, Amsterdam

    Google Scholar 

  • Cushing JM, Constantino RF, Dennis B, Desharnais RA, Henson SM (1998) Nonlinear population dynamics: models, experiments and data. J Theor Biol 194(1):1–9

    Article  Google Scholar 

  • Cushing JM, Henson SM, Desharnais RA, Dennis B, Costantino RF, King A (2001) A chaotic attractor in ecology: theory and experimental data. Chaos Solitons Fractals 12(2):219–234

    MathSciNet  Article  MATH  Google Scholar 

  • Dennis B, Desharnais RA, Cushing JM, Constantino RF (1995) Nonlinear demographic dynamics: mathematical models, statistical methods, and biological experiments. Ecol Monogr 65(3):261–282

    Article  Google Scholar 

  • Dennis B, Desharnais RA, Cushing JM, Costantino RF (1997) Transitions in population dynamics: equilibria to periodic cycles to aperiodic cycles. J Anim Ecol 66(5):704–729

    Article  Google Scholar 

  • Elaydi S (2005) An introduction to difference equations, 3rd edn. Springer, New York

    MATH  Google Scholar 

  • Izhikevich EM (2007) Dynamical systems in neuroscience. MIT press, Cambridge

    Google Scholar 

  • Kuang Y, Cushing JM (2007) Global stability in a nonlinear difference-delay equation model of flour beetle population growth. J Differ Equ Appl 2(1):31–37

    MathSciNet  Article  MATH  Google Scholar 

  • Kuznetsov YA (1998) Elements of applied bifurcation theory, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Kuznetsov YA (2013) A Tutorial for MatcontM GUI. Utretcht University, Utrecht

    Google Scholar 

  • Ludwig D, Jones DD, Holling CS (1978) Qualitative analysis of insect outbreak systems: the spruce budworm and forest. J Anim Ecol 47:315–332

    Article  Google Scholar 

  • Melbourne BA, Hastings A (2008) Extinction risk depends strongly on factors contributing to stochasticity. Nature 454(7200):100–103

    Article  Google Scholar 

  • Schreiber S, Rudolf VHW (2008) Crossing habitat boundaries: coupling dynamics of ecosystems through complex life cycles. Ecol Lett 11(6):576–587

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Lenka Přibylová.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hajnová, V., Přibylová, L. Two-parameter bifurcations in LPA model. J. Math. Biol. 75, 1235–1251 (2017). https://doi.org/10.1007/s00285-017-1115-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-017-1115-8

Keywords

  • Population dynamics
  • Two-parameter bifurcations
  • LPA model
  • Strong 1:2 resonance
  • Chenciner bifurcation

Mathematics Subject Classification

  • 92D25
  • 37N25
  • 37G10
  • 37G15
  • 34C23