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Two-parameter bifurcations in LPA model


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.

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Fig. 1

Source Elaydi (2005)

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  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.


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Correspondence to Lenka Přibylová.

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Hajnová, V., Přibylová, L. Two-parameter bifurcations in LPA model. J. Math. Biol. 75, 1235–1251 (2017).

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

Mathematics Subject Classification

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