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.