RESUMEN

In this paper we investigate long-term dynamics of the most basic model for stage-structured populations, in which the per capita transition from the juvenile into the adult class is density dependent. The model is represented by an autonomous system of two nonlinear differential equations with four parameters for a single population. We find that the interaction of intra-adult competition and intra-juvenile competition gives rise to multiple attractors, one of which can be oscillatory. A detailed numerical study reveals a rich bifurcation structure for this two-dimensional system, originating from a degenerate Bogdanov--Takens (BT) bifurcation point when one parameter is kept constant. Depending on the value of this fixed parameter, the corresponding triple critical equilibrium has either an elliptic sector or it is a topological focus, which is demonstrated by the numerical normal form analysis. It is shown that the canonical unfolding of the codimension-three BT point reveals the underlying dynamics of the model. Certain new features of this unfolding in the elliptic case, which are important in applications but have been overlooked in available theoretical studies, are established. Various three-, two-, and one-parameter bifurcation diagrams of the model are presented and interpreted in biological terms.

RESUMEN

Slowly varying conservative systems are analyzed in the case of a reverse subcritical pitchfork bifurcation in which two saddles and a center coalesce. Before the bifurcation there is a hyperbolic double-homoclinic orbit connecting a linear saddle point. At the bifurcation a double nonhyperbolic homoclinic orbit connects to a nonlinear saddle point. Strongly nonlinear oscillations obtained by the method of averaging are not valid near unperturbed homoclinic orbits. In the case in which the solution slowly passes through the nonhyperbolic homoclinic orbit associated with the subcritical pitchfork bifurcation, the solution consists of a large sequence of nonhyperbolic homoclinic orbits connecting autonomous nonlinear saddle approaches. Solutions are captured into the left and right well. Phase jumps and the boundaries of the basins of attraction are computed. It is shown that the change in action in the slow passage through the nonhyperbolic homoclinic orbits is much larger than the known change in action for the slow crossing of hyperbolic homoclinic orbits. Near the boundary of the basin of attraction, where the energy is particularly small, one of the saddle approaches is governed by the second Painlev\a'e transcendent, which is not autonomous, and the solution may oscillate around the middle center or approach the two saddles created by the subcritical pitchfork bifurcation in addition to oscillating around the left and right wells.