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

This paper explores analytically and numerically, in the context of the FitzHugh-Nagumo model of nerve membrane excitability, an interesting phenomenon that has been described as a delay or memory effect. It can occur when a parameter passes slowly through a Hopf bifurcation point and the system's response changes from a slowly varying steady state to slowly varying oscillations. On quantitative observation it is found that the transition is realized when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects; the dynamic aspect of the parameter variation. This delay and its dependence on the speed of the parameter variation are described. The model involves several parameters and particular singular limits are investigated. One in particular is the slow passage through a low frequency Hopf bifurcation where the system's response changes from a slowly varying steady state to slowly varying relaxation oscillations. We find in this case the onset of oscillations exhibits an advance rather than a delay. This paper shows that in general delays in. the onset of oscillations may be expected but that small amplitude noise and periodic environmental perturbations of near resonant frequency may decrease the delay and destroy the memory effect. This paper suggests that both deterministic and stochastic approaches will be important for comparing theoretical and experimental results in systems where slow passage through a Hopf bifurcation is the underlying mechanism for the onset of 'oscillations.