RESUMEN

The McKendrick equations model the dynamical behavior of age-dependent populations. These equations govern, at time t, the number of individuals of age a in a population, known as the population density, and arise from a conservation law subject to constitutive assumptions for the maternity and mortality rates. In this paper, we present a weakly nonlinear analysis of the McKendrick equations which describes the bifurcation to time-dependent solutions whose amplitudes are governed by a complex Landau--Stuart-type equation. The framework of the multiple scales approach is described for general assumptions on the maternity and mortality rates, and the analysis is carried out in detail for an age-independent mortality rate and a separable maternity function which includes a nonzero delay in the onset of maturation. The resulting analysis indicates that the presence of the delay significantly affects the postbifurcational dynamics and introduces a codimension two bifurcation in the system.

RESUMEN

Complex dynamics of the most frequently used tritrophic food chain model are investigated in this paper. First it is shown that the model admits a sequence of pairs of Belyakov bifurcations (codimension-two homoclinic orbits to a critical node). Then fold and period-doubling cycle bifurcation curves associated to each pair of Belyakov points are computed and analyzed. The overall bifurcation scenario explains why stable limit cycles and strange attractors with different geometries can coexist. The analysis is conducted by combining numerical continuation techniques with theoretical arguments.