Hybrid matrix models and their population dynamic consequences
Institute of Mathematics, Academy of Mathematics and System
Sciences, Academia Sinica, Beijing, 100080, P.R. China. email@example.com.
In this paper, the main purpose is to reveal what kind of qualitative dynamical changes a continuous age-structured model may undergo as continuous reproduction is replaced with an annual birth pulse. Using the discrete dynamical system determined by the stroboscopic map we obtain an exact periodic solution of system with density-dependent fertility and obtain the threshold conditions for its stability. We also present formal proofs of the supercritical flip bifurcation at the bifurcation as well as extensive analysis of dynamics in unstable parameter regions. Above this threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behavior of the single species model with birth pulses are very complex, including small-amplitude annual oscillations, large-amplitude multi-annual cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allows for a period-doubling route to chaos. Finally, we discuss the effects of generation delay on stability of positive equilibrium (or positive periodic solution), and show that generation delay is found to act both as a destabilizing and a stabilizing effect.
Mathematics Subject Classification: 58J90 / 92D40
Key words: Hybrid matrix model / birth pulse / supercritical flip bifurcation / stroboscopic map / generation delay.
© EDP Sciences, SMAI, 2003