Hybrid matrix models and their population dynamic consequences

Sanyi Tang

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 3, page 433-450
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Tang, Sanyi. "Hybrid matrix models and their population dynamic consequences." ESAIM: Mathematical Modelling and Numerical Analysis 37.3 (2010): 433-450. <http://eudml.org/doc/194172>.

@article{Tang2010,
abstract = { 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. },
author = {Tang, Sanyi},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Hybrid matrix model; birth pulse; supercritical flip bifurcation; stroboscopic map; generation delay.; hybrid matrix model; stroboscopic map; generation delay},
language = {eng},
month = {3},
number = {3},
pages = {433-450},
publisher = {EDP Sciences},
title = {Hybrid matrix models and their population dynamic consequences},
url = {http://eudml.org/doc/194172},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Tang, Sanyi
TI - Hybrid matrix models and their population dynamic consequences
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 3
SP - 433
EP - 450
AB - 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.
LA - eng
KW - Hybrid matrix model; birth pulse; supercritical flip bifurcation; stroboscopic map; generation delay.; hybrid matrix model; stroboscopic map; generation delay
UR - http://eudml.org/doc/194172
ER -

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