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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Dynamics of a logistic population model with maturation delay and nonlinear birth rate

Pages: 735 - 752, Volume 5, Issue 3, August 2005      doi:10.3934/dcdsb.2005.5.735

 
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Suqi Ma - School of Science, Beijing University of Aeronautics and Astronautics, Beijing 100083, China (email)
Qishao Lu - School of Science, Beijing University of Aeronautics and Astronautics, Beijing 100083, China (email)
Shuli Mei - School of Science, Beijing University of Aeronautics and Astronautics, Beijing 100083, China (email)

Abstract: A logistic population model with a maturation delay stage for adults is investigated. The adult population is related to its previous life stage with a maturation delay $r$, and has a non-linear exponential birth rate $be^{-pr}$ with a birth decay coefficient $p$. As $r$ increases, the unique positive equilibrium solution may experience two stability switchings, that is, from stable to unstable, and then back to stable again. The decay coefficient $p$ can also qualitatively influence the stability property of the system. Hopf bifurcation and the stability of the bifurcating periodic solution are analyzed by means of the center manifold theory and the normal form technique. By applying the integral averaging theory, phase-locked and phase-shifting solutions induced by the external excitation are also investigated and verified by numerical simulations.

Keywords:  logistic population model, maturation delay, periodic solution, quasi-periodic solution, stability, Hopf bifurcation.
Mathematics Subject Classification:  82D10, 76X05, 35B60, 35B40.

Received: January 2004;      Revised: July 2004;      Available Online: May 2005.