Bifurcation analysis in a diffusive ‘food-limited’ model with time delay

Abstract

The dynamics of a diffusive ‘food-limited’ population model with delay and Dirichlet boundary condition are investigated. The occurrence of steady state bifurcation with changes of parameter is proved by applying phase plane ideas. The existence of Hopf bifurcation at the positive steady state with the changes of specific parameter is obtained, and the phenomenon that the unstable positive equilibrium state without dispersion may become stable with dispersion under certain conditions, which is found by analysing the distribution of the eigenvalues. By the theory of normal form and centre manifold, an explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived. Examples of numerical simulation are carried out to support the analytic results.

Keywords:

• food-limited system
• diffusion
• delay
• steady state bifurcation
• Hopf bifurcation
• Dirichlet boundary condition

AMS Subject Classifications::

• 35K57
• 35B32
• 92B05

Acknowledgements

The authors wish to thank the referees for their valuable comments and suggestions that led to a truly significant improvement of the manuscript. This work was supported by the National Natural Science Foundation of China (No 10771045), Program of Excellent Team in HIT and Scientific Research of College of the Inner Mongolia (No NJ06024).