Limit Cycle in a Herbivore-Plant-Bee Model
Containing a Time Delay

Charoen Kaewpradit,^a Wannapong Triampo^b and I-Ming Tangb,^c,*

ABSTRACT: The dynamical behavior of time delay herbivore-plant-pollinator ecosystem is studied. The time delay arises from the fact that it takes time for a pollinated flower to develop into a new plant. A dynamical analysis is used to show that a stable steady state undergoes a Hopf bifurcation to a limit cycle behavior as the delay time crosses a critical value. This prediction is verified by numerically solving the set of first order differential equations. One finds that the trajectory which is spiraling into the steady state point when τ <τ₀ becomes a trajectory into a limit cycle about the state when τ > τ_c.

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a Department of Mathematics
b Department of Physics and Capability Building Unit in NanoScience & NanoTechnology Faculty of Science, Mahidol University, Bangkok 10400, Thailand
c Institute of Science & Technology for Research & Development Mahidol University, Nakhon Pathom, 73170, Thailand

*Corresponding author: e-mail address scimt@mahidol.ac.th

Received 28 Jul 2004, Accepted 23 Mar 2005