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Periodic orbits near heteroclinic cycles in a cyclic replicator system
Yuanshi Wang1, Hong Wu, Shigui Ruan
1School of Mathematics and Computational Science, SunYat-sen University, Guangzhou 510275, People's Republic of China. mcswys@mail.sysu.edu.cn
Journal of Mathematical Biology
|June 10, 2011
Summary
In semelparous species with n=4 year classes, this study reveals how competition influences periodic orbits near a heteroclinic cycle. Weak competition yields stable orbits, while strong competition leads to their disappearance, explaining subpopulation extinction.
Area of Science:
- Ecology
- Mathematical Biology
- Dynamical Systems
Background:
- Semelparous species reproduce once and die.
- Year-class systems model population dynamics with discrete age structures.
- A heteroclinic cycle exists in the n=4 year-class Lotka-Volterra model.
Purpose of the Study:
- Investigate the existence, growth, and disappearance of periodic orbits near the n=4 heteroclinic cycle.
- Test a conjecture by Diekmann and van Gils (2009).
- Explain the impact of competitive interactions on population dynamics.
Main Methods:
- Analysis of the Poincaré map near the heteroclinic cycle.
- Introduction of a metric to quantify periodic orbit size.
- Differential equation modeling of n year-class Lotka-Volterra dynamics.
Main Results:
- Weak average competitive degree supports an asymptotically stable periodic orbit.
- Periodic orbits grow with increasing competition and converge to the heteroclinic cycle.
- Strong average competitive degree eliminates asymptotically stable periodic orbits.
Conclusions:
- Competitive interactions critically determine the stability and presence of periodic orbits.
- The findings explain the expansion and extinction of periodic solutions and subpopulations.
- Results align with and extend previous conjectures on year-class dynamics.
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