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Persistence and periodic orbits of a three-competitor model with refuges
Y Takeuchi1, Y Oshime, H Matsuda
1Department of Applied Mathematics, Faculty of Engineering, Shizuoka University, Hamamatsu, Japan.
Mathematical Biosciences
|February 1, 1992
Summary
Introducing refuges into a four-patch model prevents species extinction, even with a stable competitive cycle. Diffusion allows for persistence and potential periodic orbits through Hopf bifurcation.
Area of Science:
- Mathematical Biology
- Ecology
- Dynamical Systems
Background:
- Ecological models often feature competitive exclusion.
- Heteroclinic cycles describe cyclic competition dynamics.
- Patch dynamics and species diffusion are crucial for persistence.
Purpose of the Study:
- To investigate the impact of refuges on species persistence in a competitive model.
- To analyze the conditions for stability and bifurcations in a multi-patch system.
- To explore the emergence of periodic orbits via diffusion.
Main Methods:
- Mathematical modeling of a four-patch system.
- Analysis of a heteroclinic cycle within a competitive patch.
- Incorporation of diffusion between competitive and refuge patches.
- Bifurcation analysis, including Hopf bifurcation.
Main Results:
- Refuges ensure model persistence despite an attracting heteroclinic cycle in the isolated patch.
- Varying the diffusion constant can induce Hopf bifurcation.
- Periodic orbits are shown to exist under specific diffusion conditions.
Conclusions:
- Refuge introduction is a viable strategy to maintain biodiversity in competitive ecosystems.
- Diffusion dynamics play a significant role in ecological stability and the potential for complex population dynamics.
- The model demonstrates how simple spatial structures can overcome competitive exclusion.