Discrete May–Leonard Competition Models I

Abstract

The local dynamics of two discrete-time models applicable to three competing plant species are shown to have properties similar to the May–Leonard (M-L) differential equation model. The dynamics of the two discrete models are shown to be similar. However, they are not dynamically consistent with the continuous M-L model. Unlike the continuous M-L model, the Hopf bifurcations of the discrete M-L models are not degenerate. The continuous M-L model is the limiting case of the discrete models.

Keywords:

• Lotka–Volterra
• Competition
• Difference equations
• May–Leonard systems
• Hopf bifurcation
• Primary: 39A11

Acknowledgements

We thank Jack Hale for pointing out some Hopf bifurcation results in 3-dimensional systems. We thank the referees for their helpful comments and suggestions. The research of the second author was partially supported by National Science Foundation Grant, DMS-0201105.

Notes

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