Multiple mixed-type attractors in a competition model

Abstract

We show that a discrete-time, two-species competition model with Ricker (exponential) nonlinearities can exhibit multiple mixed-type attractors. By this is meant dynamic scenarios in which there are simultaneously present both coexistence attractors (in which both species are present) and exclusion attractors (in which one species is absent). Recent studies have investigated the inclusion of life-cycle stages in competition models as a casual mechanism for the existence of these kinds of multiple attractors. In this paper we investigate the role of nonlinearities in competition models without life-cycle stages.

Keywords:

• Competitive exclusion principle
• Coexistence cycles
• Multiple attractors

Acknowledgements

We thank R.F. Constantino, J. Edmunds, S.L. Robertson and S. Arpin for helpful discussions. J.M. Cushing and S.M. Henson were supported in part by NSF grant DMS-0414142.

Notes

^†Both eigenvalues of a 2 × 2 matrix A have absolute value less than 1 if and only if, −1< det A<1 and−(1+ det A)<trA<1+ det A. At least one eigenvalue has absolute value greater 1 if and only if one of the inequalities is reversed.