Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 254, Issue 2, Pages 879-910Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2012.10.003
Keywords
Singular perturbation; Canard cycle; Limit periodic set; Predator-prey systems; Holling type response
Categories
Funding
- Faculty of Science and Engineering, York University
- NSERC of Canada
- Ministry of Research and Innovation, Ontario
- [NSFC-11171267]
- [NSFC-10831003]
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By using the singular perturbation theory developed by Dumortier and Roussarie and recent work of De Maesschalck and Dumortier, we study the canard phenomenon for predator-prey systems wills response functions of Rolling types. We first develop a formula for computing the slow divergence integrals. By using the formula we prove that for the systems with the response function of Rolling types III and IV the cyclicity of any limit periodic set is at most two, that is at most two families of hyperbolic limit cycles or at most one family of limit cycles with multiplicity two can bifurcate from the limit periodic set by small perturbations. We also indicate the regions in parameter space where the corresponding limit periodic set has cyclicity at most one or at most two. (C) 2012 Elsevier Inc. All rights reserved.
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