Canards and mixed-mode oscillations in a forest pest model.
We consider a three-variable forest pest model, proposed by Rinaldi & Muratori (1992) [Rinaldi, S., Muratori, S., 1992. Limit cycles in slow-fast forest-pest models. Theor. Popul. Biol. 41, 26-43]. The model allows relaxation oscillations where long pest-free periods are interspersed with outbreaks of high pest concentration. For small values of the timescale of the young trees, the model can be reduced to a two-dimensional model. By a geometrical analysis we identify a canard explosion in the reduced model, that is, a change over a narrow parameter interval from outbreak dynamics to small oscillations around an endemic state. For larger values of the timescale of the young trees the two-dimensional approximation breaks down, and a broader parameter interval with mixed-mode oscillations appear, replacing the simple canard explosion. The analysis only relies on simple and generic properties of the model, and is expected to be applicable in a larger class of multiple timescale dynamical models.
Brøns
Morten
M
Department of Mathematics, Technical University of Denmark, Building 303S, 2800 Kgs. Lyngby, Denmark. m.brons@mat.dtu.dk
Kaasen
Rune
R
D000818:Animals; D017753:Ecosystem; D015233:Models, Statistical; D008962:Models, Theoretical; D011157:Population Dynamics; D014197:Trees / Q000254:growth & development / Q000469:parasitology*