Long-Term and Short-Term Dynamics of Microtus epiroticus: A Yoccoz--Birkeland Model

Abstract

We study the discrete version of a dynamical system given by a model proposed by Yoccoz and Birkeland to describe the evolution in time of the population $N(t)$ of Microtus epiroticus (voles) on the Svalbard Islands. We prove that this model always has an attractor $\Lambda$ which is trivial for most of the parameter values tested but that for certain parameter values exhibits sensitivity to initial conditions. This means that, up to the validity of the model and for these particular parameter values, the behavior in time $t$ of the population $N(t)$ of Microtus epiroticus is concentrated in the attractor and varies unpredictably. This is in concordance with the experimental data collected by several researchers. In particular, for some specific choice of parameters, sustained by numerical simulations, the system is transitive and has positive order-2 Kolmogorov entropy, implying that the system presents chaos. For this choice of parameters we give numerical evidence of the existence of homoclinic points associated with a 2-periodic point presented in the attractor. We also analyze the short-term dynamics of the population of these mammals. Surprisingly, for most parameter values, the short-term behavior is almost the same as for the long-term; i.e., it seems that the behavior of the population in time is not particularly affected by transient data.