Evolution of the population of Microtus Epiroticus: the Yoccoz-Birkeland model

TL;DR

This paper analyzes a discretized dynamical system modeling Microtus Epiroticus population dynamics, demonstrating complex behaviors like chaos, mixing, and positive entropy, supported by theoretical proofs and numerical evidence.

Contribution

It provides a rigorous analysis of the Yoccoz-Birkeland model's attractor, hyperbolic periodic points, and chaotic dynamics, supported by numerical simulations.

Findings

Existence of an attractor with a hyperbolic 2-periodic point

System exhibits sensitivity to initial conditions and topological mixing

Positive Kolmogorov entropy indicating chaos

Abstract

We study the discretized version of a dynamical system given by a model proposed by Yoccoz and Birkeland to describe the evolution of the population of Microtus Epiroticus on Svalbard Islands, see http://zipcodezoo.com/Animals/M/Microtus_epiroticus . We prove that this discretized version has an attractor Lambda with a hyperbolic 2-periodic point p in it. For certain values of the parameters the system restricted to the attractor exhibits sensibility to initial conditions. Under certain assumptions that seems to be sustained by numerical simulations, the system is topologically mixing explaining some of the high oscillations observed in Nature. Moreover, we estimate its order-2 Kolmogorov entropy obtaining a positive value. Finally we give numerical evidence that there is a homoclinic point associated with p.