Evolution of the population of Microtus Epiroticus: the Yoccoz-Birkeland model.
We study the discrete 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. We provethat this discreteversionhasan attractorΛsuch that, for certain parameter values, the system restricte...
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| Format: | Text |
| Language: | English |
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2011
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| Online Access: | http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.309.9137 http://premat.fing.edu.uy/papers/2011/125.pdf |
| Summary: | We study the discrete 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. We provethat this discreteversionhasan attractorΛsuch that, for certain parameter values, the system restricted to Λ exhibits sensibility to initial conditions. This means that, up to the validity of the model, the behavior in time t of the population N(t) of Microtus Epiroticus is concentrated in Λ and varies unpredictably. The attractor contains a hyperbolic 2-periodic point p implying that for rather large periods the behavior of N(t), when near the orbit ofp, is almostthe same. This fact togetherwith sensibility to initial data imply that N(t) suddenly changes dramatically. Under certain assumptions, sustained by numerical simulations, the system is topologically mixing (see definition 4.1), implying that all possible states in the attractor are attained. This explains some of the high oscillations of N(t) observed experimentally. Moreover, we estimate the order-2 Kolmogorov entropy of the system obtaining a positive value which is usually associated with chaos. Albeit this, the system has the permanence property(see definition 2.1): the population does not extinguish. Finally we give numerical evidence of the existence of a homoclinic point associated to p in the model. This explains, from a theoretical viewpoint, the existence of positive entropy. |
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