Investigation of the Class of One-Dimensional Unimodal Mappings Obtained in the Modeling of the Lemming Population

Abstract

A model that describes periodic changes in the lemming population size as a function of the physical characteristics of ecologically significant parameters of the habitat is analyzed. Earlier studies on modeling of the population size dynamics for tundra animals substantiated a new class of one-dimensional unimodal mappings of a state space into itself. The dependence of trajectory stability on parameter variation and the bifurcation and asymptotic properties of the trajectories are in the focus of attention when this class of mappings is studied. Methods for approximating implicitly defined sets are used for numerical analysis of the maps: trajectory tubes, attractors, and bifurcation diagrams are constructed. An example of a bifurcation diagram construction along the contour in the parameter space is given for this class of mappings; the diagram enables the analysis of the dependence of trajectory behavior on changes of the modeling map parameters. Thus, the use of the mapping of the type considered for the model description of the lemming population enabled the study of the effect of long-term biospheric rhythms, which include time intervals with extreme living conditions. The study showed that such impacts do not cause degeneration of the population: the periods with chaotic dynamics that emerge under extreme conditions are replaced by well-ordered behavior in the form of small-period population cycles.