bifurcations of one-dimensional maps
nonlinear phenomena and mathematical modeling of ecological systems
Article is considered the problem of construction discrete models of dynamics of biosystems on the analysis of observational data based on the consideration of specific examples. It is necessary to classify the models of mathematical biology and do it better, depending on the features of the theoretical basis and interpretation of results, as it contributes to the possibility of modifying models and further development of modeling techniques. That is the problem of justifying the theoretical basis on which formed a dynamic model, is a common «weakest link», according to experts and practitioners in the natural sciences. Many of the most well-known mathematical models use highly simplified representations about the development of interacting populations and the existing mechanisms of internal population regulation.
We consider two examples constructing the dependence of the stock and reproduction analysis of observational data from the work of Ricker on the graphs. Examples show how involving the approximation is lost forecasting properties. For the plotted curves, as operators of dynamic systems, aperiodic dynamics is impossible. At analysis of properties of the model as the evolution operator in reality a classifying feature is a mathematical value, that have no biological interpretation. Thus, to assess opportunities of behavior change trajectory has a value of negative Schwarzian derivative and a model of fish reproduction in fact be classified as SU-map in a series of mathematical objects. All the complexity of the dynamics of SU-maps, and especially an infinite cascade of period doubling with Feigenbaum constant, expiring the formation of a chaotic attractor is homeomorphic to the Cantor set is the result of fact proved in Singer Theorem. Behavior of the two-parameter Shepard map contradictory, since the complication of behavior occurs in the amplification of the limiting factors. This property makes one wonder about the availability of essential interpretation of the principle of doubling cascade.
The second part describes the new behavior change developed by the author does notunimodal population model. Then at a certain level fisheries exposure right branch of the curve will be below bisector of the coordinate angle. Inverse tangent bifurcation occurs when the fixed points merge into one and it disappears. Dynamical system remains two unstable nontrivial fixed points. After these tangent bifurcation appears stable chaotic regime. Emerged chaotic attractor is fundamentally different from forming in the accumulation point of an infinite cascade of period doubling of Feigenbaum attractor, and is a horseshoe attractor. Sensitive dependence on initial conditions arises due to the presence of transitivity of the evolution operator of the new model.