Сriteria for Adequacy of the Cascade of Feigenbaum Bifurcations and Cycles of the Sharkovsky Ordering in Models of Transformable Biophysical Processes

Abstract The problem of development of a noncontradictory method for computer simulation for biophysical processes with clearly manifested staging and critical transformation is considered. Analogous models proposed earlier exhibit different behaviors with emergence of bifurcations, such as even and odd cycles, the coexistence of which is determined by the Sharkovsky theorem and in the limit of complication of cyclic behavior by trajectory chaotization. If, however, a model is interpreted in the field of biophysics, the coexistence of nonlinear effects becomes contradictory. In predicting the dynamics of biological resources with account for a regulating action, the iteration models generate undesirable nonlinear regimes of behavior (for example, in the case of the familiar Feigenbaum scenario). Complex effects connected with one another appear in cascades of emerging cycles with period p = 2i + 1, i → ∞, or in a cascade of cycles with p = 2i – 1, i → 0. The effects of an infinite bifurcation cascade and periodicity windows are determined by the fulfillment of the conditions of the Singer theorem. In this study, it is substantiated that bifurcations combined in one scenario cannot be explained in environmental reality and are not reflected in observed biophysical systems. Such mathematical artifacts turn out to be general for several biophysical models that differ drastically in their theoretical foundations. In this article, the insufficiently studied problem of essential interpretability (based on biological statistics) of the behavior of the class of computational models that are widely used in practical nature management. The chaotization in the real population dynamics has properties differing from those obtained in the cascade of period-doubling bifurcations. The formation of a nonattracting chaotic set in the form of a strange repeller better corresponds to actual chaos. It is shown that for describing transformation of processes in biological systems with external action (e.g., exploited population collapse), it is appropriate to use models with the emergence of alternative attractors. Such models better correspond to transitions between states of populations under exploitation than models with realization of bifurcation cycle cascades, strange Kantor attractors, and chaos regimes in the formulation of Lie and York in the form of a continuum of unstable trajectories of all periods. The hybrid models of living cycle with evolution stages that have been developed by the author earlier are in conformity with the proposed criteria for essential interpretation in ecology and in the prediction of biological systems. The correspondence analysis is based on scenarios of degradation of complexly structured population of red king crab and cod at shores of Kodiak Island (Alaska), stock crisis of salmon the Caspian Sea, and cod at the Canada shore, as well as bursts of population of insects, the models of which were described in the previous publications of the author.

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