Correspondence to biophysical criteria of nonlinear effects in the occurrence of Feigenbaum  bifurcation cascade in models of invasive processes
Authors: Perevaryukha A. Yu. 
Published: March 2023

Introduction. The problem of creating a set of criteria for practically substantiated computational modeling of a number of complex staged biophysical processes with pronounced stages and critical transformations, for example, aggressive invasions, is discussed. Known models have a variety of behavior with the occurrence of bifurcations according to the same scenarios, the appearance of cycles, the coexistence of which is determined by Sharkovskii’s theorem. In the limit of complication of cyclic behavior in such models, they often encounter chaotization of the trajectory, but with the existence of an infinite number of periodicity windows. The conditions for an infinite cascade of bifurcations for iterations are determined by the fulfillment of the conditions of Singer’s theorem. The purpose of this work is to show that most of the nonlinear effects associated with chaotization scenarios do not have an ecological interpretation, but we will propose ways to exclude non-interpretable parametric ranges.

Materials and methods. Using methods for estimating the stability of stationary states and cyclic trajectories using Singer’s theorem on the criterion for the occurrence of bifurcations for iterative models, we analyze interconnected nonlinear effects. The phenomena are considered on the example of cascades of the appearance of cycles of the period p = 2i + 1, i→∞ and a cascade of cycles p = 2i – 1, i→0 of “doubling” or “halfing” the period, which occur in ecological models often used to optimize fishing.

Results. It is confirmed that the coexistence of nonlinear effects turns out to be contradictory if the simulation results are interpreted in the field of biocybernetics, on the basis of model and real examples. Iterative models generate unnecessary non-linear modes of behavior, when predicting the dynamics of invasions or harvesting bioresources, taking into account the regulatory impact, for example, in the case of the well-known Feigenbaum scenario. It has been established that bifurcations connected in one scenario have no explanation in ecological reality and are not reflected in the observed biophysical systems. These mathematical artifacts are common to several biophysical models that are very different in their theoretical foundations. Chaotization in real population dynamics has somewhat different properties than can be obtained in a cascade of period doubling bifurcations. The formation of a non-attractive chaotic set in the form of a strange repeller is more consistent with the dynamics of the development of fast invasions.

Discussion and conclusions. It is shown that to describe the transformations of biosystemic processes with external influence, as the collapse of a commercial population, it is adequate to use models with the emergence of alternative attractors. These models correspond better to the transitions between the states of populations under the influence of fishing than models with the implementation of cascades of bifurcations of cycles, strange Cantor attractors and chaos regimes in the form of a continuum of unstable trajectories of all periods. The most promising are hybrid models of the life cycle with developmental stages for essential interpretation in ecology and forecasting of biosystems, as they allow to determine the parametric ranges of functioning and exclude unacceptable ranges of parameters where excessive nonlinear effects occur, which have no justification for population processes. The analysis of the adequacy criteria is based on degradation scenarios for a complexly structured sturgeon population in the Volga basin, cod off the coast of Canada, outbreaks of invasive insects, and the spread of the invasive ctenophore Mnemiopsis leidy in the Caspian Sea.