Theoretical Principles and Formal Criteria for Interpreting Nonlinear Effects in the Analysis of Models of Biophysical and Extreme Invasive Processes

Abstract

The construction of predictive models of biophysical processes with extreme and threshold phenomena, such as invasive processes with destruction of new habitat or sporadically renewed waves of epidemics, is an important issue. The practical application of the results of calculations with chaotic modes and bifurcations for biophysical models with nonlinear effects has no approved and generally accepted methodology. The experience of applying scenario modeling with the logic of changing the external influence in the biophysical process (for example due to the time of immune activation) led the author to develop a theory based on the interval methodology in order to limit the possibility of metamorphosis of the phase portrait of hybrid and iterative models. In terms of biophysical interpretability, the validity of behavior under parameter changes, and consistency with observations of invasion processes, many chaotic effects and periodic windows are redundant and violate the validity of the results. With the use of functional iterative designs, an overly diverse spectrum of trajectory behaviors can be realized. Principles have been developed for selecting the necessary nonlinear effects that are closely intertwined in the parameter space with redundant ones, which occurs in models based on dynamical systems with a variable evolution operator, variants of which were proposed by the author earlier. The choice of a mathematical apparatus for modeling changes in biophysics is a separate problem. The formulated theoretical framework serves as a basis for determining the applicability of models with complex behavior and extreme oscillatory dynamics. It is shown that value intervals with an internal crisis of an attractor can imperceptibly and narrowly wedge into a range of admissible variation of parameters in a computational experiment. The existence of parametric ranges in known models, for which trajectory behavior cannot be properly interpreted in biological problems, has been revealed. Signs and characteristics are determined for a multifactor iterative model of populations, which can generate excessive nonlinear effects that are incomparable with ecological reality. A method is proposed for distinguishing bifurcation-inducing model parameters with their representation as interval-defined values to exclude excessive phase-portrait metamorphosis using the example of models of dynamics of invasive processes in aquatic biosystems. The methods of the proposed theory are used by us for simulation of invasion processes in the Caspian Sea.