Understanding the evolution of pollutants via hierarchical complexity of space-time deterministic and stochastic dynamical systems

Abstract

This manuscript focuses on the implementation of the hierarchical complexity of space-time deterministic and stochastic dynamical systems to study the pollution dispersion behavior. Considering the concurrent environmental scope and requisites to understand the evolution of various types of environmentally related pollutants of high concern, herein, several suitable mathematical models are anticipated. Aiming to study the current pollution phenomenon at hand, we employed a lumped-linear or nonlinear structure and directly discussed in support of relevant equations. Up to some extent, by intuition, the researcher knows which model is more complex (suitable) than others, so the basic concepts are coated with linked references. Hence, the structural complexity features of the dynamical system are discussed in detail. The continuous dynamical system is discretized, and from the associated time series, a complexity measure can be obtained. There also exists a research gap on complexity theory, which generally deals with the behavior (solutions to the representing differential equations) in a system. Taking all these into account to cover the left behind literature gaps, herein, we propose to glimpse a family of classical models used to describe pollution and bacterial dispersion in the environment. From this review, we offer a qualitative complexity measure to each modeling paradigm by taking into account the underlying space of definition of the model and the key issue of the related differentiability. For instance, a lumped-linear set of differential equations is relatively simple with respect to its nonlinear counterpart because the former lives in the three-dimensional (3-D) real space R³, where the notion of differentiability shows up naturally. However, the latter needs to translate such conception to a manifold by means of differential geometry. Going further, we reflected on this issue for random systems where the notion of differentiability is transformed into an integral equivalence by means of Ito's lemma and so on for more exotic modeling perspectives. Moreover, the study presents a qualitative measure of complexity in terms of underlying sets and feasibility of differentiability.

Keywords: Bacterial spreading; Complexity of dynamical systems; Environmental impact; Paradigms in mathematical modeling; Pollution dispersion models.