Lagrangian fuzzy dynamics and behavior of living beings in the environment: Peace, war and ecological catastrophes

In this study, we extended our previous Lagrangian Fuzzy dynamics (S-Lagrangian dynamics) to the systems of fields. Unlike common Lagrangian dynamics, additional “ghost”-fields, which have dual roles, appear in S-Lagrangian dynamics equations. These fields design hierarchical structures of a system and generate “ghost”-forces. In addition, the dynamics of these fields could lead to time irreversibility of the dynamics equations. Because the inference of S-Lagrangian equations is based only the causality principle, the common local topology of a system’s state space, and assumptions that infinite velocities are impossible and that the system’s state can be described by a compact fuzzy set, S-Lagrangian dynamics can be applied to physical and nonphysical systems as well. We applied S-Lagrangian dynamics to describe the behavior of populations of living species in the environment, assuming that the system’s Lagrangian is defined by the dependence of velocity of changing of the population’s stress on population densities, food supply, and environmental damage. The results show that even the first approximation of the Lagrangian allows the description of almost all types of species co-evolution: competition, coexistence, prey–predator dynamics, and ecological catastrophes.