Stability and restoration phenomena in competitive systems

Abstract

A conservation law and stability, recovering phenomena and characteristic patterns of a nonlinear dynamical system have been studied and applied to biological and ecological systems. In our previous study, we proposed a system of symmetric 2n-dimensional conserved nonlinear differential equations with external perturbations. In this paper, competitive systems described by 2-dimensional nonlinear dynamical (ND) model with external perturbations are applied to population cycles and recovering phenomena of systems from microbes to mammals.The famous 10-year cycle of population density of Canadian lynx and snowshoe hare is numerically analyzed. We find that a nonlinear dynamical system with a conservation law is stable and generates a characteristic rhythm (cycle) of population density, which we call the standard rhythm of a nonlinear dynamical system. The stability and restoration phenomena are strongly related to a conservation law and balance of a system. The standard rhythm of population density is a manifestation of the survival of the fittest to the balance of a nonlinear dynamical system. 1

Noether’s Conservation Laws and Stability in Nonlinear Conservative Interactions

“…When initial starting values and nonlinear coefficients are inconsistent, one obtains no solutions and the Ψ-function diverges [8]. At present, we do not know of any theoretical relations among values of nonlinear coefficients and initial starting values in conservative nonlinear equations.…”

“…By employing the 2-variable conservative nonlinear model [7], it is possible to numerically simulate biological data such as the 10-year cycle of Canadian lynx and snowshoe hare, microbe food-web in the Okanagan lake [8]. We review briefly the 2-variable conservative nonlinear interactions, because solutions of July 2016 | Volume 3 | e2592 the 2-variable model is used to construct the solution of the 4-variable conservative nonlinear model.…”

“…Because nonlinear coefficients are not entirely free parameters, it is not possible to find solutions by selecting any nonlinear coefficients and initial conditions of scarce prey and plentiful predators, and atto-fox type problems are restricted. A specific numerical example is shown in the paper [8]. The solutions to 2n-variable conservative nonlinear equations are different from those of conventional, nonconservative nonlinear equations.…”

“…In our previous study, we proposed a conservative nonlinear dynamical model in 2n-variables which is primarily for nonlinear competitive interactions. The model provided a method for studying the relation between a conservation law and stability of a 2n-dimensional competitive system and restoration phenomena [7] [8].…”

“…There may be some conservation laws corresponding to the stability observed in microbiological systems. This is a motive that we applied nonlinear analyses based on Noether's theorem to dynamics, and interesting results for complex phenomena are discussed [7] [8].…”