Stability and restoration phenomena in competitive Systems

A conservation law along with stability, recovering phenomena, and characteristic patterns of a nonlinear dynamical system have been studied and applied to physical, biological, and ecological systems. In our previous study, we proposed a system of symmetric 2n-dimensional conserved nonlinear differential equations. In this paper, competitive systems described by a 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,whichwe call the standard rhythm of a nonlinear dynamical system. The stability and restoration phenomena are strongly related to a conservation law and the 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.