Local Stability and Lyapunov Functionals for n-Dimensional Quasipolynomial Conservative Systems Original Research Article

We present a method for determining the local stability of equilibrium points of conservative generalizations of the Lotka–Volterra equations. These generalizations incorporate both an arbitrary number of species—including odd-dimensional systems—and nonlinearities of arbitrarily high order in the interspecific interaction terms. The method combines a reformulation of the equations in terms of a Poisson structure and the construction of their Lyapunov functionals via the energy-Casimir method. These Lyapunov functionals are a generalization of those traditionally known for Lotka–Volterra systems. Examples are given.