Chapter 24. Floquet theory and averaging methods : Useful tools for equilibrium stability of dynamical model of Typha proliferation

Abstract

Motivated by the fact that Floquet theory and averaging methods used to study the stability of linear periodic systems in continuous time, we formulate and analyze the dynamics of a nonlinear and non-autonomous system of ordinary differential equations describing the dynamics of an invasive reproductive plant: the Typha. Its two modes of reproduction namely; sexual (via seeds) and asexual (via rizhomes) are included into the hybrid system which combines the features of classical continuous time and discrete time systems. Stability of the null equilibrium is investigated via the basic reproduction rate Ro of the model in the absence of Typha is computed. For $R_{0,\alpha} < 1$ the a useful too which can be applied to analyze the stability of models with seasonality. The theory of averaging is based on the construction of approximate solutions essentially first-order differential equation with rapidly oscillating ordinary. A condition of stability of the trivial equilibrium of the switching system is given. Numerical simulations to support the analytical results are provided.