Lyapunov coefficients for degenerate Hopf bifurcations and an application in a model of competing populations

This paper focuses on the study of codimension one and two Hopf bifurcations and the pertinent Lyapunov stability coefficients for a general reaction-diffusion system. We display algebraic expressions for the first and second Lyapunov coefficients for the infinite dimensional system subject to Neumann boundary conditions. As an application, a special subset of a three-dimensional Lotka-Volterra dynamical system with diffusions subject to Neumann boundary conditions is analyzed. The main goal is to perform a detailed local stability analysis for the proposed predator-prey model to show the existence of multiple spatially homogeneous and non-homogeneous periodic orbits, due to the occurrence of codimension one Hopf bifurcation.