Stability and bifurcation analysis in predator-prey system involving Holling type-II functional response

In this article, we focused on the study of codimension-one Hopf bifurcations and the associated Lyapunov stability coefficients in the context of general two-dimensional reaction-diffusion systems defined on a finite fixed-length segment. Algebraic expressions for the first Lyapunov coefficients are provided for the infinite-dimensional system subject to Neumann boundary conditions. As an application, a diffusive predator-prey system modeling competing populations with a Holling type-II functional response for the predator was analyzed and studied under Neumann boundary conditions. Our main goal is to perform a detailed, local stability analysis of the proposed model, showing the existence of multiple spatially homogeneous and non-homogeneous periodic orbits, arising from the occurrence of a codimension-one Hopf bifurcation.