Local bifurcation analysis of a five-species food web model with two coupled food chains
Local bifurcations of a steady state of a five-species food web model with two top-predator-coupled food chains were studied by a revised Gross's Method. The system was normalized at an interior equilibrium point and new general parameters describing the behavior of the normalized system were defined. Local bifurcation diagrams of the normalized model in various 3D general parameter settings were then obtained. A number of codimension-1 and codimension-2 bifurcations including the Double-Hopf (DH) bifurcation were identified. Multiple examples of the Whitney umbrella around a codimension-3 1:1 resonant DH point were shown for the first time to exist in ecologically plausible parameter regions. The results showed that breaking the symmetry between parameter pairs and varying parameters from both branches are necessary conditions to generate the DH bifurcation, which indicates possible coupling of two oscillations in the two branches. A stable 2-torus arising from such a coupling was also detected.