Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web

Complex dynamics of a four-species food web with two preys, one middle predator, and one top predator are investigated. Via the method of Jacobian matrix, the stability of coexisting equilibrium for all populations is determined. Based on this equilibrium, three bifurcations, i.e., Hopf bifurcation, Hopf-Hopf bifurcation, and period-doubling bifurcation, are analyzed by center manifold theorem, bifurcation theorem, and numerical simulations. We reveal that, influenced by the three bifurcations, the food web can exhibit very complex dynamical behaviors, including limit cycles, quasiperiodic behaviors, chaotic attractors, route to chaos, period-doubling cascade in orbits of period 2, 4, and 8 and period 3, 6, and 12, periodic windows, intermittent period, and chaos crisis. However, the complex dynamics may disappear with the extinction of one of the four populations, which may also lead to collapse of the food web. It suggests that the dynamical complexity and food web stability are determined by the food web structure and existing populations.