Bifurcation Structure of a Three-Species Food-Chain Model

A three-species food chain model utilizing type II functional responses and allometric relationships is analyzed mathematically. A reduction of the two-dimensional nullsurfaces to a set of one-dimensional curves allows for an intuitive understanding of the equilibria structure. With the reduction in hand, we then perform a local bifurcation analysis around an organizing center and categorize the entire parameter space into twelve different regions of dynamic behaviour. These regions in parameter space are characterized by an extremely rich set of dynamical behaviours, including multiple domains of attraction, quasi-periodicity, chaos, homoclinic events, and transient chaos. From this mathematical analysis it is possible to qualify the type of population dynamics under any given parameter set.