NUMERICAL REVISIT TO A CLASS OF ONE-PREDATOR, TWO-PREY MODELS

Abstract

Some observations are made on a class of one-predator, two-prey models via numerical analysis. The simulations are performed with the aid of an adaptive grid method for constructing bifurcation diagrams and cell-to-cell mapping for global analysis. A two-dimensional bifurcation diagram is constructed to show that regions of coexistence of all three species, which imply the balance of competitive and predatory forces, are surrounded by regions of extinction of one or two species. Two or three coexisting attractors which may have a chaotic member are found in some regions of the bifurcation diagram. Their separatrices are computed to show the domains of attraction. The bifurcation diagram also contains codimension-two bifurcation points including Bogdanov–Takens, Gavrilov–Guckenheimer, and Bautin bifurcations. The dynamics in the vicinity of these codimension-two bifurcation points are discussed. Some global bifurcations including homoclinic and heteroclinic bifurcations are investigated. They can account for the disappearance of chaotic attractors and limit cycles. Bifurcations of limit cycles such as transcritical and saddle-node bifurcations are also studied in this work. Finally, some relevant calculations of Lyapunov exponents and power spectra are included to support the chaotic properties.