Global bifurcations of domains of feasible trajectories: An analysis of a discrete predator-prey model

This paper aims to provide some new results, by a computer-assisted study, on some global bifurcations that change the domains of feasible trajectories (bounded discrete trajectories having an ecological sense). It is shown that the domain boundaries of two-dimensional maps can be generally obtained by the union of all rank preimages of two axes. A discrete predator-prey ecosystem is employed to demonstrate how the global properties of persistence in a biological model can be analyzed. The main results of this paper are from the study of some global bifurcations that change the stable manifold of a saddle fixed point belonging to the domain boundary. The basin fractalization is explained by using two types of nonclassical singular sets. The first one, critical curve, separates the plane into two regions having a different number of real inverses (here zero and two). The second one is a line of nondefinition for one of the two inverses of the map, i.e. this inverse has a vanishing denominator on this line.