The bifurcations and dynamics of certain quadratic maps of the plane

Discrete models of density-dependent population growth provide simpleexamples of dynamical systems which exhibit complicated dynamics. Singleage-class models lead to the study of maps of an interval into itself andwe outline the main results which are known in this case.Here our main concern is with a two age-class model, due to Maynard Smith,which takes the form of a two-parameter family of maps of the plane with aquadratic non-linearity. After a description of the local bifurcations ofgeneral two-parameter families .in the plane we give a linear stabilityanalysis for the fixed points in our model and analyse their bifurcations.This local theory is extended by finding numerically the boundaries of theregions in the parameter space where the map has a periodic orbit of lowperiod created by resonant Hopf bifurcation. A series of computer-drawnphase portraits is presented for a one-parameter path through the parameterplane, showing .the creation of an attracting invariant circle by a Hopfbifurcation followed by a passage to a more complicated attractor.We examine a three-parameter family connecting our map to the quadraticdiffeomorphism studied by Henon and conclude with a discussion of a morerealistic model which, however, contains the same complexities associatedwith resonant Hopf bifurcation found in the simpler model.