Codimension-Two Bifurcations of Fixed Points in a Class of Discrete Prey-Predator Systems

The dynamic behaviour of a Lotka-Volterra system, describedby a planar map, is analytically and numerically investigated. We deriveanalytical conditions for stability and bifurcation of the fixed points ofthe system and compute analytically the normal form coefficients for thecodimension 1 bifurcation points (flip and Neimark-Sacker), and so establishsub- or supercriticality of these bifurcation points. Furthermore,by using numerical continuation methods, we compute bifurcation curvesof fixed points and cycles with periods up to 16 under variation of oneand two parameters, and compute all codimension 1 and codimension 2bifurcations on the corresponding curves. For the bifurcation points, wecompute the corresponding normal form coefficients. These quantitiesenable us to compute curves of codimension 1 bifurcations that branchoff from the detected codimension 2 bifurcation points. These curvesform stability boundaries of various types of cycles which emerge aroundcodimension 1 and 2 bifurcation points. Numerical simulations confirmour results and reveal further complex dynamical behaviours