Stability in discrete dynamical systems containing independent subsystems

Abstract

Motivated by problems frequently occurring in the analysis of evolutionary ecological and population genetic models, discrete dynamical systems on the product of compact topological spaces are considered which contain an independent subsystem on one component space. If this subsystem possesses an equilibrium point, then a dynamical system is defined on the other component space whenever the independent system is in equilibrium. It is shown that in the presence of global asymptotic stability in the two component systems, perturbation will result in a return of the entire system to equilibrium (i.e. in particular, the equilibrium is globally asymptotically stable). A weakening of these conditions to combinations of global attractivity and global asymptotic stability either no longer guarantees convergence, or else the equilibrium is globally attractive but unstable, and thus sensitive to further perturbation.