Permanence for replicator equations

Many dynamical systems display strange attractors and hence orbits that are so sensitive to initial conditions as to make any long-term prediction (except on a statistical basis) a hopeless task. Such a lack of Ljapunov stability is not always crucial, however: Lagrange stability may be more relevant. Thus, for some models the precise asymptotic behavior — whether it settles down to an equilibrium or keeps oscillating in a regular or irregular fashion — is less important than the fact that all orbits wind up in some preassigned bounded set. The former problem can be impossibly hard to solve and the latter one easy to handle