A Problem on Invariant Manifolds arising in Population Genetics

In the Fisher — Wright — Haidane model from population genetics (see Crow and Kimura [6], Edwards [7], Hadeler [9], [10]) the mean fitness of a population of diploid individuals increases as time evolves and remains constant only when the population is in equilibrium. In mathematical terms this means that the mean fitness is a global Ljapunov function, which ensures that any solution of the underlying differential equation or difference equation, respectively, approaches the set Eof stationary solutions. Both in the continuous time and the discrete time version of the Fisher-Wright-Haldane model the set Egenerally is the union of linear submanifolds of the state space (see Hughes and Seneta [15] and Aulbach and Hadeler [5]) and so the following problem arises. IF A SOLUTION APPROACHES A CONTINUUM C OF STATIONARY SOLUTIONS, DOES IT THEN CONVERGE TO A POINT ON C OR MAY IT IGNORE THE STATIONARY FLOW ON C BY CREEPING ALONG C FOR ALL FUTURE TIME?As far as the concrete equations of the Fisher-Wright-Haldane model are concerned thi s question has found partial answers by Feller [8], an der Heiden [12], and Aulbach and Hadeler [5]. Meanwhile the general answer has been given by Losert and Akin [17].