Stable limit cycles of time dependent multispecies interactions

The standard differential system which models the interaction of n species is considered under the assumption that the coefficients (i.e., the net birth rates, the self-inhibition coefficients and the interaction coefficients) are all periodic functions of time. Conditions are given which guarantee the existence of a stable periodic limit cycle. The basic result implies, roughly, that if an n - 1 species subcommunity with a stable periodic limit cycle exists and if the interactions of the system are sufficiently weak, then the addition of the nth species will result in a stable periodic limit cycle provided its average net birth rate is larger than and close to a specified critical value; on the other hand, if this average is less than but close to the critical value, then the nth species will not survive and the system will stabilize on the limit cycle of the subcommunity. Starting with basic results concerning periodic solutions of the one species case, we apply our basic result in a "bootstrapping" manner to derive a corollary which, roughly speaking, states that if at least one species has a positive average net birth rate and if the interactions are sufficiently weak, then a multispecies community will have a stable, positive limit cycle provided that the average net birth rates of the remaining species lie in certain specified ranges.