We present a general framework to study compact limit sets of trajectories for a class of nonautonomous systems, including asymptotically autonomous differential equations, certain stochastic differential equations, stochastic approximation processes with decreasing gain, and fictitious plays in game theory. Such limit sets are shown to be internally chain recurrent, and conversely. © 1996 Plenum Publishing Corporation.

Smale's approach [13] to the classical two-players repeated Prisoner's Dilemma game is revisited here for N-players and Network games in the framework of Blackwell's approachability, stochastic approximations and differential inclusions.

Understanding under what conditions interacting populations, whether they be plants, animals, or viral particles, coexist is a question of theoretical and practical importance in population biology. Both biotic interactions and environmental fluctuations are key factors that can facilitate or disrupt coexistence. To better understand this interplay between these deterministic and stochastic forces, we develop a mathematical theory extending the nonlinear theory of permanence for deterministic systems to stochastic difference and differential equations. Our condition for coexistence requires that there is a fixed set of weights associated with the interacting populations and this weighted combination of populations' invasion rates is positive for any (ergodic) stationary distribution associated with a subcollection of populations. Here, an invasion rate corresponds to an average per-capita growth rate along a stationary distribution. When this condition holds and there is sufficient noise in the system, we show that the populations approach a unique positive stationary distribution. Moreover, we show that our coexistence criterion is robust to small perturbations of the model functions. Using this theory, we illustrate that (i) environmental noise enhances or inhibits coexistence in communities with rock-paper-scissor dynamics depending on correlations between interspecific demographic rates, (ii) stochastic variation in mortality rates has no effect on the coexistence criteria for discrete-time Lotka-Volterra communities, and (iii) random forcing can promote genetic diversity in the presence of exploitative interactions.