Different entry and exit times and overlapping generations of players are common in real-life game situations. In addition, durable strategies which have effects over a period of time are no less common than nondurable strategies which have only one-shot effects. This paper develops a new class of dynamic games which contains durable strategies with asynchronous players' horizons. The optimization techniques for solving asynchronous horizons durable strategies control are derived. Noncooperative game equilibria and cooperative optimal solution are presented. An asynchronous horizons durable strategies dynamic environmental game is provided to analyze the seemingly catastrophe-bound environmental degradation problem. The Price of Anarchy (PoA) in cross-generational exploitation of environmental commons is calibrated. A cooperative solution with a dynamically stable compensatory scheme is presented to alleviate the problem.