In this paper, a class of infinite-horizon, nonzero-sum differential games and their Nash equilibria are studied and the notion of epsilon(alpha)-Nash equilibrium strategies is introduced. Dynamic strategies satisfying partial differential inequalities in place of the Hamilton-Jacobi-Isaacs partial differential equations associated with the differential games are constructed. These strategies constitute (local) epsilon(alpha)-Nash equilibrium strategies for the differential game. The proposed methods are illustrated on a differential game for which the Nash equilibrium strategies are known and on a Lotka-Volterra model, with two competing species. Simulations indicate that both dynamic strategies yield better performance than the strategies resulting from the solution of the linear-quadratic approximation of the problem.