The purpose of this article is to contribute to the exploration of non-convex dynamics in coupled human-nature systems. We study welfare issues associated with the management of a human-nature complex adaptive system with a threshold and a stochastic driver. We exemplify with a specific system where we link changes in the number and diversity of birds to the abundance of a pest (insects) that causes damages to goods and services valuable to human beings. We present a method that simplifies the analysis and helps us discuss different management models that combine direct and indirect controls of the pest. This allows us to show that 1) the choice of control method depends in a highly non-linear way on biodiversity characteristics and 2) the socially optimal outcome may not be reachable using price instruments. Hence the price vs. quantity debate needs to be revisited using a complex adaptive system lens.