Geometrical methods for analyzing the optimal management of tipping point dynamics

Abstract

AbstractNatural resources are not infinitely resilient and should not be modeled as being such. Finitely resilient resources feature tipping points and history dependence. This paper provides a didactical discussion of mathematical methods that are needed to understand the optimal management of such resources: viscosity solutions of Hamilton–Jacobi–Bellman equations, the costate equation and the associated canonical equations, exact root counting, and geometrical methods to analyze the geometry of the invariant manifolds of the canonical equations.Recommendations for Resource Managers Management of natural resources has to take into account the possible breakdown of resilience and induced regime shifts. Depending on the characteristics of the resource and on its present and future economic importance, either for all initial states the same kind of management policy is optimal, or the type of the optimal management policy depends on the initial state. Modeling should reflect the finiteness of the data.