Diffusion-Induced Instability and Pattern Formation in Infinite Horizon Recursive Optimal Control

Abstract

This paper develops local stability analysis for deterministic optimal control theory for recursive infinite horizon intertemporal optimization problems where there is a continuum of spatial sites and the state variable can diffuse over these sites. We identify sufficient conditions for a type of local instability which emerges from the interaction of the discount rate on the future, the curvature of the Hamiltonian, and the spatial features of the problem. We call this phenomenon diffusion-induced instability. We give a self-contained treatment of the control theory needed for our particular context because, although it is covered in the mathematical literature, our treatment is much more transparent for economists readers. We illustrate our analytical methods with three stylized applications. The first application is the optimal management of spatially connected human dominated ecosystems. The second and third applications are harvesting of spatially interconnected renewable resources.

Keywords: Spatial analysis, Pattern formation, Optimal Control, Diffusion-Induced Instability.

JEL Classification: C6, Q2