Centralized and Competitive Extraction for Distributed Renewable Resources with Nonlinear Reproduction

TL;DR

This paper develops explicit policies and equilibria for the optimal and strategic extraction of distributed renewable resources with nonlinear growth on networks, combining spatial dynamics, migration, and nonlinear reproduction laws.

Contribution

It introduces the first explicit policies and Markov equilibria for spatial resource extraction with nonlinear growth on general networks.

Findings

Closed-form value functions for three growth laws.

Explicit feedback rules for the planner.

Symmetric Markov equilibrium constructed on strongly connected networks.

Abstract

We study optimal and strategic extraction of a renewable resource that is distributed over a network, migrates mass-conservatively across nodes, and evolves under nonlinear (concave) growth. A subset of nodes hosts extractors while the remaining nodes serve as reserves. We analyze a centralized planner and a non-cooperative game with stationary Markov strategies. The migration operator transports shadow values along the network so that Perron-Frobenius geometry governs long-run spatial allocations, while nonlinear growth couples aggregate biomass with its spatial distribution and bounds global dynamics. For three canonical growth families, logistic, power, and log-type saturating laws, under related utilities, we derive closed-form value functions and feedback rules for the planner and construct a symmetric Markov equilibrium on strongly connected networks. To our knowledge, this is the first paper to obtain explicit policies for spatial resource extraction with nonlinear growth and, a fortiori, closed-form Markov equilibria, on general networks.