Centralized and Competitive Extraction for Distributed Renewable Resources with Nonlinear Reproduction

TL;DR

The paper addresses optimal and strategic extraction of a spatially distributed renewable resource with mass-conserving migration on a strongly connected network and nonlinear concave growth. Using dynamic programming, it derives closed-form value functions and optimal feedback rules for a central planner and constructs a symmetric Markovian Nash equilibrium for a decentralized game; long-run spatial allocations are governed by the dominant eigenvector $\zeta$ of $D^\top$ and the aggregate mass $m$ couples to the distribution through the growth function $\varphi(m)$. For three canonical growth families, logistic, power, and log-type saturating laws, the authors obtain explicit policies with $c_i^*(x)=(\partial v/\partial x_i)^{-1/\sigma}$ (planner) and a symmetric per-site coefficient $\hat{\theta}$ (equilibrium), along with $V(x)=A\,u(\langle \mathbf e,x\rangle)+B$ and convergence results for stock shares toward $\zeta_\theta$. The work highlights a clear efficiency wedge between centralized and decentralized extraction, quantifies network-structure effects via Perron–Frobenius geometry, and provides a rigorous benchmark for spatial resource management under nonlinear population dynamics.

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.

Theorems & Definitions (29)

• Example 1: Symmetric diffusion of Fick's type
• Remark 1
• Remark 2
• Lemma 1
• Lemma 2
• Corollary 1
• Theorem 1: Existence and Uniqueness of the Optimal Strategy
• Remark 3
• Corollary 2
• Proposition 1: Evolution of the Total Mass