Symmetric Equilibria in Spatially Distributed Extraction Games with Nonlinear Growth

We study optimal and strategic extraction of a renewable resource that is distributed over a network, migrates mass-conservingly 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 noncooperative 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.