Game-environment feedback dynamics in growing population: Effect of finite carrying capacity

TL;DR

This paper develops a mathematical model incorporating population growth and resource limitations to study the tragedy of the commons, revealing how finite carrying capacity can either prevent or cause resource depletion depending on initial conditions.

Contribution

It introduces a novel eco-evolutionary framework that includes density-dependent payoffs and logistic growth, extending prior models to account for finite population effects on resource dynamics.

Findings

Finite carrying capacity can prevent or cause the tragedy of the commons.

Bistability in dynamics depends on initial resource and cooperation levels.

The impact of carrying capacity varies with the type of strategic interaction.

Abstract

The tragedy of the commons (TOC) is an unfortunate situation where a shared resource is exhausted due to uncontrolled exploitation by the selfish individuals of a population. Recently, the paradigmatic replicator equation has been used in conjunction with a phenomenological equation for the state of the shared resource to gain insight into the influence of the games on the TOC. The replicator equation, by construction, models a fixed infinite population undergoing microevolution. Thus, it is unable to capture any effect of the population growth and the carrying capacity of the population although the TOC is expected to be dependent on the size of the population. Therefore, in this paper, we present a mathematical framework that incorporates the density dependent payoffs and the logistic growth of the population in the eco-evolutionary dynamics modelling the game-resource feedback. We discover a bistability in the dynamics: a finite carrying capacity can either avert or cause the TOC depending on the initial states of the resource and the initial fraction of cooperators. In fact, depending on the type of strategic game-theoretic interaction, a finite carrying capacity can either avert or cause the TOC when it is exactly the opposite for the corresponding case with infinite carrying capacity.