# Steering Eco-Evolutionary Games Dynamics with Manifold Control

## TL;DR

This paper develops a mathematical framework to control eco-evolutionary dynamics in multi-player games with nonlinear environmental feedbacks, enabling steering populations to desired states, especially relevant for microbial systems.

## Contribution

It introduces a novel manifold control approach for eco-evolutionary games with nonlinear feedback, extending previous models to more complex multi-player interactions.

## Key findings

- Multiple equilibrium manifolds can emerge from nonlinear feedback dynamics.
- Asymmetrical feedback speeds influence convergence and attraction basins.
- Switching control laws can steer populations to targeted states.

## Abstract

Feedback loops between population dynamics of individuals and their ecological environment are ubiquitously found in nature, and have shown profound effects on the resulting eco-evolutionary dynamics. Incorporating linear environmental feedback law into replicator dynamics of two-player games, recent theoretical studies shed light on understanding the oscillating dynamics of social dilemma. However, detailed effects of more general nonlinear feedback loops in multi-player games, which is more common especially in microbial systems, remain unclear. Here, we focus on ecological public goods games with environmental feedbacks driven by nonlinear selection gradient. Unlike previous models, multiple segments of stable and unstable equilibrium manifolds can emerge from the population dynamical systems. We find that a larger relative asymmetrical feedback speed for group interactions centered on cooperators not only accelerates the convergence of stable manifolds, but also increases the attraction basin of these stable manifolds. Furthermore, our work offers an innovative manifold control approach: by designing appropriate switching control laws, we are able to steer the eco-evolutionary dynamics to any desired population states. Our mathematical framework is an important generalization and complement to coevolutionary game dynamics, and also fills the theoretical gap in guiding the widespread problem of population state control in microbial experiments.