# Pattern Formation in Agent-Based and PDE Models for Evolutionary Games with Payoff-Driven Motion

## TL;DR

This paper studies how movement patterns affect cooperation and population success in evolutionary games using mathematical models.

## Contribution

The paper introduces and compares stochastic and deterministic models to analyze pattern formation and population dynamics in evolutionary games with payoff-driven motion.

## Key findings

- Turing patterns emerge when hawks move more than doves in diffusive motion models.
- Population size and average payoff increase with higher hawk mobility in diffusive motion.
- Payoff-driven motion leads to complex behaviors and potential mathematical instabilities in PDE models.

## Abstract

Spatial structure can play an important role in the evolution of cooperative behavior and the achievement of collective success of a population. In this paper, we explore the role of random and directed motion on spatial pattern formation and the payoff achieved by populations in both stochastic and deterministic models of spatial populations who engage in social interactions following a hawk-dove game. For the case of purely diffusive motion, both a stochastic spatial model and a partial differential equation model show that Turing patterns can emerge when hawks have a greater movement rate than doves, and in both models hawks and doves see an increase in population size and average payoff as hawk mobility increases. For the case of the payoff-driven motion, the stochastic model shows an overall decrease in population size and average payoff, but the PDE model displays more subtle behavior in this setting and will depend on the relative diffusivities of the two strategies. The PDE model also displays a biologically infeasible short-wave instability in the case of payoff-driven motion and equal diffusivities, indicating that we need to be careful about the mathematical properties of PDE models with payoff-driven directed motion and indicating potential use for nonlocal PDE models for spatial patterns in evolutionary games with directed motion.