Evolutionary Stable Strategies and Cubic Vector Fields

TL;DR

This paper studies a class of cubic vector fields arising from evolutionary game theory, analyzing their global phase portraits and introducing the concept of genericity to understand their dynamics.

Contribution

It introduces the notion of genericity for these vector fields and provides comprehensive global phase portraits for systems with a central singularity.

Findings

Characterization of global phase portraits for generic systems

Identification of invariant structures within the vector fields

Insights into the stability and dynamics near singularities

Abstract

The introduction of concepts of Game Theory and Ordinary Differential Equations into Biology gave birth to the field of Evolutionary Stable Strategies, with applications in Biology, Genetics, Politics, Economics and others. In special, the model composed by two players having two pure strategies each results in a planar cubic vector field with an invariant octothorpe. Therefore, in this paper we study such class of vector fields, suggesting the notion of genericity and providing the global phase portraits of the generic systems with a singularity at the central region of the octothorpe.