Eco-Evolutionary game dynamics with Three Strategies: how geometric analysis provides insights into stability

In this paper we extend a modeling framework that embeds ecological realism by examining howdemographic factors determine payoffs, which are subsequently incorporated into evolutionarygame models. In particular, it segregates births and deaths, as well as activities in additionto the between player interactions central to the classical theory. In the standard examples ofevolutionary games so far considered such as the Hawk-Dove game, a two strategy populationhas been studied.The above demographic models are extended from two strategies to three strategies, giving athorough analysis of the mechanistic framework, which lays down the necessary conditions onthe existence of steady states for a constrained system. We show that population trajectoriesare drawn toward the density manifold by logistic suppression, which in turn structures theinteraction between frequency dynamics and fixed points. We also introduce a novel approachwhich helps derive stability from geometric insights through phase space elements (nullclines andtheir intersections) and streamlines calculations by decomposing gradients into the componentsthat contribute to directional change, allowing dynamic and stability conditions to be more easilyinterpreted directly from the figures.