Nullclines, Subnullclines and the Asymptotic and Transient Attractors in Eco-Evolutionary Dynamics

Abstract

In the demographic framework, mortality payoff function describes the cost of an interaction and fertility payoff function describes its reward. So while mortality cost depends on opponent's strategy, fertility reward can be affected by the density-dependent juvenile recruitment survival. This motivates an analysis of the eco-evolutionary dynamics of the classical Hawk-Dove game. It is shown that the stable and unstable equilibria (determined by the intersections of frequency and density nullclines) are connected by heteroclinic orbits, which attract nearby trajectories. The resulting bundle of trajectories leads to the discovery of the so-called subnullcines (manifolds placed between frequency and density nullcline) before they converge to the stable rest point. The initial isolated system is then extended by adding environmental seasonality (periodic background mortality), which acts as an external factor. This leads to complex cycling behavior and the subnullclines act as barriers to the propagation of the perturbation (resilience/resistance threshold). Thus, in a way, this paper completes, yet extends, previous works on the eco-evolutionary dynamics of games with demographic payoffs.