Figure 6

Exploring the role of system parameters in eco-evolutionary dynamics: Subfigure (a) shows that as the mortality rate increases, all strategies face extinction unless free space benefits defectors substantially, leading to their dominance. A significant portion of the parameter space allows for the coexistence of cooperators and defectors, while another portion enables the survival of cooperators alone. Similarly, in subfigure (b), escalating mortality rates result in the extinction of all strategies, except in a region where only cooperators survive. However, this region diminishes with higher temptation. This parameter space exhibits the coexistence of cooperators and defectors, also. In subfigure (c), increasing benefits for cooperators, while keeping other parameters fixed, leads to a surge in their density. Higher mortality rates pose a challenge, but cooperators can persist due to their enhanced benefits. A small region, shown in white, exhibits unbounded dynamics in subfigures (a) and (c). Subfigure (d) illustrates the delicate balance between benefits for defectors and cooperators. When defectors have a greater advantage and exceed the mortality rate, they become the sole survivors. Cooperators can coexist with defectors or thrive as the sole strategy when their benefits surpass both the mortality rate and defectors’ benefits. Punishers are absent in the whole figure due to chosen parameters and initial conditions, highlighting the intriguing dynamics. Parameter values: \(\sigma _1=1.2\), \(\sigma _2=1.5\), \(\sigma _3=1.4\), \(\xi =1.1\), \(\beta =1.5\), \(\delta =0.5\), and \(\tau =0.2\), unless those parameters are varied. Initial conditions for delayed variables: (0.25, 0.25, 0.25), and for non-delayed variables: (0.3, 0.3, 0.3).