Figure 8

Exploring the impact of initial conditions on eco-evolutionary dynamics: Varying initial conditions within the constraint \(x_0+y_0+z_0=0.9\), we uncover diverse steady states, highlighting the profound influence of initial conditions on emergent dynamics. This multistability provides valuable insights into system stability, robustness, and regime shifts. Notably, our findings reveal four distinct attractors representing different stable strategies, including coexistence of punishers and defectors. Depending on the chosen initial conditions, the system can exhibit stable states with cooperators, punishers, or defectors alone. These discoveries deepen our comprehension of the intricate dynamics and potential outcomes in our eco-evolutionary model, underscoring the critical interplay between initial conditions and steady states. Parameter values: \(\sigma _1=1.2\), \(\sigma _2=1.5\), \(\sigma _3=1.4\), \(\xi =1.1\), \(\beta =1.5\), \(\delta =0.5\), and \(\tau =0.2\). Initial conditions for delayed variables: (0.25, 0.25, 0.25).