Evolutionary games with two species and delayed reciprocity

In the rich variety of biological interaction patterns, the state of an individual often does not depend solely on immediate factors but is significantly associated also with interactions or circumstances from the past. In evolutionary game theory with the focus on the evolution of cooperation, these phenomena frequently fall under the umbrella of delayed reciprocity. Especially in populations or systems where multiple species are considered there is significant asymmetry in the different types of interaction returns, but also in various delayed effects incurred by different species. With this motivation, this paper studies three different two-species evolutionary models: the intraspecific payoff delay model, the interspecific payoff delay model, and the all-payoff delay model. It is found that if the delay applies only to interspecific interactions, it only affects the convergence time of the cooperation rate but not the stability of the equilibrium points. In contrast, once the delay applies to intraspecific interactions, then the system transitions from asymptotic stability to oscillations around the equilibrium point as the time delay period increases. Lastly, increasing the delay value postpones convergence when the internal equilibrium point is asymptotically stable, and increases the amplitude when the system is oscillating.