Effect of chaotic agent dynamics on coevolution of cooperation and synchronization

Abstract

The effect of the chaotic dynamical states of the agents on the coevolution of cooperation and synchronization in a structured population of the agents remains unexplored. With a view to gaining insights into this problem, we construct a coupled map lattice of the paradigmatic chaotic logistic map by adopting the Watts-Strogatz network algorithm. The map models the agent's chaotic state dynamics. In the model, an agent benefits by synchronizing with its neighbours and in the process of doing so, it pays a cost. The agents update their strategies (cooperation or defection) by using either a stochastic or a deterministic rule in an attempt to fetch themselves higher payoffs than what they already have. Among some other interesting results, we find that beyond a critical coupling strength, that increases with the rewiring probability parameter of the Watts-Strogatz model, the coupled map lattice is spatiotemporally synchronized regardless of the rewiring probability. Moreover, we observe that the population does not desynchronize completely-and hence finite level of cooperation is sustained-even when the average degree of the coupled map lattice is very high. These results are at odds with how a population of the non-chaotic Kuramoto oscillators as agents would behave. Our model also brings forth the possibility of the emergence of cooperation through synchronization onto a dynamical state that is a periodic orbit attractor.

Cooperators overcome migration dilemma through synchronization

“…We must restrict between 0 to 1 so that x i n doesn't become either negative or greater than one and work with only those values [50] of S and T for which 0 ≤ x i n ≤ 1 for all values of i and n. In our model with same A at all the nodes-all the demes having same game (say, the LG; see later)-there is a possibility that the dynamics at all the demes may be completely synchronized to an interior fixed point, i.e., x i = x j = x * for all i and j. One could do a linear stability analysis [53,54] to find whether this synchronized state is at all stable and hence attainable. As is shown in the next paragraph, such a stable state in fact exists when ≥ crit := [(|df /dx| − 1)/(|df /dx| − 1 + p)] x * .…”

Amplitude death in coupled replicator map lattice: averting migration dilemma

“…Of course, in order for s i n to be a non-trivial game theoretic action, one has to associate some payoffs for the corresponding player. In the present context, it is straightforward because on choosing to cooperate (or equivalently, participate in migration), the deme has to pay a cost that we take as the rate of the deviation [18,21] from its state:…”

“…The nonlinear dynamics and network dynamics of the evolutionary systems in the context of the interplay between synchronization and cooperation have motivated quite a few recent studies, e.g., the ones on the evolutionary Kuramoto dilemma [18][19][20] and the one on chaotic agent dynamics [21]. In the setting of the CML with chaotic replicator maps, the amplitude variations of the chaotic oscillations of the fraction of the cooperators in the subpopulations are suppressed due to synchronization onto a fixed point of the CML [15].…”

“…Thirdly, how is cooperation across the population supported as the degree of the network varies? Furthermore, costly interactions in networks while studying the coevolution of synchronization and cooperation has gathered a lot of recent interest [18,20,21]. In such a scenario, the deme-based on the payoff it receivescan decide whether to participate in the process of migration or not.…”

Deciphering chaos in evolutionary games

“…Of course, we are not implying that the chaos has not been explored in the context of the evolutionary game dynamics. Within the paradigms of the game theory and the theory of evolution, issues related to chaos have been presented in the context of learning [6][7][8][9], emergence of cooperation [10][11][12][13][14], mutation [15,16], fictitious play [17], imitation game of bird song [18], Darwinian evolution [19,20], consciousness [21], law and economics [22], and language acquisition [23].…”