Delayed evolutionary model for public goods competition with policing in phenotypically variant bacterial biofilms

The emergence and abundance of public goods cooperation in the face of selfish "cheaters" is an enduring puzzle in the realm of evolutionary biology. The synergistic and antagonistic interactions within and between cooperators and free riders have an immense impact on the sustenance of cooperation, which often maintains biodiversity in nature. We, therefore, address the problem to formulate and theoretically analyze the effects of social interactions on population dynamics of cooperators in a primitive natural population, such as phenotypically variant monospecies pathogenic microbial biofilms, where cooperators often inhibit the defective exploitation of produced goods by active punishment (policing). To make the proposed model more realistic, we introduce two discrete time delays, which are a combined form of quorum sensing delay and the period between secretion and reception of exoenzymes. In the present letter, our primary objectives are to illustrate i) the significance of the policing effect towards the evolutionary stability of cooperation and ii) the impact of delay(s) on the nature of coexistence. We find that the varying costs of policing can divulge four well-known cases, namely, domination of defection, domination of cooperation, bistability, and the stable coexistence of cooperative and defective individuals. Interestingly, the consideration of asymmetric non-zero time delays in receiving the cooperative exoproducts towards kin and non-kin additionally unveils the occurrence of chaotic coexistence between cooperators and free riders which happens through interior crisis.