Using G-Functions to Investigate the Evolutionary Stability of Bacterial Quorum Sensing

Abstract

In ecology, G-functions can be employed to define a growth function G for a population b, which can then be universally applied to all individuals or groups b i within this population. We can further define a strategy v i for every group b i . Examples for strategies include diverse behaviour such as number of offspring, habitat choice, and time of nesting for birds. In this work, we employ G-functions to investigate the evolutionary stability of the bacterial cooperation process known as quorum sensing. We employ the G-function ansatz to model both the population dynamics and the resulting evolutionary pressure in order to find evolutionary stable states. This results in a semi-linear parabolic system of equations, where cost and benefit are taken into account separately. Depending on different biological assumptions, we analyse a variety of typical model functions. These translate into different long-term scenarios for different functional responses, ranging from single-strategy states to coexistence. As a special feature, we distinguish between the production of public goods, available for all subpopulations, and private goods, from which only the producers can benefit.