Stability Analysis of Population Dynamics Model in Microbial Biofilms with Non-participating Strains

Abstract

The existence of phenotypic heterogeneity in single-species bacterial biofilms is well-established in the published literature. However, the modeling of population dynamics in biofilms from the viewpoint of social interactions, i.e. interplay between heterotypic strains, and the analysis of this kind using control theory are not addressed significantly. Therefore, in this paper, we theoretically analyze the population dynamics model in microbial biofilms with non-participating strains (coexisting with public goods producers and non-producers) in the context of evolutionary game theory and nonlinear dynamics. Our analysis of the replicator dynamics model is twofold: first without the inclusion of spatial pattern, and second with the consideration of degree of assortment. In the first case, Lyapunov stability analysis of the stable equilibrium point of the proposed replicator system determines (1,0) ('full dominance of cooperators') as a global asymptotic stable equilibrium whenever the return exceeds the metabolic cost of cooperation. Hence, the global asymptotic stable nature of (1,0) in the context of non-consideration of spatial pattern helps to justify mathematically the adversity in the eradication of cooperative enterprise that is an infectious biofilm. In the second case, we found non-existence of global asymptotic stability in the system, and it unveils two additional phenomena - bistability and coexistence. In this context, two inequality conditions are derived for the 'full dominance of cooperators' and coexistence. Therefore, the inclusion of spatial pattern in biofilms with non-competing strains intends conditional dominance of pathogenic (with respect to the hosts) public goods producers which can be an effective strategy towards the control of an infectious biofilm with the drug-dependent regulation of degree of segregation. Furthermore, the simulation results of the proposed dynamics for both the discussed scenario confirm the results of the analysis of equilibrium points. The proposed stability analysis not only demonstrate a mathematical framework to analyze the population dynamics in biofilms but also gives a clue to control an infectious biofilm, where phenotypic and spatial heterogeneity exist.