  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.
