Optimization and exact solutions for biofilm model of bacterial communities

In this study, the optimization of the fixed point in function spaces and exact solutions to the biofilm model are studied to describe growth, and bacteria working together, including quorum sensing. Engineering applications that involve biotechnology, environmental engineering, and the creation of medical devices are particularly pertinent to biofilm models that include quorum sensing processes. This model is applied to cellular processes such as cell growth in mathematical biology. The optimization of the solution is also discussed by using the Banach space in an optimal ball. The existence of a solution is shown by using the Schauder fixed point theorem. Diverse exact solutions are extracted in hyperbolic, trigonometric, and plane waveforms are archive by using two techniques such as modified extended direct algebraic (MEDA) and G '/G-expansion technique. Additionally, the 3D plots and their corresponding contour graphs are also drawn to show the physical behavior of the solutions.