Comparisons of Numerical and Solitary Wave Solutions for the Stochastic Reaction–Diffusion Biofilm Model including Quorum Sensing

Muhammad Zafarullah Baber, Nauman Ahmed, Muhammad Waqas Yasin, Muhammad Sajid Iqbal, Ali Akgül, Alicia Cordero () and Juan R. Torregrosa
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Muhammad Zafarullah Baber: Department of Mathematics and Statistics, The University of Lahore, Lahore 54590, Pakistan
Nauman Ahmed: Department of Mathematics and Statistics, The University of Lahore, Lahore 54590, Pakistan
Muhammad Waqas Yasin: Department of Mathematics, University of Narowal, Narowal 51600, Pakistan
Muhammad Sajid Iqbal: School of Foundation Studies and Mathematics, OUC with Liverpool John Moores University (UK), Doha P.O. Box 12253, Qatar
Ali Akgül: Department of Computer Science and Mathematics, Lebanese American University, Beirut P.O. Box 13-5053, Lebanon
Alicia Cordero: Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, 46022 València, Spain
Juan R. Torregrosa: Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, 46022 València, Spain

Mathematics, 2024, vol. 12, issue 9, 1-30

Abstract: This study deals with a stochastic reaction–diffusion biofilm model under quorum sensing. Quorum sensing is a process of communication between cells that permits bacterial communication about cell density and alterations in gene expression. This model produces two results: the bacterial concentration, which over time demonstrates the development and decomposition of the biofilm, and the biofilm bacteria collaboration, which demonstrates the potency of resistance and defense against environmental stimuli. In this study, we investigate numerical solutions and exact solitary wave solutions with the presence of randomness. The finite difference scheme is proposed for the sake of numerical solutions while the generalized Riccati equation mapping method is applied to construct exact solitary wave solutions. The numerical scheme is analyzed by checking consistency and stability. The consistency of the scheme is gained under the mean square sense while the stability condition is gained by the help of the Von Neumann criteria. Exact stochastic solitary wave solutions are constructed in the form of hyperbolic, trigonometric, and rational forms. Some solutions are plots in 3D and 2D form to show dark, bright and solitary wave solutions and the effects of noise as well. Mainly, the numerical results are compared with the exact solitary wave solutions with the help of unique physical problems. The comparison plots are dispatched in three dimensions and line representations as well as by selecting different values of parameters.

Keywords: reaction–diffusion biofilm model; multiplicative time noise; finite difference scheme; stochastic solitary wave solutions; generalized Riccati equation mapping method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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