Global bifurcation and stability of steady states for a bacterial colony model with density-suppressed motility |
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| Affiliation: | 1. Department of Mathematics, School of Sciences, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, 310018, China;2. Department of Mathematics and Statistics, University of Turku, Turku FIN-20014, Finland;1. Department of Mathematics, Faculty of Sciences and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia;2. Department of Basic Engineering Science, Faculty of Engineering, Menofia University, Shebin El-Kom 32511, Egypt;3. Department of Mathematics, College of Science and Humanities at Howtat Sudair, Majmaah University, Majmaah 11952, Saudi Arabia;4. Department of Mathematics, Faculty of Science, University of Tabuk, P.O.Box 741, Tabuk 71491, Saudi Arabia;5. Institute of Engineering, Polytechnic of Porto, Rua Dr. António Bernardino de Almeida, 431, Porto 4249-015, Portugal;1. Department of Mathematics, Shaheed Bhagat Singh College, University of Delhi, India;2. Government P.G. College, Mushafirkhana, Uttar Pradesh, India;1. Department of Astronautic Science and Mechanics, Harbin Institute of Technology, Harbin 150001, China;2. College of Mathematics, Sichuan University, Chengdu 610043, China;1. Center of Materials Science and Optoelectronics Engineering, College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China;2. Qiushi Honors College, Tianjin University, Tian jin 300350, China;3. McGill Metals Processing Centre, McGill University, Montreal, Quebec H3A 2B2 Canada;1. School of Reliability and Systems Engineering, Beihang University, Beijing 100083, China;2. Science and Technology on Reliability and Environmental Engineering Laboratory, Beijing 100083, China |
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| Abstract: | We investigate the structure and stability of the steady states for a bacterial colony model with density-suppressed motility. We treat the growth rate of bacteria as a bifurcation parameter to explore the local and global structure of the steady states. Relying on asymptotic analysis and the theory of Fredholm solvability, we derive the second-order approximate expression of the steady states. We analytically establish the stability criterion of the bifurcation solutions, and show that sufficiently large growth rate of bacteria leads to a stable uniform steady state. While the growth rate of bacteria is less than some certain value, there is pattern formation with the admissible wave mode. All the analytical results are corroborated by numerical simulations from different stages. |
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