Hopf-Hopf bifurcation in the delayed nutrient-microorganism model |
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| Institution: | 1. School of Mathematical Sciences, Anhui University, Hefei 230601, China;2. School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China;3. School of Electrical Engineering and Automation, Hefei University of Technology, Hefei 230009, China;1. School of Automation, China University of Geosciences, Wuhan 430074, PR China;2. School of Automation and Electrical Engineering, Linyi University, Linyi 276005, PR China;3. Hubei Key Laboratory of Advanced Control and Intelligent Automation of Complex Systems, Wuhan 430074, PR China;4. College of Mechatronics and Control Engineering, Hubei Normal University, Huangshi 435002, PR China;5. School of Science, Hubei University of Technology, Wuhan 430068, PR China;1. School of Mathematical Sciences, Anhui University, Hefei 230601, China;2. School of Electrical Engineering and Automation, Hefei University of Technology, Hefei 230009, China;1. Department of Mathematics, Hangzhou Normal University, Hangzhou 311121, China;2. School of Mathematical Sciences, Tongji University, Shanghai 200092, China;3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada;1. Department of Mathematics, Indian Institute of Technology, Ropar 140001, Punjab, India;2. School of Basic Sciences, Indian Institute of Technology, Mandi 175005, H.P., India;3. Department of Mathematics, Yasar University, Izmir, Turkey |
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| Abstract: | In view of time delay in the transport of nutrients, a delayed reaction-diffusion system with homogeneous Neumann boundary conditions is presented to understand the formation of the heterogeneous distribution of bacteria and nutrients in the sediment. With the effects of time delay and diffusion, the system will experience various dynamical behaviors, such as stability, the Turing instability, successive switches of stability of equilibria, the Hopf and the Hopf-Hopf bifurcations. To further understand the dynamics of the Hopf-Hopf bifurcation, the multiple time scale (MTS) technique is employed to derive the amplitude equations at this co-dimensional bifurcation point, and the dynamical classification near such bifurcation point is also identified by analyzing the obtained amplitude equations. Some numerical simulations are carried out to demonstrate the validity of the theoretical analysis. |
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