Global dynamics analysis of a water hyacinth fish ecological system under impulsive control |
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| Institution: | 1. Department of Systems and Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650500, PR China;2. School of Mechanical and Mechatronic Engineering, University of Technology Sydney, 15 Broadway, Ultimo, NSW 2007, Australia;3. College of Mathematics and Computer Science, Changsha UniversityChangsha, Hunan 410022, PR China;4. Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha, Hunan 410114, China;1. School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China;2. Key Laboratory of Knowledge Automation for Industrial Processes, Ministry of Education, Beijing 100083, China;1. School of Mathematics and Statistics & FJKLMAA, Fujian Normal University, Fuzhou 350117, PR China;2. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, PR China;1. School of Electrical and Information Engineering, Wuhan Institute of Technology, Wuhan 430205, China;2. Hubei Key Laboratory of Optical Information and Pattern Recognition, Wuhan 430205, China;3. Hubei Province Key Laboratory of Systems Science in Metallurgical Process(Wuhan University of Science and Technology), Wuhan 430081, China;1. College of Automation and College of Artificial Intelligence, Nanjing University of Posts and Telecommunications, Nanjing 210023, China;2. Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education, Anhui Polytechnic University, Wuhu, 241000, China;3. School of Marine Engineering, Jimei University, China;4. NARI Group Corporation (State Grid Electric Power Research Institute), Nanjing 211000, China |
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| Abstract: | Control of a water hyacinth-fish ecological system is required for a healthy and sustainable environment. This paper aims to investigate the global dynamics of a water hyacinth fish ecological system under ratio-dependent state impulsive control. First, we study the positivity and boundedness of the solution of the controlled system. By studying the local stability of the equilibrium, we find that the system has two situations. One is that there are two equilibria, namely a saddle point and a boundary equilibrium. In the second case, there are four equilibria, namely, two saddle points, a boundary equilibrium, and a focus point. For the first case, when we select an appropriate ratio-dependent control threshold, the trajectory will globally converge to the boundary equilibrium. For the second case, when the control line is located below the focus point, by using Poincare mapping method, flip bifurcation theory, and vector field analysis techniques, we find that the solution of the controlled system either globally converges to the boundary equilibrium, order-1 periodic solution, or order-2 periodic solution under certain conditions. When the control line is located above the focus point, the solution of the controlled system either globally converges to the focus point, order-1 or order-2 periodic solution. Finally, we use examples to verify the correctness and validity of the theoretical results. |
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