A study of predator–prey models with the Beddington–DeAnglis functional response and impulsive effect |
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| Institution: | 1. Department of Engineering Science, Kermanshah University of Technology, Kermanshah, Iran;2. Department of Mathematics, Faculty of Engineering and Natural Sciences, Bahçe?ehir University, Istanbul 34349, Turkey;3. Laboratoire d’Analyse Non Linéaire et Mathématiques Appliquées, Université de Tlemcen, Tlemcen, Algeria;4. Department of Mathematics, Faculty of Exact Sciences and Computer Science, Hassiba Benbouali University, Chlef, Algeria;1. Department of Mathematics, Central University of Rajasthan NH-8, Bandarsindri, Kishangarh-305801, Distt.-Ajmer, Rajasthan, India;2. School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, 175001, H.P., India;1. School of Mathematics and Information Science, Guangxi Universities Key Lab of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, Guangxi 537000, PR China;2. College of Mathematics and Statistics, Hainan Normal University, Haikou, Hainan 571158, PR China;3. School of Science, China University of Petroleum (East China), Qingdao 266580, PR China;4. Nonlinear Analysis and Applied Mathematics (NAAM) – Research Group, King Abdulaziz University, Jeddah, Saudi Arabia;1. Departamento de Matemáticas, Facultad de Ciencias, UNAM., C. Universitaria, D.F. 04510, México;2. Grupo de Investigación en Sistemas Dinámicos y Aplicaciones (GISDA), Departamento de Matemática, Facultad de Ciencias, Universidad de Bío-Bío, Casilla 5–C, Concepción, VIII–región, Chile |
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| Abstract: | In this paper, the predator–prey system with the Beddington–DeAngelis functional response is developed, by introducing a proportional periodic impulsive catching or poisoning for the prey populations and a constant periodic releasing for the predator. The Beddington–DeAngelis functional response is similar to the Holling type II functional response but contains an extra term describing mutual interference by predators. This model has the potential to protect predator from extinction, but under some conditions may also lead to extinction of the prey. That is, the system exists a locally stable prey-eradication periodic solution when the impulsive period satisfies an inequality. The condition for permanence is established via the method of comparison involving multiple Liapunov? functions. Further, by numerical simulation method the influences of the impulsive perturbations and mutual interference by predators on the inherent oscillation are investigated. With the increasing of releasing for the predator, the system appears a series of complex phenomenon, which include (1) period-doubling, (2) chaos attractor, (3) period-halfing. (4) non-unique dynamics (meaning that several attractors coexist). |
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