Extinction and permanence of the predator-prey system with general functional response and impulsive control |
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| Institution: | 1. College of Arts and Sciences, Shanxi Agricultural University, Taigu 030801, PR China;2. School of software, Shanxi Agricultural University, Taigu 030801, PR China;3. Centre for Electronics Warfare, Information and Cyber (CEWIC)Cranfield University, Shrivenham, Swindon SN6 8LA U.K.;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. Computational Mechanics and Scientific Computing Group, Technical University of Cartagena, Campus Muralla del Mar, Cartagena 30202, Murcia, Spain;2. INEI, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain;1. National Research University Higher School of Economics, Moscow, Russia;2. Hydrometeorological Centre of Russia, Moscow 123242, Russia;1. School of Science, Henan Institute of Technology, Xinxiang 453003, PR China;2. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China;3. School of Ocean Engineering, Guangdong Ocean University, Zhanjiang 524088, PR China;4. School of Naval Architecture, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, PR China;5. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, PR China |
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| Abstract: | Traditional approach for modelling the evolution of populations in the predator-prey ecosystem has commonly been undertaken using specific impulsive response function, and this kind of modelling is applicable only for a specific ecosystem under certain environmental situations only. This paper attempts to fill the gap by modelling the predator-prey ecosystem using a ‘generalized’ impulsive response function for the first time. Different from previous research, the present work develops the modelling for an integrated pest management (IPM) especially when the stocking of predator (natural enemy) and the harvesting of prey (pest) occur impulsively and at different instances of time. The paper firstly establishes the sufficient conditions for the local and the global stabilities of prey eradication periodic solution by applying the Floquet theorem of the Impulsive different equation and small amplitude perturbation under a ‘generalized’ impulsive response function. Subsequently the sufficient condition for the permanence of the system is given through the comparison techniques. The corollaries of the theorems that are established by using the ‘general impulsive response function’ under the locally asymptotically stable condition are found to be in excellent agreement with those reported previously. Theoretical results that are obtained in this work is then validated by using a typical impulsive response function (Holling type-II) as an example, and the outcome is shown to be consistent with the previously reported results. Finally, the implication of the developed theories for practical pest management is illustrated through numerical simulation. It is shown that the elimination of either the preys or the pest can be effectively deployed by making use of the theoretical model established in this work. The developed model is capable to predict the population evolutions of the predator-prey ecosystem to accommodate requirements such as: the combinations of the biological control, chemical control, any functional response function, the moderate impulsive period, the harvest rate for the prey and predator parameter and the incremental stocking of the predator parameter. |
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