Dynamical response of an eco-epidemiological system with harvesting
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Dynamical response of an eco-epidemiological system with harvesting Harekrishna Das1 · Absos Ali Shaikh1 Received: 17 October 2019 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract This article presents a study of Leslie–Gower predator–prey system to investigate the dynamics of disease transmission among predator species. The system includes the harvesting of infected predator. The positivity, boundedness of the solutions and permanence of the system are taken into consideration. The stability and Hopf bifurcation analyses around biologically feasible equilibria are scrutinized. The harvesting of infected predator plays a crucial role for the occurrence of limit cycle oscillations and stability around the interior equilibrium point. Our results disclose that infected predator harvesting has a considerable consequence on the eco-epidemiological system. The optimal control theory has been applied to investigate optimal strategies for controlling the infection. Analytical findings are confirmed through numerical simulations. Keywords Eco-epidemiological system · Harvesting · Stability · Persistence · Hopf bifurcation Mathematics Subject Classification 92D30 · 92D40 · 65L07 · 49J15 · 34K18
1 Introduction Ecology is the part of biology that studies the interactions among organisms and their environment. Epidemiology deals with the incidence, distribution and possible control of diseases and other factors relating to health. Eco-epidemic models deal with ecosystems of interacting populations among which a disease spreads [3,25,33,34]. A mathematical model describes a real-world phenomenon by means of mathematical language to understand the behaviour of a natural or physical system. Two impor-
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Absos Ali Shaikh [email protected] Harekrishna Das [email protected]
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Department of Mathematics, The University of Burdwan, Burdwan 713104, West Bengal, India
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H. Das, A. A. Shaikh
tant areas viz., epidemiology and theoretical ecology exist in the literature and were studied separately for decades. Lotka and Volterra [22,36] were the forerunners in the study of the interaction of biological species. Kermack and McKendrick [20] were the first who studied the spread of diseases mathematically. Since then many research works have been done independently for years. Anyway, in the late 1990s, these two fields came closer, and a new field arised called eco-epidemiology which studied both epidemiology and ecological problems together. Anderson and May [2] initiated the field of eco-epidemiology where the interaction between predator and infected prey species was considered with some disease. Prey-predator models are divided into three cases by the infectious disease. First, the infected prey is assumed in models [6,7,15,25]. Johri et al. [18] studied a Lotka– Volterra type predator-prey model without harvesting and disease in prey and it is assumed that the conversion rate of the susceptible prey is equal to that of the infected prey. An eco-epidemiology model is studied by Sharma and
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