Modeling and analysis of a fractional-order prey-predator system incorporating harvesting
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ORIGINAL ARTICLE
Modeling and analysis of a fractional‑order prey‑predator system incorporating harvesting Manotosh Mandal1,4 · Soovoojeet Jana2 · Swapan Kumar Nandi3 · T. K. Kar4 Received: 2 July 2020 / Accepted: 5 September 2020 © Springer Nature Switzerland AG 2020
Abstract In the present paper, we propose and study a fractional-order prey-predator type ecological model with the functional response of Holling type II and the effect of harvesting. In addition, we have also considered the influence of a super-predator on the conventional predator. The complex dynamical behaviour of the proposed model system including existence and uniqueness criteria, finiteness and nonnegativity of the solutions have studied rigorously. In addition, we have determined existence criteria of several equilibria and analyzed the asymptotic nature of those equilibria. We have enriched our analysis with the inclusion of fractional Hopf bifurcation and the criteria of global stability of the equilibrium points. Finally, some numerical simulation works have been incorporated to validate the analytical analysis. Keywords Fractional differential · Harvesting · Global asymptotic stability · Fractional Hopf bifurcation · Super-predator · Functional response
Introduction Prey-predator models are very useful for acquiring the knowledge of dynamics of interacting populations of prey-predators and hence the prey-predator model acts a key role in both theoretical as well as experimental ecology and mathematical ecology. This type of ecological model will go forward as one of the key ideas due to its importance and simplicity * Soovoojeet Jana [email protected] Manotosh Mandal [email protected] Swapan Kumar Nandi [email protected] T. K. Kar [email protected] 1
Department of Mathematics, Tamralipta Mahavidyalaya, Tamluk, West Bengal 721636, India
2
Department of Mathematics, Ramsaday College, Amta, Howrah, West Bengal 711401, India
3
Department of Mathematics, Nayabasat P.M.Sikshaniketan, Paschim Medinipur, West Bengal 721253, India
4
Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal 711103, India
(see Kar and Jana 2012; Fussmann et al. 2005; Chakraborty et al. 2012a; Mishra et al. 2016; Sahoo et al. 2016; Shaikh et al. 2018; Foutayeni et al. 2020; Khatua et al. 2020; Thakur et al. 2020). In the last few decades, these prey-predator theory have progressed very much but still there are lots of mathematical and ecological problems that can be studied through this tool. The dynamics of a prey-predator model are influenced by many constituents, e.g., the densities of prey and predators, harvesting of preys or predators or both etc. But the proposed model should be possible easiest biologically applicable form, even if the dynamics of the model may suggest difficult behaviour, e.g., stationary oscillations of the sizes of population, or suggest the dependency on some parameters, e.g., the “paradox of enrichment” (see for details Rosenzweig 1971) which indi
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