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On fractional predator and prey models with mutualistic predation including non-local and nonsingular kernels

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  • Ghanbari, Behzad
  • Cattani, Carlo

Abstract

In a biological system, discovering the interactions between species is of high importance in the preservation and maintenance of these rare species. In this research work, we have utilized powerful mathematical tools to describe two Lotka–Volterra models with mutualistic predation. The kernel used in the derivative is a non-singular and non-local type that gives us many benefits in practice. Besides, this type of derivative is capable of storing critical system information, which is an essential feature of studying biological models. The equilibrium points of the two dynamical systems are determined. Moreover, the necessary criteria for establishing the stability of points are investigated in terms of the model parameters. These relationships provide significant and applicable results in the maintenance or extinction of systems that are highly valuable in biological and functional aspects. The necessary conditions for the existence and uniqueness of the solutions are provided. To verify the theoretical results, many practical simulations have been performed in various cases. The approximate technique employed is a very efficient and accurate method for solving such biological systems, which can be utilized to solve similar biological problems. Along with approximate numerical solutions, the ”memory effects” on these fractional models are also perused. The results confirm that the use of fractional derivatives is one of the crucial requirements in biological models.

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  • Ghanbari, Behzad & Cattani, Carlo, 2020. "On fractional predator and prey models with mutualistic predation including non-local and nonsingular kernels," Chaos, Solitons & Fractals, Elsevier, vol. 136(C).
    • Handle: RePEc:eee:chsofr:v:136:y:2020:i:c:s096007792030223x
    DOI: 10.1016/j.chaos.2020.109823
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    References listed on IDEAS

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    1. Doungmo Goufo, Emile F. & Kumar, Sunil & Mugisha, S.B., 2020. "Similarities in a fifth-order evolution equation with and with no singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    2. Djilali, Salih, 2019. "Impact of prey herd shape on the predator-prey interaction," Chaos, Solitons & Fractals, Elsevier, vol. 120(C), pages 139-148.
    3. Jajarmi, Amin & Arshad, Sadia & Baleanu, Dumitru, 2019. "A new fractional modelling and control strategy for the outbreak of dengue fever," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 535(C).
    4. Ghanbari, Behzad & Kumar, Sunil & Kumar, Ranbir, 2020. "A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    5. Ghanbari, Behzad & Gómez-Aguilar, J.F., 2018. "Modeling the dynamics of nutrient–phytoplankton–zooplankton system with variable-order fractional derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 114-120.
    6. Cressman, Ross & Garay, József, 2009. "A predator–prey refuge system: Evolutionary stability in ecological systems," Theoretical Population Biology, Elsevier, vol. 76(4), pages 248-257.
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    1. Ghanbari, Behzad & Günerhan, Hatıra & Srivastava, H.M., 2020. "An application of the Atangana-Baleanu fractional derivative in mathematical biology: A three-species predator-prey model," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    2. Attia, Nourhane & Akgül, Ali & Seba, Djamila & Nour, Abdelkader, 2020. "An efficient numerical technique for a biological population model of fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).

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