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

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.

[1]  James D. Murray.  Mathematical Biology: I. An Introduction , 2007 .

[2]  Tugba Akman Yildiz,et al.  New aspects of time fractional optimal control problems within operators with nonsingular kernel , 2020, Discrete & Continuous Dynamical Systems - S.

[3]  Dumitru Baleanu,et al.  A new and efficient numerical method for the fractional modeling and optimal control of diabetes and tuberculosis co-existence. , 2019, Chaos.

[4]  M. A. Morales.  MECHANISMS AND DENSITY DEPENDENCE OF BENEFIT IN AN ANT–MEMBRACID MUTUALISM , 2000 .

[5]  Carlos Castillo-Chavez,et al.  Role of Prey Dispersal and Refuges on Predator-Prey Dynamics , 2010, SIAM J. Appl. Math..

[6]  S. Djilali.  Impact of prey herd shape on the predator-prey interaction , 2019, Chaos, Solitons & Fractals.

[7]  Behzad Ghanbari,et al.  Analysis of two avian influenza epidemic models involving fractal-fractional derivatives with power and Mittag-Leffler memories. , 2019, Chaos.

[8]  C. Cattani,et al.  HYBRID TWO SCALES MATHEMATICAL TOOLS FOR ACTIVE PARTICLES MODELLING COMPLEX SYSTEMS WITH LEARNING HIDING DYNAMICS , 2007 .

[9]  A. Atangana,et al.  New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , 2016, 1602.03408.

[10]  A. Atangana,et al.  New numerical approach for fractional differential equations , 2017, 1707.08177.

[11]  S. S. Sajjadi,et al.  A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator. , 2019, Chaos.

[12]  Alexandre G. Ciancio,et al.  Qualitative analysis of second-order models of tumor-immune system competition , 2008, Math. Comput. Model..

[13]  Carlo Cattani,et al.  Metamodeling the learning-hiding competition between tumours and the immune system: A kinematic approach , 2010, Math. Comput. Model..

[14]  José Francisco Gómez-Aguilar,et al.  Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations , 2018 .

[15]  B. Ghanbari,et al.  Mathematical and numerical analysis of a three‐species predator‐prey model with herd behavior and time fractional‐order derivative , 2019, Mathematical Methods in the Applied Sciences.

[16]  A. Atangana,et al.  New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models , 2017 .

[17]  Z. Hammouch,et al.  Mathematical modeling and analysis of two-variable system with noninteger-order derivative. , 2019, Chaos.

[18]  M. Caputo,et al.  A new Definition of Fractional Derivative without Singular Kernel , 2015 .

[19]  Devendra Kumar,et al.  Numerical solution of predator-prey model with Beddington-DeAngelis functional response and fractional derivatives with Mittag-Leffler kernel. , 2019, Chaos.

[20]  Behzad Ghanbari,et al.  Modeling the dynamics of nutrient–phytoplankton–zooplankton system with variable-order fractional derivatives , 2018, Chaos, Solitons & Fractals.

[21]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[22]  D. Baleanu,et al.  A new fractional modelling and control strategy for the outbreak of dengue fever , 2019 .

[23]  S. Djilali.  Herd behavior in a predator–prey model with spatial diffusion: bifurcation analysis and Turing instability , 2018 .

[24]  Sunil Kumar,et al.  Similarities in a fifth-order evolution equation with and with no singular kernel , 2020 .

[25]  Junjie Wei,et al.  Stationary patterns of a diffusive predator–prey model with Crowley–Martin functional response , 2017, 1701.02619.

[26]  Behzad Ghanbari,et al.  A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative , 2020 .

[27]  Eric A. Butcher,et al.  Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods , 2018 .

[28]  Roberto Garrappa,et al.  Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial , 2018 .

[29]  Ross Cressman,et al.  A predator-prey refuge system: Evolutionary stability in ecological systems. , 2009, Theoretical population biology.

[30]  Sunil Kumar,et al.  A Robust Computational Algorithm of Homotopy Asymptotic Method for Solving Systems of Fractional Differential Equations , 2019, Journal of Computational and Nonlinear Dynamics.

[31]  Roy M. Anderson,et al.  The Population Dynamics of Microparasites and Their Invertebrate Hosts , 1981 .