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Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case

## An efficient numerical method for fractional model of allelopathic stimulatory phytoplankton species with Mittag-Leffler law

 1 Department of Engineering Science, Kermanshah University of Technology, Kermanshah, Iran, Department of Mathematics, Faculty of Engineering and Natural Sciences, Bahçeşehir University, 34349 Istanbul, Turkey 2 Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India 3 Department of Mathematics, JECRC University, Jaipur-303905, Rajasthan, India

* Corresponding author: jagdevsinghrathore@gmail.com

Received  December 2019 Revised  March 2020 Published  September 2020

The principal aim of the present article is to study a mathematical pattern of interacting phytoplankton species. The considered model involves a fractional derivative which enjoys a nonlocal and nonsingular kernel. We first show that the problem has a solution, then the proof of the uniqueness is included by means of the fixed point theory. The numerical solution of the mathematical model is also obtained by employing an efficient numerical scheme. From numerical simulations, one can see that this is a very efficient method and provides precise and outstanding results.

Citation: Behzad Ghanbari, Devendra Kumar, Jagdev Singh. An efficient numerical method for fractional model of allelopathic stimulatory phytoplankton species with Mittag-Leffler law. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020428
##### References:
 [1] S. Abbas, L. Mahto, A. Favini and M. Hafayed, Dynamical study of fractional model of allelopathic stimulatory phytoplankton species, Differ. Equ. Dyn. Syst., 24 (2016), 267-280.  doi: 10.1007/s12591-014-0219-5.  Google Scholar [2] D. M. Anderson, Toxic algae bloojpgms and red tides: A global perspective, In: Okaichi, T., Anderson, D.M., Nemoto, T. (eds.) Red Tides: Biology, Environmental Science and Toxicology, Elsevier, New York (1989) 11–21. Google Scholar [3] C. Arora, V. Kumar and S. Kant, Dynamics of a high-dimensional stage-structured prey predator model, Int. J. Appl. Comput. Math., 3 (2017), 427-445.  doi: 10.1007/s40819-017-0363-z.  Google Scholar [4] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.  Google Scholar [5] A. Atangana and R. T. Alqahtani, New numerical method and application to Keller-Segel model with fractional order derivative, Chaos, Solitons & Fractals, 116 (2018), 14-21.  doi: 10.1016/j.chaos.2018.09.013.  Google Scholar [6] A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order, Chaos, Solitons & Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.  Google Scholar [7] D. Baleanu, A. Jajarmi, S. S. Sajjadi and D. Mozyrska, A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator, Chaos, 29 (2019), 083127. doi: 10.1063/1.5096159.  Google Scholar [8] D. Baleanu, B. Shiri, H. M. Srivastava and M. Al Qurashi, A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel, Advances in Difference Equations, 2018 (2018), 353. doi: 10.1186/s13662-018-1822-5.  Google Scholar [9] D. Baleanu and B. Shiri, Collocation methods for fractional differential equations involving non-singular kernel, Chaos, Solitons & Fractals, 116 (2018), 136-145.  doi: 10.1016/j.chaos.2018.09.020.  Google Scholar [10] R. G. Batogna and A. Atangana, Generalised class of time fractional Black Scholes equation and numerical analysis, Discrete & Continuous Dynamical Systems - Series S, 12 (2019), 435-445.  doi: 10.3934/dcdss.2019028.  Google Scholar [11] H. Berglund, Stimulation of growth of two marine green algae by organic substances excreted by enteromorphalinza in unialgal and axenic cultures, Physiol., Plant, 22 (2006), 1069-1073.   Google Scholar [12] R. Garrappa, Numerical solution of fractional differential equations: A survey and a software tutorial, Mathematics, 6 (2018), 16. doi: 10.3390/math6020016.  Google Scholar [13] R. Caponetto, G. Dongola, L. Fortuna and I. Petrás, Fractional Order Systems Modeling and Control Applications, World Scientific Series on Nonlinear Science Series A, (2010). Google Scholar [14] M. Caputo, Elasticita e Dissipazione, Zani-Chelli, Bologna, 1969. Google Scholar [15] M. Caputo and M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., (2015) 73–85. Google Scholar [16] B. Edvarsen and E. Paasche, Bloom dynamics and physiology of Primnesium and Chrysochromulina, Physiological Ecology of Harmful Algal Bloom, Springer, Berlin (1998). Google Scholar [17] B. Ghanbari and D. Kumar, Numerical solution of predator-prey model with Beddington-DeAngelis functional response and fractional derivatives with Mittag-Leffler kernel, Chaos, 29 (2019), 063103. doi: 10.1063/1.5094546.  Google Scholar [18] A. Jajarmi, S. Arshad and D. Baleanu, A new fractional modelling and control strategy for the outbreak of dengue fever, Physica A, 535 (2019) 122524. doi: 10.1016/j.physa.2019.122524.  Google Scholar [19] A. Jajarmi, D. Baleanu, S. S. Sajjadi and J. H. Asad, A new feature of the fractional Euler-Lagrange equations for a coupled oscillator using a nonsingular operator approach, Frontiers in Physics, 7 (2019), 196. doi: 10.3389/fphy.2019.00196.  Google Scholar [20] D. Kumar, J. Singh, D. Baleanu and Sushila, Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel, Physica A, 492 (2018) 155–167. doi: 10.1016/j.physa.2017.10.002.  Google Scholar [21] D. Kumar, J. Singh and D. Baleanu, A new analysis of Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler type kernel, European Journal of Physical Plus, 133 (2018), 70. Google Scholar [22] D. Kumar, J. Singh, K. Tanwar and D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler Laws, International Journal of Heat and Mass Transfer, 138 (2019), 1222-1227.  doi: 10.1016/j.ijheatmasstransfer.2019.04.094.  Google Scholar [23] K. M Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov–Zhabotinskii reaction systems with Atangana–Baleanu fractional order derivative., Physica A, 523 (2019), 1072-1090.  doi: 10.1016/j.physa.2019.04.017.  Google Scholar [24] K. M. Owolabi, Numerical patterns in reaction–diffusion system with the Caputo and Atangana–Baleanu fractional derivatives, Chaos, Solitons & Fractals, 115 (2018), 160-169.  doi: 10.1016/j.chaos.2018.08.025.  Google Scholar [25] R. Pratt, Influence of the size of the inoculum on the growth of Chlorella vulgaris in freshly prepared culture medium, Am. J. Bot., 27 (1940), 52-67.  doi: 10.1002/j.1537-2197.1940.tb14214.x.  Google Scholar [26] E. Rice, Allelopathy, Academic Press, New York, 1984.   Google Scholar [27] K. M Saad, M. M. Khader, J. F. Gómez-Aguilar and D. Baleanu, Numerical solutions of the fractional Fisher's type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods, Chaos, 29 (2019), 023116. doi: 10.1063/1.5086771.  Google Scholar [28] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.  Google Scholar [29] B. Shiri and D. Baleanu, System of fractional differential algebraic equations with applications, Chaos, Solitons & Fractals, 120 (2019), 203-212.  doi: 10.1016/j.chaos.2019.01.028.  Google Scholar [30] B. Shiri and D. Baleanu, Numerical solution of some fractional dynamical systems in medicine involving non-singular kernel with vector order, Results in Nonlinear Analysis, 2 (2019), 160-168.   Google Scholar [31] J. Singh, D. Kumar and D. Baleanu, New aspects of fractional Biswas-Milovic model with Mittag-Leffler law, Mathematical Modelling of Natural Phenomena, 14 (2019), 303. doi: 10.1051/mmnp/2018068.  Google Scholar [32] J. Singh, A new analysis for fractional rumor spreading dynamical model in a social network with Mittag-Leffler law, Chaos, 29 (2019), 013137. doi: 10.1063/1.5080691.  Google Scholar [33] T. Smayda, Novel and nuisance phytoplankton blooms in the sea: evidence for a global epidemic. In: Graneli, E., Sundstrom, B., Edler, L., Anderson, D.M. (eds.), Toxic Marine Phytoplankton, Elsevier, New York (1990), 29–40. Google Scholar [34] J. M. Smith, Mathematical Models in Biology, Cambridge University Press, Cambridge, 1968.   Google Scholar [35] S. Uçar, E. Uçar, N. Özdemir and Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana–Baleanu derivative, Chaos, Solitons & Fractals, 118 (2019), 300-306.  doi: 10.1016/j.chaos.2018.12.003.  Google Scholar [36] R. H. Whittaker and P. P. Feeny, Allelochemics: chemical interactions between species, Science, 171 (1971), 757-770.  doi: 10.1126/science.171.3973.757.  Google Scholar

show all references

##### References:
 [1] S. Abbas, L. Mahto, A. Favini and M. Hafayed, Dynamical study of fractional model of allelopathic stimulatory phytoplankton species, Differ. Equ. Dyn. Syst., 24 (2016), 267-280.  doi: 10.1007/s12591-014-0219-5.  Google Scholar [2] D. M. Anderson, Toxic algae bloojpgms and red tides: A global perspective, In: Okaichi, T., Anderson, D.M., Nemoto, T. (eds.) Red Tides: Biology, Environmental Science and Toxicology, Elsevier, New York (1989) 11–21. Google Scholar [3] C. Arora, V. Kumar and S. Kant, Dynamics of a high-dimensional stage-structured prey predator model, Int. J. Appl. Comput. Math., 3 (2017), 427-445.  doi: 10.1007/s40819-017-0363-z.  Google Scholar [4] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.  Google Scholar [5] A. Atangana and R. T. Alqahtani, New numerical method and application to Keller-Segel model with fractional order derivative, Chaos, Solitons & Fractals, 116 (2018), 14-21.  doi: 10.1016/j.chaos.2018.09.013.  Google Scholar [6] A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order, Chaos, Solitons & Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.  Google Scholar [7] D. Baleanu, A. Jajarmi, S. S. Sajjadi and D. Mozyrska, A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator, Chaos, 29 (2019), 083127. doi: 10.1063/1.5096159.  Google Scholar [8] D. Baleanu, B. Shiri, H. M. Srivastava and M. Al Qurashi, A Chebyshev spectral method based on operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel, Advances in Difference Equations, 2018 (2018), 353. doi: 10.1186/s13662-018-1822-5.  Google Scholar [9] D. Baleanu and B. Shiri, Collocation methods for fractional differential equations involving non-singular kernel, Chaos, Solitons & Fractals, 116 (2018), 136-145.  doi: 10.1016/j.chaos.2018.09.020.  Google Scholar [10] R. G. Batogna and A. Atangana, Generalised class of time fractional Black Scholes equation and numerical analysis, Discrete & Continuous Dynamical Systems - Series S, 12 (2019), 435-445.  doi: 10.3934/dcdss.2019028.  Google Scholar [11] H. Berglund, Stimulation of growth of two marine green algae by organic substances excreted by enteromorphalinza in unialgal and axenic cultures, Physiol., Plant, 22 (2006), 1069-1073.   Google Scholar [12] R. Garrappa, Numerical solution of fractional differential equations: A survey and a software tutorial, Mathematics, 6 (2018), 16. doi: 10.3390/math6020016.  Google Scholar [13] R. Caponetto, G. Dongola, L. Fortuna and I. Petrás, Fractional Order Systems Modeling and Control Applications, World Scientific Series on Nonlinear Science Series A, (2010). Google Scholar [14] M. Caputo, Elasticita e Dissipazione, Zani-Chelli, Bologna, 1969. Google Scholar [15] M. Caputo and M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., (2015) 73–85. Google Scholar [16] B. Edvarsen and E. Paasche, Bloom dynamics and physiology of Primnesium and Chrysochromulina, Physiological Ecology of Harmful Algal Bloom, Springer, Berlin (1998). Google Scholar [17] B. Ghanbari and D. Kumar, Numerical solution of predator-prey model with Beddington-DeAngelis functional response and fractional derivatives with Mittag-Leffler kernel, Chaos, 29 (2019), 063103. doi: 10.1063/1.5094546.  Google Scholar [18] A. Jajarmi, S. Arshad and D. Baleanu, A new fractional modelling and control strategy for the outbreak of dengue fever, Physica A, 535 (2019) 122524. doi: 10.1016/j.physa.2019.122524.  Google Scholar [19] A. Jajarmi, D. Baleanu, S. S. Sajjadi and J. H. Asad, A new feature of the fractional Euler-Lagrange equations for a coupled oscillator using a nonsingular operator approach, Frontiers in Physics, 7 (2019), 196. doi: 10.3389/fphy.2019.00196.  Google Scholar [20] D. Kumar, J. Singh, D. Baleanu and Sushila, Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel, Physica A, 492 (2018) 155–167. doi: 10.1016/j.physa.2017.10.002.  Google Scholar [21] D. Kumar, J. Singh and D. Baleanu, A new analysis of Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler type kernel, European Journal of Physical Plus, 133 (2018), 70. Google Scholar [22] D. Kumar, J. Singh, K. Tanwar and D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler Laws, International Journal of Heat and Mass Transfer, 138 (2019), 1222-1227.  doi: 10.1016/j.ijheatmasstransfer.2019.04.094.  Google Scholar [23] K. M Owolabi and Z. Hammouch, Spatiotemporal patterns in the Belousov–Zhabotinskii reaction systems with Atangana–Baleanu fractional order derivative., Physica A, 523 (2019), 1072-1090.  doi: 10.1016/j.physa.2019.04.017.  Google Scholar [24] K. M. Owolabi, Numerical patterns in reaction–diffusion system with the Caputo and Atangana–Baleanu fractional derivatives, Chaos, Solitons & Fractals, 115 (2018), 160-169.  doi: 10.1016/j.chaos.2018.08.025.  Google Scholar [25] R. Pratt, Influence of the size of the inoculum on the growth of Chlorella vulgaris in freshly prepared culture medium, Am. J. Bot., 27 (1940), 52-67.  doi: 10.1002/j.1537-2197.1940.tb14214.x.  Google Scholar [26] E. Rice, Allelopathy, Academic Press, New York, 1984.   Google Scholar [27] K. M Saad, M. M. Khader, J. F. Gómez-Aguilar and D. Baleanu, Numerical solutions of the fractional Fisher's type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods, Chaos, 29 (2019), 023116. doi: 10.1063/1.5086771.  Google Scholar [28] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.  Google Scholar [29] B. Shiri and D. Baleanu, System of fractional differential algebraic equations with applications, Chaos, Solitons & Fractals, 120 (2019), 203-212.  doi: 10.1016/j.chaos.2019.01.028.  Google Scholar [30] B. Shiri and D. Baleanu, Numerical solution of some fractional dynamical systems in medicine involving non-singular kernel with vector order, Results in Nonlinear Analysis, 2 (2019), 160-168.   Google Scholar [31] J. Singh, D. Kumar and D. Baleanu, New aspects of fractional Biswas-Milovic model with Mittag-Leffler law, Mathematical Modelling of Natural Phenomena, 14 (2019), 303. doi: 10.1051/mmnp/2018068.  Google Scholar [32] J. Singh, A new analysis for fractional rumor spreading dynamical model in a social network with Mittag-Leffler law, Chaos, 29 (2019), 013137. doi: 10.1063/1.5080691.  Google Scholar [33] T. Smayda, Novel and nuisance phytoplankton blooms in the sea: evidence for a global epidemic. In: Graneli, E., Sundstrom, B., Edler, L., Anderson, D.M. (eds.), Toxic Marine Phytoplankton, Elsevier, New York (1990), 29–40. Google Scholar [34] J. M. Smith, Mathematical Models in Biology, Cambridge University Press, Cambridge, 1968.   Google Scholar [35] S. Uçar, E. Uçar, N. Özdemir and Z. Hammouch, Mathematical analysis and numerical simulation for a smoking model with Atangana–Baleanu derivative, Chaos, Solitons & Fractals, 118 (2019), 300-306.  doi: 10.1016/j.chaos.2018.12.003.  Google Scholar [36] R. H. Whittaker and P. P. Feeny, Allelochemics: chemical interactions between species, Science, 171 (1971), 757-770.  doi: 10.1126/science.171.3973.757.  Google Scholar
Influence of $\rho$ on response behavior of solutions when $\gamma = 0.001$.
Influence of $\rho$ on response behavior of solutions when $\gamma = 0.002$.
Influence of $\rho$ on response behavior of solutions when $\gamma = 0.003$.
Influence of $\rho$ on response behavior of solutions when $\gamma = 0.005$.
Influence of $\rho$ on response behavior of solutions when $\gamma = 0.05$.
Influence of $\rho$ on response behavior of solutions when $\gamma = 0.01$.
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