# American Institute of Mathematical Sciences

October  2021, 14(10): 3577-3587. doi: 10.3934/dcdss.2020428

## 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  October 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (10) : 3577-3587. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] R. Garrappa, Numerical solution of fractional differential equations: A survey and a software tutorial, Mathematics, 6 (2018), 16. doi: 10.3390/math6020016. [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). [14] M. Caputo, Elasticita e Dissipazione, Zani-Chelli, Bologna, 1969. [15] M. Caputo and M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., (2015) 73–85. [16] B. Edvarsen and E. Paasche, Bloom dynamics and physiology of Primnesium and Chrysochromulina, Physiological Ecology of Harmful Algal Bloom, Springer, Berlin (1998). [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. [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. [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. [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. [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. [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. [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. [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. [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. [26] E. Rice, Allelopathy, Academic Press, New York, 1984. [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. [28] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993. [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. [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. [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. [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. [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. [34] J. M. Smith, Mathematical Models in Biology, Cambridge University Press, Cambridge, 1968. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] R. Garrappa, Numerical solution of fractional differential equations: A survey and a software tutorial, Mathematics, 6 (2018), 16. doi: 10.3390/math6020016. [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). [14] M. Caputo, Elasticita e Dissipazione, Zani-Chelli, Bologna, 1969. [15] M. Caputo and M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., (2015) 73–85. [16] B. Edvarsen and E. Paasche, Bloom dynamics and physiology of Primnesium and Chrysochromulina, Physiological Ecology of Harmful Algal Bloom, Springer, Berlin (1998). [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. [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. [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. [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. [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. [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. [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. [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. [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. [26] E. Rice, Allelopathy, Academic Press, New York, 1984. [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. [28] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993. [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. [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. [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. [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. [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. [34] J. M. Smith, Mathematical Models in Biology, Cambridge University Press, Cambridge, 1968. [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. [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.
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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