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June  2020, 13(3): 823-834. doi: 10.3934/dcdss.2020047

Dynamical behaviour of fractional-order predator-prey system of Holling-type

Kolade M. Owolabi 1,2,,

1. 

Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa
2. 

Department of Mathematical Sciences, Federal University of Technology, PMB 704, Akure, Ondo State, Nigeria

* Corresponding author: mkowolax@yahoo.com (K. M. Owolabi)

Received  April 2018 Revised  May 2018 Published  March 2019

Fund Project: The research contained in this report is supported by South African National Research Foundation

In this paper, the local derivative in time is replaced with the Caputo-Fabrizio fractional derivative of order $ \alpha\in(0, 1) $. A two-step fractional version of the Adams-Bashforth method is formulated for the approximation of this derivative. To enhance the correct choice of parameters when numerically simulating the full-system, we examine the stability analysis of the main equation. Two important examples are drawn to explore the dynamic richness of the predator-prey model with Holling type. Simulation results at different instances of $ \alpha $ is in agreement with the theoretical findings.

Keywords: Caputo-Fabrizio fractional derivative, equilibrium points, spatiotemporal oscillations, Holling-type Ⅳ, predator-prey, stability analysis.
Mathematics Subject Classification: Primary: 65L05, 65L06, 93C10; Secondary: 34A34, 49M25, 65M70.
Citation: Kolade M. Owolabi. Dynamical behaviour of fractional-order predator-prey system of Holling-type. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 823-834. doi: 10.3934/dcdss.2020047
References:
[1]

H. I. Abdel-Gawad and K. M. Saad, On the behaviour of solutions of the two-cell cubic autocatalator, ANZIAM, 44 (2002), E1–E32. doi: 10.1017/S1446181100007859.  Google Scholar

[2]

P. Agarwal, A. Berdyshev and E. Karimov, Solvability of a non-local problem with integral transmitting condition for mixed type equation with Caputo fractional derivative, Results in Mathematics, 17 (2017), 1235-1257.  doi: 10.1007/s00025-016-0620-1.  Google Scholar

[3]

P. Agarwal and A. A. El-Sayed, Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation, Physica A: Statistical Mechanics and its Applications, 500 (2018), 40-49.  doi: 10.1016/j.physa.2018.02.014.  Google Scholar

[4]

P. Agarwal, S. Jain and T. Mansour, Further extended Caputo fractional derivative operator and its applications, Russian Journal of Mathematical Physics, 24 (2017), 415-425.  doi: 10.1134/S106192081704001X.  Google Scholar

[5]

A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, Journal of Computational Physics, 293 (2015), 104-114.  doi: 10.1016/j.jcp.2014.12.043.  Google Scholar

[6]

A. Atangana, On the new fractional derivative and application to Fisher's reaction-diffusion, Applied Mathematics and Computation, 273 (2016), 948-956.  doi: 10.1016/j.amc.2015.10.021.  Google Scholar

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A. Atangana and B. S. T. Alkahtani, New model of groundwater owing within a confine aquifer: Application of Caputo-Fabrizio derivative, Arabian Journal of Geosciences, 9 (2016), 1-6.   Google Scholar

[8]

A. Atangana and R. T. Alqahtani, Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation, Advances in Difference Equations, 2016 (2016), Paper No. 156, 13 pp. doi: 10.1186/s13662-016-0871-x.  Google Scholar

[9]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.  Google Scholar

[10]

A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 3, 21 pp, https://doi.org/10.1051/mmnp/2018010 doi: 10.1051/mmnp/2018010.  Google Scholar

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M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 73-85.   Google Scholar

[12]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progress in Fractional Differentiation and Applications, 2 (2016), 1-11.  doi: 10.18576/pfda/020101.  Google Scholar

[13]

J. F. Gómez-Aguilar, H. Yépez-Martínez, J. Torres-Jiménez, T. Córdova-Fraga, R. F. Escobar-Jiménez and V. H. Olivares-Peregrino, Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel, Advances in Difference Equations, 2017 (2017), Paper No. 68, 18 pp. doi: 10.1186/s13662-017-1120-7.  Google Scholar

[14]

J. F. Gómez-Aguilar, Analytical and numerical solutions of the telegraph equation using the Atangana-Caputo fractional order derivative, Journal of Electromagnetic Waves and Applications, 32 (2018), 695-712.   Google Scholar

[15]

J. F. Gómez-Aguilar, Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations, Physica A: Statistical Mechanics and its Applications, 494 (2018), 52-75.  doi: 10.1016/j.physa.2017.12.007.  Google Scholar

[16]

J. F. Gómez-Aguilar, R. F. Escobar-Jiménez, M. G. López-López and V. M. Alvarado-Martínez, Analysis of projectile motion: A comparative study using fractional operators with power law, exponential decay and Mittag-Leffler kernel, The European Physical Journal Plus, 133 (2018), 1-14.   Google Scholar

[17]

J. Huang and D. Xiao, Analyses of bifurcations and stability in a predator-prey system with Holling type-Ⅳ functional response, Acta Mathematicae Applicatae Sinica, English Series, 20 (2004), 167-178.  doi: 10.1007/s10255-004-0159-x.  Google Scholar

[18]

V. F. Morales-Delgado, J. F. Gómez-Aguilar, H. Yépez-Martínez, D. Baleanu, R. F. Escobar-Jimenez and V. H. Olivares-Peregrino, Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Advances in Difference Equations, 2016 (2016), Paper No. 164, 17 pp. doi: 10.1186/s13662-016-0891-6.  Google Scholar

[19]

K. M. Owolabi, Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems, Chaos, Solitons and Fractals, 93 (2016), 89-98.  doi: 10.1016/j.chaos.2016.10.005.  Google Scholar

[20]

K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Communications in Nonlinear Science and Numerical Simulation, 44 (2017), 304-317.  doi: 10.1016/j.cnsns.2016.08.021.  Google Scholar

[21]

K. M. Owolabi and A. Atangana, Analysis of mathematics and numerical pattern formation in superdiffusive fractional multicomponent system, Advances in Applied Mathematics and Mechanics, 9 (2017), 1438-1460.  doi: 10.4208/aamm.OA-2016-0115.  Google Scholar

[22]

K. M. Owolabi and A. Atangana, Numerical simulation of noninteger order system in subdiffusive, diffusive, and superdiffusive scenarios, Journal of Computational and Nonlinear Dynamics, 12 (2017), 031010-1, 7pages. Google Scholar

[23]

K. M. Owolabi and A. Atangana, Numerical simulations of chaotic and complex spatiotemporal patterns in fractional reaction-diffusion systems, Computational and Applied Mathematics, 37 (2018), 2166-2189.  doi: 10.1007/s40314-017-0445-x.  Google Scholar

[24]

K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Chaos, Solitons and Fractals, 103 (2017), 544-554.  doi: 10.1016/j.chaos.2017.07.013.  Google Scholar

[25]

K. M. Owolabi and A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo-Fabrizio derivative in Riemann-Liouville sense, Chaos, Solitons and Fractals, 99 (2017), 171-179.  doi: 10.1016/j.chaos.2017.04.008.  Google Scholar

[26]

K. M. Owolabi and A. Atangana, Analysis and application of new fractional Adams-Bashforth scheme with Caputo-Fabrizio derivative, Chaos, Solitons and Fractals, 105 (2017), 111-119.  doi: 10.1016/j.chaos.2017.10.020.  Google Scholar

[27]

K. M. Owolabi, Mathematical analysis and numerical simulation of chaotic noninteger order Differential systems with Riemann-Liouville derivative, Numerical Methods for Partial Differential Equations, 34 (2018), 274-295.  doi: 10.1002/num.22197.  Google Scholar

[28]

K. M. Owolabi, Riemann-Liouville fractional derivative and application to model chaotic differential equations, Progress in Fractional Differentiation and Applications, 4 (2018), 99-110.  doi: 10.18576/pfda/040204.  Google Scholar

[29]

K. M. Owolabi, Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 7, 17 pp. doi: 10.1051/mmnp/2018006.  Google Scholar

[30]

K. M. Owolabi, Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator, The European Physical Journal Plus, 133 (2018), 98. doi: 10.1140/epjp/i2018-11951-x.  Google Scholar

[31]

K. M. Owolabi and A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos, Solitons and Fractals, 111 (2018), 119-127.  doi: 10.1016/j.chaos.2018.04.019.  Google Scholar

[32]

K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, The European physical Journal Plus, 133 (2018), 15. doi: 10.1140/epjp/i2018-11863-9.  Google Scholar

[33]

E. Pindza and K. M. Owolabi, Fourier spectral method for higher order space fractional reaction-diffusion equations, Communications in Nonlinear Science and Numerical Simulation, 40 (2016), 112-128.  doi: 10.1016/j.cnsns.2016.04.020.  Google Scholar

[34]

M. V. Ruzhansky, Y. J. Cho, P. Agarwal and I. Area, Advances in Real and Complex Analysis with Applications, Birkhuser, 2017.  Google Scholar

[35]

K. M. Saad, An approximate analytical solution of coupled nonlinear fractional diffusion equations, Journal of Fractional Calculus and Applications, 5 (2014), 58-72.   Google Scholar

[36]

K. M. Saad, Comparing the Caputo, Caputo-Fabrizio and Atangana-Baleanu derivative with fractional order: Fractional cubic isothermal auto-catalytic chemical system, The European Physical Journal Plus, 133 (2018), 94. doi: 10.1140/epjp/i2018-11947-6.  Google Scholar

[37]

X. J. Yang, Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems, Thermal Science, 21 (2017), 1161-1171.  doi: 10.2298/TSCI161216326Y.  Google Scholar

[38]

X. J. Yang, F. Gao and H. M. Srivastava, A new computational approach for solving nonlinear local fractional PDEs, Journal of Computational and Applied Mathematics, 339 (2018), 285-296.  doi: 10.1016/j.cam.2017.10.007.  Google Scholar

[39]

H. Yépez-Martínez, J. F. Gómez-Aguilar, I. O. Sosa, J. M. Reyes and J. Torres-Jiménez, The Feng's first integral method applied to the nonlinear mKdV space-time fractional partial differential equation, Revista Mexicana de Física, 62 (2016), 310-316.   Google Scholar

show all references

References:
[1]

H. I. Abdel-Gawad and K. M. Saad, On the behaviour of solutions of the two-cell cubic autocatalator, ANZIAM, 44 (2002), E1–E32. doi: 10.1017/S1446181100007859.  Google Scholar

[2]

P. Agarwal, A. Berdyshev and E. Karimov, Solvability of a non-local problem with integral transmitting condition for mixed type equation with Caputo fractional derivative, Results in Mathematics, 17 (2017), 1235-1257.  doi: 10.1007/s00025-016-0620-1.  Google Scholar

[3]

P. Agarwal and A. A. El-Sayed, Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation, Physica A: Statistical Mechanics and its Applications, 500 (2018), 40-49.  doi: 10.1016/j.physa.2018.02.014.  Google Scholar

[4]

P. Agarwal, S. Jain and T. Mansour, Further extended Caputo fractional derivative operator and its applications, Russian Journal of Mathematical Physics, 24 (2017), 415-425.  doi: 10.1134/S106192081704001X.  Google Scholar

[5]

A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, Journal of Computational Physics, 293 (2015), 104-114.  doi: 10.1016/j.jcp.2014.12.043.  Google Scholar

[6]

A. Atangana, On the new fractional derivative and application to Fisher's reaction-diffusion, Applied Mathematics and Computation, 273 (2016), 948-956.  doi: 10.1016/j.amc.2015.10.021.  Google Scholar

[7]

A. Atangana and B. S. T. Alkahtani, New model of groundwater owing within a confine aquifer: Application of Caputo-Fabrizio derivative, Arabian Journal of Geosciences, 9 (2016), 1-6.   Google Scholar

[8]

A. Atangana and R. T. Alqahtani, Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation, Advances in Difference Equations, 2016 (2016), Paper No. 156, 13 pp. doi: 10.1186/s13662-016-0871-x.  Google Scholar

[9]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.  Google Scholar

[10]

A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 3, 21 pp, https://doi.org/10.1051/mmnp/2018010 doi: 10.1051/mmnp/2018010.  Google Scholar

[11]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 73-85.   Google Scholar

[12]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progress in Fractional Differentiation and Applications, 2 (2016), 1-11.  doi: 10.18576/pfda/020101.  Google Scholar

[13]

J. F. Gómez-Aguilar, H. Yépez-Martínez, J. Torres-Jiménez, T. Córdova-Fraga, R. F. Escobar-Jiménez and V. H. Olivares-Peregrino, Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel, Advances in Difference Equations, 2017 (2017), Paper No. 68, 18 pp. doi: 10.1186/s13662-017-1120-7.  Google Scholar

[14]

J. F. Gómez-Aguilar, Analytical and numerical solutions of the telegraph equation using the Atangana-Caputo fractional order derivative, Journal of Electromagnetic Waves and Applications, 32 (2018), 695-712.   Google Scholar

[15]

J. F. Gómez-Aguilar, Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations, Physica A: Statistical Mechanics and its Applications, 494 (2018), 52-75.  doi: 10.1016/j.physa.2017.12.007.  Google Scholar

[16]

J. F. Gómez-Aguilar, R. F. Escobar-Jiménez, M. G. López-López and V. M. Alvarado-Martínez, Analysis of projectile motion: A comparative study using fractional operators with power law, exponential decay and Mittag-Leffler kernel, The European Physical Journal Plus, 133 (2018), 1-14.   Google Scholar

[17]

J. Huang and D. Xiao, Analyses of bifurcations and stability in a predator-prey system with Holling type-Ⅳ functional response, Acta Mathematicae Applicatae Sinica, English Series, 20 (2004), 167-178.  doi: 10.1007/s10255-004-0159-x.  Google Scholar

[18]

V. F. Morales-Delgado, J. F. Gómez-Aguilar, H. Yépez-Martínez, D. Baleanu, R. F. Escobar-Jimenez and V. H. Olivares-Peregrino, Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Advances in Difference Equations, 2016 (2016), Paper No. 164, 17 pp. doi: 10.1186/s13662-016-0891-6.  Google Scholar

[19]

K. M. Owolabi, Mathematical analysis and numerical simulation of patterns in fractional and classical reaction-diffusion systems, Chaos, Solitons and Fractals, 93 (2016), 89-98.  doi: 10.1016/j.chaos.2016.10.005.  Google Scholar

[20]

K. M. Owolabi, Robust and adaptive techniques for numerical simulation of nonlinear partial differential equations of fractional order, Communications in Nonlinear Science and Numerical Simulation, 44 (2017), 304-317.  doi: 10.1016/j.cnsns.2016.08.021.  Google Scholar

[21]

K. M. Owolabi and A. Atangana, Analysis of mathematics and numerical pattern formation in superdiffusive fractional multicomponent system, Advances in Applied Mathematics and Mechanics, 9 (2017), 1438-1460.  doi: 10.4208/aamm.OA-2016-0115.  Google Scholar

[22]

K. M. Owolabi and A. Atangana, Numerical simulation of noninteger order system in subdiffusive, diffusive, and superdiffusive scenarios, Journal of Computational and Nonlinear Dynamics, 12 (2017), 031010-1, 7pages. Google Scholar

[23]

K. M. Owolabi and A. Atangana, Numerical simulations of chaotic and complex spatiotemporal patterns in fractional reaction-diffusion systems, Computational and Applied Mathematics, 37 (2018), 2166-2189.  doi: 10.1007/s40314-017-0445-x.  Google Scholar

[24]

K. M. Owolabi, Mathematical modelling and analysis of two-component system with Caputo fractional derivative order, Chaos, Solitons and Fractals, 103 (2017), 544-554.  doi: 10.1016/j.chaos.2017.07.013.  Google Scholar

[25]

K. M. Owolabi and A. Atangana, Numerical approximation of nonlinear fractional parabolic differential equations with Caputo-Fabrizio derivative in Riemann-Liouville sense, Chaos, Solitons and Fractals, 99 (2017), 171-179.  doi: 10.1016/j.chaos.2017.04.008.  Google Scholar

[26]

K. M. Owolabi and A. Atangana, Analysis and application of new fractional Adams-Bashforth scheme with Caputo-Fabrizio derivative, Chaos, Solitons and Fractals, 105 (2017), 111-119.  doi: 10.1016/j.chaos.2017.10.020.  Google Scholar

[27]

K. M. Owolabi, Mathematical analysis and numerical simulation of chaotic noninteger order Differential systems with Riemann-Liouville derivative, Numerical Methods for Partial Differential Equations, 34 (2018), 274-295.  doi: 10.1002/num.22197.  Google Scholar

[28]

K. M. Owolabi, Riemann-Liouville fractional derivative and application to model chaotic differential equations, Progress in Fractional Differentiation and Applications, 4 (2018), 99-110.  doi: 10.18576/pfda/040204.  Google Scholar

[29]

K. M. Owolabi, Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 7, 17 pp. doi: 10.1051/mmnp/2018006.  Google Scholar

[30]

K. M. Owolabi, Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator, The European Physical Journal Plus, 133 (2018), 98. doi: 10.1140/epjp/i2018-11951-x.  Google Scholar

[31]

K. M. Owolabi and A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos, Solitons and Fractals, 111 (2018), 119-127.  doi: 10.1016/j.chaos.2018.04.019.  Google Scholar

[32]

K. M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, The European physical Journal Plus, 133 (2018), 15. doi: 10.1140/epjp/i2018-11863-9.  Google Scholar

[33]

E. Pindza and K. M. Owolabi, Fourier spectral method for higher order space fractional reaction-diffusion equations, Communications in Nonlinear Science and Numerical Simulation, 40 (2016), 112-128.  doi: 10.1016/j.cnsns.2016.04.020.  Google Scholar

[34]

M. V. Ruzhansky, Y. J. Cho, P. Agarwal and I. Area, Advances in Real and Complex Analysis with Applications, Birkhuser, 2017.  Google Scholar

[35]

K. M. Saad, An approximate analytical solution of coupled nonlinear fractional diffusion equations, Journal of Fractional Calculus and Applications, 5 (2014), 58-72.   Google Scholar

[36]

K. M. Saad, Comparing the Caputo, Caputo-Fabrizio and Atangana-Baleanu derivative with fractional order: Fractional cubic isothermal auto-catalytic chemical system, The European Physical Journal Plus, 133 (2018), 94. doi: 10.1140/epjp/i2018-11947-6.  Google Scholar

[37]

X. J. Yang, Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems, Thermal Science, 21 (2017), 1161-1171.  doi: 10.2298/TSCI161216326Y.  Google Scholar

[38]

X. J. Yang, F. Gao and H. M. Srivastava, A new computational approach for solving nonlinear local fractional PDEs, Journal of Computational and Applied Mathematics, 339 (2018), 285-296.  doi: 10.1016/j.cam.2017.10.007.  Google Scholar

[39]

H. Yépez-Martínez, J. F. Gómez-Aguilar, I. O. Sosa, J. M. Reyes and J. Torres-Jiménez, The Feng's first integral method applied to the nonlinear mKdV space-time fractional partial differential equation, Revista Mexicana de Física, 62 (2016), 310-316.   Google Scholar

Figure 1.  Dynamic behaviour of fractional system (16) with $ \alpha = 0.50 $. Other parameters are as fixed in (17)
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Figure 2.  Dynamic behaviour of fractional system (16) with α = 0:79. Other parameters are as fixed in (17).
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Figure 3.  Dynamic behaviour of fractional system (16) with $ \alpha = 0.91 $. Other parameters are as fixed in (17)
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Figure 4.  A strange attractor for dynamic system (16) with $ \alpha = 0.48 $
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Figure 5.  One-dimensional distribution of time-fractional reaction-diffusion system (19) for $ \alpha = 0.11 $
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Figure 6.  One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:25
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Figure 7.  One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:45
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Figure 8.  One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:79
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Figure 9.  One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:91
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