# American Institute of Mathematical Sciences

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

 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.

Citation: Kolade M. Owolabi. Dynamical behaviour of fractional-order predator-prey system of Holling-type. Discrete & Continuous Dynamical Systems - S, 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [34] M. V. Ruzhansky, Y. J. Cho, P. Agarwal and I. Area, Advances in Real and Complex Analysis with Applications, Birkhuser, 2017. [35] K. M. Saad, An approximate analytical solution of coupled nonlinear fractional diffusion equations, Journal of Fractional Calculus and Applications, 5 (2014), 58-72. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [34] M. V. Ruzhansky, Y. J. Cho, P. Agarwal and I. Area, Advances in Real and Complex Analysis with Applications, Birkhuser, 2017. [35] K. M. Saad, An approximate analytical solution of coupled nonlinear fractional diffusion equations, Journal of Fractional Calculus and Applications, 5 (2014), 58-72. [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. [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. [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. [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.
Dynamic behaviour of fractional system (16) with $\alpha = 0.50$. Other parameters are as fixed in (17)
Dynamic behaviour of fractional system (16) with α = 0:79. Other parameters are as fixed in (17).
Dynamic behaviour of fractional system (16) with $\alpha = 0.91$. Other parameters are as fixed in (17)
A strange attractor for dynamic system (16) with $\alpha = 0.48$
One-dimensional distribution of time-fractional reaction-diffusion system (19) for $\alpha = 0.11$
One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:25
One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:45
One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:79
One-dimensional distribution of time-fractional reaction-diffusion system (18) for α = 0:91
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