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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 |
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.
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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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.
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