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March  2020, 13(3): 429-442. doi: 10.3934/dcdss.2020024

## A new numerical scheme applied on re-visited nonlinear model of predator-prey based on derivative with non-local and non-singular kernel

 1 Department of mathematics, Riyadh, 11989, colle of science, King Saud University, P.O. Box 1142, Saudi Arabia 2 Mehmet Akif Ersoy University, Department of Mathematics, Faculty of Sciences, 15100, Burdur, Turkey

Received  May 2018 Revised  May 2018 Published  March 2019

A new concept of dynamical system of predator-prey model is presented in this paper. The model takes into account the memory of interaction between the prey and predator due to the inclusion of fractional differentiation. In addition, the model takes into account the inherent disposition of a prey or predator toward hunting or defending in time. Analysis of existence and uniqueness of the solutions is presented. A numerical method is used to generate some simulations as the fractional orders change from one to zero. A new traveling waves patterns are obtained.

Citation: Badr Saad T. Alkahtani, Ilknur Koca. A new numerical scheme applied on re-visited nonlinear model of predator-prey based on derivative with non-local and non-singular kernel. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 429-442. doi: 10.3934/dcdss.2020024
##### References:
 [1] P. A. Abrams and L. R. Ginzburg, The nature of predation: Prey dependent, ratio dependent or neither?, Trends in Ecology & Evolution, 15 (2000), 337-341.  doi: 10.1016/S0169-5347(00)01908-X. [2] B. S. T. Alkahtani, Chua's circuit model with Atangana–Baleanu derivative with fractional order, Chaos, Solitons and Fractals, 89 (2016), 547-551.  doi: 10.1016/j.chaos.2016.03.020. [3] O. J. J. Alkahtani, Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model, Chaos, Solitons and Fractals, 89 (2016), 552-559.  doi: 10.1016/j.chaos.2016.03.026. [4] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio dependence, Journal of Theoretical Biology, 139 (1989), 311-326. [5] 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. [6] A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractionalorder, Chaos Solitons Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012. [7] A. Atangana and D. Baleanu, Caputo-Fabrizio applied to groundwater flow within a confined aquifer, J Eng Mech, 143 (2016), D4016005. doi: 10.1061/(ASCE)EM.1943-7889.0001091. [8] A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056. [9] A. Atangana and J. F. Gomez Aguila, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 166. [10] G. Gandolfo, Giuseppe Palomba and the Lotka–Volterra equations, Rendiconti Lincei, 19 (2008), 347-257. [11] A. J. Lotka, Contribution to the theory of periodic reaction, J. Phys. Chem., 14 (1910), 271-274.  doi: 10.1021/j150111a004. [12] P. H. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Corresp. Mathématique et Physique, 10 (1838), 113-121. [13] V. Volterra, Variations and Fluctuations of the Number of Individuals in Animal Species Living Together in Animal Ecology, Chapman, R.N. (ed), McGraw–Hill, 1931. [14] V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. Acad. Lincei Roma, 2 (1926), 31-113. [15] T. Yamamoto and X. Chen, An existence and nonexistence theorem for solutions of nonlinear systems and its application to algebraic equations, Journal of Computational and Applied Mathematics, 30 (1990), 87-97.  doi: 10.1016/0377-0427(90)90008-N. [16] P. Zhuang, F. Liu, I. Turner and V. Anh, Galerkin finite element method and error analysis for the fractional cable equation, Appl. Math. Comput., 217 (2010), 2534-2545.  doi: 10.1016/j.amc.2010.07.066.

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##### References:
 [1] P. A. Abrams and L. R. Ginzburg, The nature of predation: Prey dependent, ratio dependent or neither?, Trends in Ecology & Evolution, 15 (2000), 337-341.  doi: 10.1016/S0169-5347(00)01908-X. [2] B. S. T. Alkahtani, Chua's circuit model with Atangana–Baleanu derivative with fractional order, Chaos, Solitons and Fractals, 89 (2016), 547-551.  doi: 10.1016/j.chaos.2016.03.020. [3] O. J. J. Alkahtani, Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model, Chaos, Solitons and Fractals, 89 (2016), 552-559.  doi: 10.1016/j.chaos.2016.03.026. [4] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio dependence, Journal of Theoretical Biology, 139 (1989), 311-326. [5] 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. [6] A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractionalorder, Chaos Solitons Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012. [7] A. Atangana and D. Baleanu, Caputo-Fabrizio applied to groundwater flow within a confined aquifer, J Eng Mech, 143 (2016), D4016005. doi: 10.1061/(ASCE)EM.1943-7889.0001091. [8] A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056. [9] A. Atangana and J. F. Gomez Aguila, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 166. [10] G. Gandolfo, Giuseppe Palomba and the Lotka–Volterra equations, Rendiconti Lincei, 19 (2008), 347-257. [11] A. J. Lotka, Contribution to the theory of periodic reaction, J. Phys. Chem., 14 (1910), 271-274.  doi: 10.1021/j150111a004. [12] P. H. Verhulst, Notice sur la loi que la population poursuit dans son accroissement, Corresp. Mathématique et Physique, 10 (1838), 113-121. [13] V. Volterra, Variations and Fluctuations of the Number of Individuals in Animal Species Living Together in Animal Ecology, Chapman, R.N. (ed), McGraw–Hill, 1931. [14] V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. Acad. Lincei Roma, 2 (1926), 31-113. [15] T. Yamamoto and X. Chen, An existence and nonexistence theorem for solutions of nonlinear systems and its application to algebraic equations, Journal of Computational and Applied Mathematics, 30 (1990), 87-97.  doi: 10.1016/0377-0427(90)90008-N. [16] P. Zhuang, F. Liu, I. Turner and V. Anh, Galerkin finite element method and error analysis for the fractional cable equation, Appl. Math. Comput., 217 (2010), 2534-2545.  doi: 10.1016/j.amc.2010.07.066.
Numerical solution for $\alpha = 0.05.$
Numerical solution for $\alpha = 0.5.$
Numerical solution for $\alpha = 0.8.$
Numerical solution for $\alpha = 1.$
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