On discrete fractional-order Lotka-Volterra model based on the Caputo difference discrete operator. (English) Zbl 1516.39009
Summary: This work aims at introducing a new discrete fractional order model based on Lotka-Volterra prey-predator model with logistic growth of prey species. The proposed model is a generalization of the standard integer order discrete-time Lotka-Volterra model to its fractional-order version while incorporating also logistic growth for prey population. The equilibrium points of the presented model are firstly obtained, and their stability analysis is conducted. Then, the nonlinear dynamics of the proposed model and possible occurrence of chaotic behavior are explored. The effects of fractional order along with other key parameters in the model are investigated using several techniques. Thorough numerical simulations are carried out where Lyapunov exponents, bifurcation diagrams, phase portraits and as well as \(C_0\) complexity measure are obtained to analyze the dynamics of the proposed model and confirm theoretical results.
MSC:
| 39A70 | Difference operators |
| 39A13 | Difference equations, scaling (\(q\)-differences) |
| 39A33 | Chaotic behavior of solutions of difference equations |
| 26A33 | Fractional derivatives and integrals |
Keywords:
fractional Caputo difference operator; Lotka-Volterra system; logistic growth; chaotic dynamicsSoftware:
Fractional Order Chaotic SystemsReferences:
| [1] | Lotka, A., Elements of physical biology (1925), Baltimore, Md: Williams and Wilkins, Baltimore, Md · JFM 51.0416.06 |
| [2] | Volterra,V.: Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. (Societá anonima tipografica” Leonardo da Vinci, (1927)) · JFM 52.0450.06 |
| [3] | Holling, CS, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 97, S45, 5-60 (1965) · doi:10.4039/entm9745fv |
| [4] | Edelstein-Keshet, L.: Mathematical models in biology. SIAM (2005) · Zbl 1100.92001 |
| [5] | Berryman, AA, The orgins and evolution of predator-prey theory, Ecology, 73, 5, 1530-1535 (1992) · doi:10.2307/1940005 |
| [6] | Odum, EP; Barrett, GW, Fundamentals of ecology (1971), Philadelphia: Saunders, Philadelphia |
| [7] | May, RM; Hunt, BR; Kennedy, JA, Simple mathematical models with very complicated dynamics, The theory of chaotic attractors, 85-93 (2004), Berlin: Springer, Berlin · doi:10.1007/978-0-387-21830-4_7 |
| [8] | Hus, S.; Hwang, T., Global stability for a class of predator-prey system, SIAM J. Appl. Math., 55, 763-783 (1995) · Zbl 0832.34035 · doi:10.1137/S0036139993253201 |
| [9] | Danca, M.; Codreanu, S.; Bako, B., Detailed analysis of a nonlinear prey-predator model, J. Biol Phys., 23, 1, 1-11 (1997) · doi:10.1023/A:1004918920121 |
| [10] | Kot, M., Elements of mathematical ecology (2001), Cambridge: Cambridge University Press, Cambridge · Zbl 1060.92058 · doi:10.1017/CBO9780511608520 |
| [11] | Xiao, D.; Ruan, S., Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61, 4, 1445-1472 (2001) · Zbl 0986.34045 · doi:10.1137/S0036139999361896 |
| [12] | Xiao, Y.; Cheng, D.; Tang, S., Dynamic complexities in predator-prey ecosystem models with age-structure for predator, Chaos, Solitons & Fractals, 14, 9, 1403-1411 (2002) · Zbl 1032.92033 · doi:10.1016/S0960-0779(02)00061-9 |
| [13] | Jing, Z.; Yang, J., Bifurcation and chaos in discrete-time predator-prey system, Chaos, Solitons & Fractals, 27, 1, 259-277 (2006) · Zbl 1085.92045 · doi:10.1016/j.chaos.2005.03.040 |
| [14] | Agiza, HN; Elabbasy, EM; El-Metwally, H.; Elsadany, AA, Chaotic dynamics of a discrete prey-predator model with Holling type II, Nonlinear Anal. Real World Appl., 10, 1, 116-129 (2009) · Zbl 1154.37335 · doi:10.1016/j.nonrwa.2007.08.029 |
| [15] | Elsadany, AA; El-Metwally, H.; Elabbasy, EM; Agiza, HN, Chaos and bifurcation of a nonlinear discrete prey-predator system, Comput. Ecol. Softw., 2, 3, 169-180 (2012) |
| [16] | Yousef, A.; Salman, S.; Elsadany, AA, Stability and bifurcation analysis of a delayed discrete predator-prey model, Int. J. Bifurc. Chaos, 28, 9, 1850116 (2018) · Zbl 1401.39021 · doi:10.1142/S021812741850116X |
| [17] | Lin, Y.; Din, Q.; Rafaqat, M.; Elsadany, AA; Zeng, Y., Dynamics and chaos control for a discrete-time Lotka-Volterra model, IEEE Access, 8, 126760-126775 (2020) · doi:10.1109/ACCESS.2020.3008522 |
| [18] | Oldham, K.; Spanier, J., The fractional calculus theory and applications of differentiation and integration to arbitrary order (1974), London: Elsevier, London · Zbl 0292.26011 |
| [19] | Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (1999), London: Elsevier, London · Zbl 0924.34008 |
| [20] | Kilbas, AA; Srivastava, HM; Trujillo, JJ, Theory and applications of fractional differential equations (2006), London: Elsevier, London · Zbl 1092.45003 · doi:10.1016/S0304-0208(06)80001-0 |
| [21] | Diethelm, K., A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dyn., 71, 4, 613-619 (2013) · doi:10.1007/s11071-012-0475-2 |
| [22] | El-Misiery, A.; Ahmed, E., On a fractional model for earthquakes, Appl. Math. Comput., 178, 2, 207-211 (2006) · Zbl 1130.86309 |
| [23] | Petráš, I., Fractional-order nonlinear systems: modeling, analysis and simulation (2011), Berlin: Springer, Berlin · Zbl 1228.34002 · doi:10.1007/978-3-642-18101-6 |
| [24] | Meral, F.; Royston, T.; Magin, R., Fractional calculus in viscoelasticity: an experimental study, Commun. Nonlinear Sci. Numer. Simul., 15, 4, 939-945 (2010) · Zbl 1221.74012 · doi:10.1016/j.cnsns.2009.05.004 |
| [25] | Sun, H.; Zhang, Y.; Baleanu, D.; Chen, W.; Chen, Y., A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64, 213-231 (2018) · Zbl 1509.26005 · doi:10.1016/j.cnsns.2018.04.019 |
| [26] | Hilfer, R., Applications of fractional calculus in physics (2000), Singapore: World Scientific, Singapore · Zbl 0998.26002 · doi:10.1142/3779 |
| [27] | Tenreiro Machado, JA; Silva, MF; Barbosa, RS; Jesus, IS; Reis, CM; Marcos, MG; Galhano, AF, Some applications of fractional calculus in engineering, Math. Problem Eng., 2010, 639-801 (2010) · Zbl 1191.26004 · doi:10.1155/2010/639801 |
| [28] | Tarasov, VE, Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media (2011), Berlin: Springer, Berlin |
| [29] | Herrmann, R., Fractional calculus: an introduction for physicists (2014), Singapore: World Scientific, Singapore · Zbl 1293.26001 · doi:10.1142/8934 |
| [30] | Goodrich, C.; Peterson, AC, Discrete fractional calculus (2015), Berlin: Springer, Berlin · Zbl 1350.39001 · doi:10.1007/978-3-319-25562-0 |
| [31] | Diaz, JB; Olser, TJ, Differences of fractional order, Math. Comput., 28, 185-202 (1974) · Zbl 0282.26007 · doi:10.1090/S0025-5718-1974-0346352-5 |
| [32] | Huang, LL; Wu, GC; Baleanu, D.; Wang, HY, Discrete fractional calculus for interval-valued systems, Fuzzy Sets Syst., 404, 141-158 (2021) · Zbl 1464.39007 · doi:10.1016/j.fss.2020.04.008 |
| [33] | Wu, GC; Luo, M.; Huang, LL; Banerjee, S., Short memory fractional differential equations for new memristor and neural network design, Nonlinear Dyn., 100, 3611-3623 (2020) · Zbl 1516.34022 · doi:10.1007/s11071-020-05572-z |
| [34] | Abdeljawad, T.; Banerjee, S.; Wu, GC, Discrete tempered fractional calculus for new chaotic systems with short memory and image encryption, Optik, 218, 163698 (2020) · doi:10.1016/j.ijleo.2019.163698 |
| [35] | Ostalczyk, P., Discrete fractional calculus: applications in control and image processing (2015), Singapore: World Scientific, Singapore |
| [36] | Huang, LL; Park, JH; Wu, GC; Mo, ZW, Variable-order fractional discrete-time recurrent neural networks, J. Comput. Appl. Math., 370, 112633 (2020) · Zbl 1432.39012 · doi:10.1016/j.cam.2019.112633 |
| [37] | Wu, GC; Deng, ZG; Baleanu, D.; Zeng, DQ, New variable-order fractional chaotic systems for fast image encryption, Chaos Interdiscip. J. Nonlinear Sci., 29, 8, 083103 (2019) · Zbl 1420.34028 · doi:10.1063/1.5096645 |
| [38] | Li, Y.; Sun, C.; Ling, H.; Lu, A.; Liu, Y., Oligopolies price game in fractional order system, Chaos Solitons & Fractals, 132, 109583 (2020) · Zbl 1434.91043 · doi:10.1016/j.chaos.2019.109583 |
| [39] | Xin, B.; Peng, W.; Kwon, Y., A discrete fractional-order cournot duopoly game, Physica A Stat. Mech. Appl., 558, 124993 (2020) · Zbl 07531995 · doi:10.1016/j.physa.2020.124993 |
| [40] | Danca, MF, Puu system of fractional order and its chaos suppression, Symmetry, 12, 3, 340 (2020) · doi:10.3390/sym12030340 |
| [41] | Ouannas, A.; Khennaoui, AA; Momani, S.; Pham, V., The discrete fractional duffing system: chaos, 0-1 test, C-0 complexity, entropy, and control, Chaos, 30, 083131 (2020) · Zbl 1445.39004 · doi:10.1063/5.0005059 |
| [42] | AKhennaoui,A.A., Almatroud, A.O., Ouannas, A., Al-sawalha, M.M., Grassi, G., Pham, V.T., Batiha,I.M.: An unprecedented 2-dimensional discrete-time fractional-order system and its hidden chaotic attractors. Math. Problems Eng. 2021 (2021) · Zbl 1535.39005 |
| [43] | Wu, GC; Baleanu, D.; Luo, WH, Lyapunov functions for Riemann-liouville-like fractional difference equations, Appl. Math. Comput., 314, 228-236 (2017) · Zbl 1426.39010 |
| [44] | Wang, ZR; Shiri, B.; Baleanu, D., Discrete fractional watermark technique, Front. Inf. Technol. Electr. Eng., 21, 6, 880-883 (2020) · doi:10.1631/FITEE.2000133 |
| [45] | Atici, FM; Eloe, P., Discrete fractional calculus with the nabla operator, Electron. J. Qualit. Theory Differ. Equ., 2009, 1-12 (2009) · Zbl 1189.39004 |
| [46] | Abdeljawad, T., On Riemann and Caputo fractional differences, Comput. Math. Appl., 62, 3, 1602-1611 (2011) · Zbl 1228.26008 · doi:10.1016/j.camwa.2011.03.036 |
| [47] | Cermák, J.; Gyori, I.; Nechvatal, L., On explicit stability conditions for a linear fractional difference system, Fract. Calculus Appl. Anal., 18, 3, 651 (2015) · Zbl 1320.39004 · doi:10.1515/fca-2015-0040 |
| [48] | Wu, GC; Baleanu, D., Jacobian matrix algorithm for lyapunov exponents of the discrete fractional maps, Commun. Nonlinear Sci. Numer. Simul., 22, 1-3, 95-100 (2015) · Zbl 1329.65309 · doi:10.1016/j.cnsns.2014.06.042 |
| [49] | Almatroud, AO; Khennaoui, AA; Ouannas, A.; Grassi, G.; Al- Sawalha, MM; Gasri, A., Dynamical analysis of a new chaotic fractional discrete-time system and its control, Entropy, 22, 12, 1344 (2020) · doi:10.3390/e22121344 |
| [50] | Ott, E., Chaos in dynamical systems (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 1006.37001 · doi:10.1017/CBO9780511803260 |
| [51] | Sandri, M., Numerical calculation of Lyapunov exponents, Math. J., 6, 3, 78-84 (1996) |
| [52] | Ran, J., Discrete chaos in a novel two-dimensional fractional chaotic map, Adv. Diff. Equ., 294, 1-12 (2018) · Zbl 1448.37043 |
| [53] | Shen, EH; Cai, ZJ; Gu, FJ, Mathematical foundation of a new complexity measure, Appl. Math. Mech., 26, 1188-1196 (2005) · Zbl 1144.94361 · doi:10.1007/BF02507729 |
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