{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,12]],"date-time":"2026-04-12T15:38:41Z","timestamp":1776008321704,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,6]],"date-time":"2023-06-06T00:00:00Z","timestamp":1686009600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The paper introduces a novel two-dimensional fractional discrete-time predator\u2013prey Leslie\u2013Gower model with an Allee effect on the predator population. The model\u2019s nonlinear dynamics are explored using various numerical techniques, including phase portraits, bifurcations and maximum Lyapunov exponent, with consideration given to both commensurate and incommensurate fractional orders. These techniques reveal that the fractional-order predator\u2013prey Leslie\u2013Gower model exhibits intricate and diverse dynamical characteristics, including stable trajectories, periodic motion, and chaotic attractors, which are affected by the variance of the system parameters, the commensurate fractional order, and the incommensurate fractional order. Finally, we employ the 0\u20131 method, the approximate entropy test and the\u00a0C0\u00a0algorithm to measure complexity and confirm chaos in the proposed system.<\/jats:p>","DOI":"10.3390\/axioms12060561","type":"journal-article","created":{"date-parts":[[2023,6,7]],"date-time":"2023-06-07T02:02:15Z","timestamp":1686103335000},"page":"561","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["Complexity and Chaos Analysis for Two-Dimensional Discrete-Time Predator\u2013Prey Leslie\u2013Gower Model with Fractional Orders"],"prefix":"10.3390","volume":"12","author":[{"given":"Tareq","family":"Hamadneh","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al Zaytoonah University of Jordan, Amman 11733, Jordan"}]},{"given":"Abderrahmane","family":"Abbes","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University Badji Mokhtar, Annaba 23000, Algeria"},{"name":"Laboratory of Mathematics, Dynamics and Modelization, University Badji Mokhtar, Annaba 23000, Algeria"}]},{"given":"Ibraheem Abu","family":"Falahah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, The Hashemite University, Zarqa 13133, Jordan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8683-6698","authenticated-orcid":false,"given":"Yazan Alaya","family":"AL-Khassawneh","sequence":"additional","affiliation":[{"name":"Data Science and Artificial Intelligence Department, Zarqa University, Zarqa 13133, Jordan"}]},{"given":"Ahmed Salem","family":"Heilat","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Information Technology, Jadara University, Irbid 21110, Jordan"}]},{"given":"Abdallah","family":"Al-Husban","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Technology, Irbid National University, Irbid 21110, Jordan"}]},{"given":"Adel","family":"Ouannas","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Larbi Ben M\u2019hidi, Oum El Bouaghi 04000, Algeria"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ali, I., and Saleem, M.T. (2023). Spatiotemporal Dynamics of Reaction\u2013Diffusion System and Its Application to Turing Pattern Formation in a Gray\u2013Scott Model. Mathematics, 11.","DOI":"10.3390\/math11061459"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14232\/ejqtde.2009.4.3","article-title":"Discrete fractional calculus with the nabla operator","volume":"3","author":"Atici","year":"2009","journal-title":"Electron. J. Qual. Theory Differ. Equ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1016\/j.mcm.2010.03.055","article-title":"Principles of delta fractional calculus on time scales and inequalities","volume":"52","author":"Anastassiou","year":"2010","journal-title":"Math. Comput. Model."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1602","DOI":"10.1016\/j.camwa.2011.03.036","article-title":"On Riemann and Caputo fractional differences","volume":"62","author":"Abdeljawad","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"9751","DOI":"10.1002\/mma.9085","article-title":"Numerical approach to solve Caputo-Fabrizio-fractional model of Corona pandemic with Optimal Control Design and analysis","volume":"46","author":"Hanif","year":"2023","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"He, Z.Y., Abbes, A., Jahanshahi, H., Alotaibi, N.D., and Wang, Y. (2022). Fractional-order discrete-time SIR epidemic model with vaccination: Chaos and complexity. Mathematics, 10.","DOI":"10.3390\/math10020165"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"015208","DOI":"10.1088\/1402-4896\/aca531","article-title":"A new two-dimensional fractional discrete rational map: Chaos and complexity","volume":"98","author":"Shatnawi","year":"2022","journal-title":"Phys. Scr."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4501","DOI":"10.1007\/s11071-022-08086-y","article-title":"Dynamical analysis of a fractional discrete-time vocal system","volume":"111","author":"Vignesh","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1140\/epjp\/s13360-022-02472-6","article-title":"Incommensurate Fractional Discrete Neural Network: Chaos and complexity","volume":"137","author":"Abbes","year":"2022","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"035213","DOI":"10.1088\/1402-4896\/acafac","article-title":"Hidden multistability of fractional discrete non-equilibrium point memristor based map","volume":"98","author":"Shatnawi","year":"2023","journal-title":"Phys. Scr."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"965","DOI":"10.1007\/s11071-022-07766-z","article-title":"The fractional-order discrete COVID-19 pandemic model: Stability and chaos","volume":"111","author":"Abbes","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"9520","DOI":"10.3934\/math.2023480","article-title":"Modified 5-point fractional formula with Richardson extrapolation","volume":"8","author":"Batiha","year":"2023","journal-title":"AIMS Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Butt, A.I.K., Imran, M., Batool, S., and Nuwairan, M.A. (2023). Theoretical Analysis of a COVID-19 CF-Fractional Model to Optimally Control the Spread of Pandemic. Symmetry, 15.","DOI":"10.3390\/sym15020380"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"030203","DOI":"10.1088\/1674-1056\/ac7296","article-title":"The incommensurate fractional discrete macroeconomic system: Bifurcation, chaos and complexity","volume":"32","author":"Abbes","year":"2023","journal-title":"Chin. Phys. B"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.chaos.2018.12.019","article-title":"On fractional\u2013order discrete\u2013time systems: Chaos, stabilization and synchronization","volume":"119","author":"Khennaoui","year":"2019","journal-title":"Chaos Solitons Fractals"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"045310","DOI":"10.1063\/5.0004884","article-title":"Chaos and control of a three-dimensional fractional order discrete-time system with no equilibrium and its synchronization","volume":"10","author":"Ouannas","year":"2020","journal-title":"AIP Adv."},{"key":"ref_17","unstructured":"Radwan, A.G., Khanday, F.A., and Said, L.A. (2022). Fractional-Order Design, Academic Press. Volume 3 in Emerging Methodologies and Applications in Modelling."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Saadeh, R., Abbes, A., Al-Husban, A., Ouannas, A., and Grassi, G. (2023). The Fractional Discrete Predator\u2013Prey Model: Chaos, Control and Synchronization. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7020120"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Rahmi, E., Darti, I., Suryanto, A. (2021). A modified Leslie\u2013Gower model incorporating Beddington\u2013deangelis functional response, double Allee effect and memory effect. Fractal Fract., 5.","DOI":"10.3390\/fractalfract5030084"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Lin, S., Chen, F., Li, Z., and Chen, L. (2022). Complex dynamic behaviors of a modified discrete Leslie\u2013Gower Predator\u2013prey system with fear effect on prey species. Axioms, 11.","DOI":"10.3390\/axioms11100520"},{"key":"ref_21","first-page":"198","article-title":"A modified Leslie\u2013Gower fractional order prey-predator interaction model incorporating the effect of fear on prey","volume":"13","author":"Mondal","year":"2023","journal-title":"J. Appl. Anal. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2345","DOI":"10.1016\/j.apm.2014.10.040","article-title":"Bifurcation, invariant curve and hybrid control in a discrete-time predator-prey system","volume":"39","author":"Yuan","year":"2015","journal-title":"Appl. Math. Model."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1186\/s13662-020-2498-1","article-title":"Bifurcation analysis and chaos control in discrete-time modified Leslie\u2013Gower prey harvesting model","volume":"2020","author":"Ajaz","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.cnsns.2017.01.025","article-title":"Complexity and chaos control in a discrete-time prey-predator model","volume":"49","author":"Din","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"11391","DOI":"10.1016\/j.aej.2022.04.042","article-title":"Global dynamics, Neimark\u2013Sacker bifurcation and hybrid control in a Leslie\u2019s prey-predator model","volume":"61","author":"Khan","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6931354","DOI":"10.1155\/2022\/6931354","article-title":"A novel discrete-time Leslie\u2013Gower model with the impact of Allee effect in predator population","volume":"2022","author":"Vinoth","year":"2022","journal-title":"Complexity"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Allee, W.C. (1931). Animal Aggregations, a Study in General Sociology, The University of Chicago Press.","DOI":"10.5962\/bhl.title.7313"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1069","DOI":"10.1016\/S0893-9659(03)90096-6","article-title":"Boundedness and global stability for a predator-prey model with modified Leslie\u2013Gower and Holling-type II schemes","volume":"16","author":"Okiye","year":"2003","journal-title":"Appl. Math. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/s11071-013-1065-7","article-title":"Discrete fractional logistic map and its chaos","volume":"75","author":"Wu","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.cnsns.2014.06.042","article-title":"Jacobian matrix algorithm for Lyapunov exponents of the discrete fractional maps","volume":"22","author":"Wu","year":"2015","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_31","unstructured":"Gottwald, G., and Melbourne, I. (2016). Chaos Detection and Predictability, Springer."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.1073\/pnas.88.6.2297","article-title":"Approximate entropy as a measure of system complexity","volume":"88","author":"Pincus","year":"1991","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1188","DOI":"10.1007\/BF02507729","article-title":"Mathematical foundation of a new complexity measure","volume":"26","author":"Shen","year":"2005","journal-title":"Appl. Math. Mech."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"8299","DOI":"10.3390\/e17127882","article-title":"Complexity analysis and DSP implementation of the fractional-order Lorenz hyperchaotic system","volume":"17","author":"He","year":"2015","journal-title":"Entropy"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/6\/561\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:49:28Z","timestamp":1760125768000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/6\/561"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,6]]},"references-count":34,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["axioms12060561"],"URL":"https:\/\/doi.org\/10.3390\/axioms12060561","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,6]]}}}