{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T20:41:07Z","timestamp":1778013667348,"version":"3.51.4"},"reference-count":22,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,21]],"date-time":"2024-01-21T00:00:00Z","timestamp":1705795200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Council of Science and Technology of Mexico (CONACYT)","award":["A1-S-45928"],"award-info":[{"award-number":["A1-S-45928"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we study the fractional form of a well-known dynamical system from mathematical biology, namely, the Lotka\u2013Volterra model. This mathematical model describes the dynamics of a predator and prey, and we consider here the fractional form using the Rabotnov fractional-exponential (RFE) kernel. In this work, we derive an approximate formula of the fractional derivative of a power function \u03b6p in terms of the RFE kernel. Next, by using the spectral collocation method (SCM) based on the shifted Vieta\u2013Lucas polynomials (VLPs), the fractional differential system is reduced to a set of algebraic equations. We provide a theoretical convergence analysis for the numerical approach, and the accuracy is verified by evaluating the residual error function through some concrete examples. The results are then contrasted with those derived using the fourth-order Runge-Kutta (RK4) method.<\/jats:p>","DOI":"10.3390\/axioms13010071","type":"journal-article","created":{"date-parts":[[2024,1,22]],"date-time":"2024-01-22T12:01:13Z","timestamp":1705924873000},"page":"71","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A Note on a Fractional Extension of the Lotka\u2013Volterra Model Using the Rabotnov Exponential Kernel"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3844-139X","authenticated-orcid":false,"given":"Mohamed M.","family":"Khader","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11566, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7580-7533","authenticated-orcid":false,"given":"Jorge E.","family":"Mac\u00edas-D\u00edaz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, 10120 Tallinn, Estonia"},{"name":"Department of Mathematics and Physics, Autonomous University of Aguascalientes, Ave. Universidad 940, Ciudad Universitaria, Aguascalientes 20100, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5532-362X","authenticated-orcid":false,"given":"Alejandro","family":"Rom\u00e1n-Loera","sequence":"additional","affiliation":[{"name":"Department of Electronics, Autonomous University of Aguascalientes, Ave. Universidad 940, Ciudad Universitaria, Aguascalientes 20100, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6381-6806","authenticated-orcid":false,"given":"Khaled M.","family":"Saad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences and Arts, Najran University, Najran P.O. Box 1988, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz P.O. Box 6803, Yemen"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,21]]},"reference":[{"key":"ref_1","unstructured":"Kilbas, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordon & Breach."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1140\/epjp\/i2018-12191-x","article-title":"On the numerical evaluation for studying the fractional KdV, KdV-Burger\u2019s, and Burger\u2019s equations","volume":"133","author":"Khader","year":"2018","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_3","first-page":"4460","article-title":"A new Rabotnov fractional-exponential function-based fractional derivative for diffusion equation under external force","volume":"47","author":"Kumar","year":"2020","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1677","DOI":"10.2298\/TSCI180320239Y","article-title":"A new general fractional-order derivative with Rabotnov fractional-exponential kernel applied to model the anomalous heat transfer","volume":"23","author":"Yang","year":"2019","journal-title":"Therm. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.chaos.2019.07.037","article-title":"New numerical simulations for some real-world problems with Atangana-Baleanu fractional derivative","volume":"128","author":"Gao","year":"2019","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1167","DOI":"10.1002\/mma.5421","article-title":"Application of the Caputo-Fabrizio and Atangana-Baleanu fractional derivatives to the mathematical model of cancer chemotherapy effect","volume":"42","author":"Saad","year":"2019","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1140\/epjp\/i2017-11717-0","article-title":"New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models","volume":"132","author":"Toufik","year":"2017","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1002\/mma.5903","article-title":"On the analysis of vibration equation involving a derivative with Mittag-Leffler law","volume":"43","author":"Kumar","year":"2020","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/s40819-020-0799-4","article-title":"Connection formulae between generalized Lucas polynomials and some Jacobi polynomials: Application to certain types of fourth-order BVPs","volume":"6","author":"Youssri","year":"2020","journal-title":"Int. J. Appl. Comput. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"626","DOI":"10.1186\/s13662-020-03085-y","article-title":"Vieta\u2013Lucas polynomials for solving a fractional-order mathematical physics model","volume":"2020","author":"Agarwal","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1007\/BF02458847","article-title":"Explosive route to chaos through a fractal torus in a generalized Lotka\u2013Volterra model","volume":"50","author":"Samardzija","year":"1988","journal-title":"Bull. Math. Biol."},{"key":"ref_12","first-page":"1","article-title":"An algorithm for the numerical solution of differential equations of fractional order","volume":"5","author":"Diethelm","year":"1997","journal-title":"Electron. Trans. Numer. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1007\/s40314-013-0079-6","article-title":"Numerical treatment for solving fractional SIRC model and influenza A","volume":"33","author":"Khader","year":"2014","journal-title":"Comput. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10440-018-0171-4","article-title":"Chebyshev wavelet procedure for solving FLDEs","volume":"158","author":"Khader","year":"2018","journal-title":"Acta Appl. Math."},{"key":"ref_15","unstructured":"Horadam, A.F. (2000). Vieta Polynomials, The University of New England."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1901131","DOI":"10.1155\/2022\/1901131","article-title":"Solving fractional generalized Fisher-Kolmogorov-Petrovsky-Piskunov\u2019s equation using compact finite difference method together with spectral collocation algorithms","volume":"2022","author":"Zakaria","year":"2022","journal-title":"J. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"5","DOI":"10.17512\/jamcm.2023.4.01","article-title":"Semi-analytical scheme with its stability analysis for solving the fractional-order predator\u2013prey equations by using Laplace-VIM","volume":"22","author":"Adel","year":"2023","journal-title":"J. Appl. Math. Comput. Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1023\/A:1022463810376","article-title":"Ultraspherical integral method for optimal control problems governed by ordinary differential equations","volume":"25","author":"Salim","year":"2003","journal-title":"J. Glob. Optim."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1085","DOI":"10.1016\/S0898-1221(00)85018-X","article-title":"The LFOPC Leap-Frog algorithm for constrained optimization","volume":"40","author":"Snyman","year":"2000","journal-title":"Comput. Math. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/S0898-1221(97)00101-6","article-title":"An ODE method for solving constrained optimization","volume":"3","author":"Zhou","year":"1997","journal-title":"Comput. Math. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1006\/jmaa.1997.5791","article-title":"A convergence of ODE method in constrained optimization","volume":"218","author":"Zongfang","year":"1998","journal-title":"J. Math. Anal. Appl."},{"key":"ref_22","first-page":"895","article-title":"Operational matrices to solve nonlinear Volterra-Fredholm integrodifferential equations of multi-arbitrary order","volume":"29","author":"Parand","year":"2016","journal-title":"Gazi Univ. J. Sci."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/1\/71\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:46:43Z","timestamp":1760104003000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/1\/71"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,21]]},"references-count":22,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["axioms13010071"],"URL":"https:\/\/doi.org\/10.3390\/axioms13010071","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,21]]}}}