Stability Analysis of a Lotka-Volterra Type Biological Model in Its Fractional Version
Abstract
The stability of systems of ordinary differential equations of fractional order has been a study subject over the last two decades within fractional analysis. This study presented some fundamental results on the stability of linear fractional systems, which has allowed, in a certain sense, the generalization of some results for nonlinear fractional differential equations. This paper aims to present the fundamental results of the stability of systems of nonlinear ordinary differential equations of fractional order. These results are specified and demonstrated for a particular type of nonlinear differential equations of fractional order and applied to the analysis of the dynamics of a Lotka–Volterra-type biological model, which allows an analysis of the interaction between three species, some acting as prey and others as predator. The stability analysis of the prey–predator biological model used fractional operators in the sense of Caputo, and simulations verified these results.
Keywords: fractional calculus, fractional differential equations, stability of fractional differential equations, biological models, Lotka-Volterra type model.
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TOLEDO S.F. Stability analysis of systems of differential equations using Hurwitz polynomials. Master's thesis in mathematics. Technological University of Pereira, Pereira, 2020.
LYAPUNOV A.M. The general problem of the stability of motion. Taylor and Francis, London, 1992.
MAGIN M., ORTIGUEIRA D., PODLUBNY I., and TRUJILLO J.J. On the fractional signals and systems. Signal Processing, 2011, 91(3): 350-371.
ORTIGUEIRA M.D., and BENGOCHEA G. A Simple Solution for the General Fractional Ambartsumian Equation. Applied Sciences, 2023, 13(2): 871.
BENGOCHEA G., and ORTIGUEIRA M.D. Operational Calculus with Applications to Generalized Two-Sided Fractional Derivative. In: Proceedings of the International Conference on Fractional Differentiation and its Applications (ICFDA’21). 2022: 452.
DUARTE ORTIGUEIRA M. Fractional Calculus for Scientists and Engineers. Springer, Dordrecht, London, New York, 2011.
BALEANU D., DIETHELM K., SCALAS E., and TRUJILLO J.J. Fractional Calculus Models and Numerical Methods. Vol. 3. Series on Complexity, Nonlinearity and Chaos. World Scientific Publishing, 2012.
ATANGANA A., and BALEANU D. New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model. Thermal Science, 2016, 20: 763-769.
http://dx.doi.org/10.2298/TSCI160111018A
SUN H.G., ZHANG Y., BALEANU D., CHEN W., and CHEN Y.Q. A new collection of real world applications of fractional calculus in science and engineering. Communications in Nonlinear Science and Numerical Simulation, 2018, 64: 213-231.
BALEANU D., TENREIRO J.A., MACHADO, and LUO A.C.J. Fractional dynamics and control. Springer Science and Business Media, New York, 2011.
BALEANU D., JAJARMI A., MOHAMMADI H., and REZAPOUR S. A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative. Chaos, Solitons and Fractals, 2020, 134, 109705.
BALEANU D., MOHAMMADI H., and REZAPOUR H. A fractional differential equation model for the COVID-19 transmission by using the Caputo-Fabrizio derivative. Advances in Difference Equations, 2020, 299.
BALEANU D., MOHAMMADI H., and REZAPOUR H. Analysis of the model of HIV-1 infection of CD4+T-cell with a new approach of fractional derivative. Advances in Difference Equations, 2020, 71.
KILBAS A., SRIVASTAVA H., and TRUJILLO J.J. Theory and Applications of Fractional Differential Equations. North Holland Mathematics Studies, Elseiver, Amsterdam, 2006.
MATIGNON D. Stability results for fractional differential equations with applications to control processing. Computational Engineering in Systems Applications, 1996, 2: 963-968. https://scirp.org/reference/referencespapers.aspx?referenceid=649253
PETRAS I. Stability of Fractional-Order Systems with Rational Orders. Fractional Calculus and Applied Analysis, 2009, 12(3).
https://doi.org/10.48550/arXiv.0811.4102
LI C.P., and ZHANG F.R. A survey on the stability of fractional differential equations. European Physical Journal: Special Topics, 2011, 193(1): 27-47. https://doi.org/10.1140/epjst/e2011-01379-1
TAVAZOEI M.S., and HAERI M. A note on the stability of fractional order systems. Mathematics and Computers in Simulation. Mathematics and Computers in Simulation, 2009, 79(5): 1566-1576. https://doi.org/10.1016/j.matcom.2008.07.003
SABATIER J., MOZE M., and FARGES C. LMI stability conditions for fractional order systems. Computers and Mathematics with Applications, 2010, 59(5): 1594-1609.
https://doi.org/10.1016/j.camwa.2009.08.003
TOLEDO F.F., CÁRDENAS A.P.P., and ESCUDERO S.C. A note on the stability of a modified Lotka-Volterra model using Hurwitz polynomials. WSEAS Transactions on Mathematics, 2021, 20: 431-441. https://doi.org/10.37394/23206.2021.20.44
RIVERO M., TRUJILLO J.J., VÁZQUEZ L., and PILAR VELASCO M. Fractional dynamics of populations. Applied Mathematics and Computation, 2011, 218: 1089-1095. https://doi.org/10.1016/j.amc.2011.03.017
SUN G., and MAI A. Stability analysis of a two-patch predator–prey model with two dispersal delays. Advances in Difference Equations, 2018, Article number: 373. https://doi.org/10.1186/s13662-018-1833-2
TANG B. Dynamics for a fractional-order predator-prey model with group defense. Scientific Reports, 2020, 10(1). https://doi.org/10.1038/s41598-020-61468-3
WANG X., TAN Y., CAI Y., and WANG W. Impact of the fear effect on the stability and bifurcation of a Leslie-Gower predator-prey model. International Journal of Bifurcation and Chaos, 2020, 30(14). https://doi.org/10.1142/S0218127420502107
AHMED E., EL-SAYED A.M.A., and EL-SAKA H.A. A. Equilibrium points, stability and numerical solutions of fractional order predator-prey and rabies models. Journal of Mathematical Analysis and Applications, 2007, 325(1): 542-553. https://doi.org/10.1016/j.jmaa.2006.01.087
PERUMAL R., MUNIGOUNDER S., MOHD M.H., and BALACHANDRAN K. Stability analysis of the fractional-order prey-predator model with infection. International Journal of Modelling and Simulation, 2020, 41(6): 434-450. https://doi.org/10.1080/02286203.2020.1783131
SAMBATH M., RAMESH P., and BALACHANDRAN K. Asymptotic Behavior of the Fractional Order three Species Prey-Predator Model. International Journal of Nonlinear Sciences and Numerical Simulation, 2018, 19(7-8): 721-733. https://doi:10.1515/ijnsns-2017-0273
TOLEDO S.F., SEPÚLVEDA C.A.R. and CÁRDENAS A.P.P. On the Stability of Systems of Nonlinear Ordinary Differential Equations of Fractional Order. Journal of Hunan University Natural Sciences, 2023, 50(7).
TOLEDO S.F., SEPÚLVEDA C.A.R. and CÁRDENAS A.P.P. An Approach to the Stability of Systems of Differential Equations of Fractional Order with Time Delay. Journal of Hunan University Natural Sciences, 2023, 50(6).
BERTRAM R. Fractional Calculus and Its Applications (Lecture Notes in Mathematics). Springer-Verlag, New York, 1975. https://catalogosiidca.csuca.org/Record/UCR.000279697
SÁNCHEZ J.M. Genesis and development of Fractional Calculus. Mathematical Thinking, 2011, 1: 1-15.
http://www2.caminos.upm.es/Departamentos/matematicas/revistapm/revista_impresa/numero_1/genesis_y_desarrollo_del_calculo_fraccional.pdf
PODLUBNY I. Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation. Fractional Calculus and Applied Analysis, 2002, 5(4): 367-386. https://arxiv.org/abs/math/0110241
HILFER R. Applications of Fractional Calculus in Physics. World Scientific Publishing, Mainz University, 2000, 473. https://www.worldscientific.com/worldscibooks/10.1142/3779#t=aboutBook
LOVERRO A. Fractional Calculus: History, Definitions and Applications for the Engineer. Department of Aerospace and Mechanical Engineering, University of Notre Dame Notre, 2004. https://www.semanticscholar.org/paper/Fractional-Calculus-%3A-History-%2C-Definitions-and-for-Loverro/6256fee0c10bdb7096df51ca8e64df58414ed026
KILBAS A., BLANCA A., and TRUJILLO, J.J. Fractional calculus and fractional differential equations. National Distance Education University, vol. 1, 2003.
SHANTANU D. Functional Fractional Calculus. 2nd ed. Springer-Verlag, Berlin, Heidelberg, 2011.
https://link.springer.com/book/10.1007/978-3-540-72703-3
KENNETH M., and BERTRAM R. An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, New York, 1993. https://www.semanticscholar.org/paper/An-Introduction-to-the-Fractional-Calculus-and-Miller-Ross/d1ada669360efa7be4cbba3c50f414bcf864d6f3
PODLUBNY I. Fractional Differential Equations, Mathematics in Science and Engineering. Technical University of Kosice, Slovak Republic, 1999. https://www.elsevier.com/books/fractional-differential-equations/podlubny/978-0-12-558840-9I
LI C., and ZENG F. Numerical methods for fractional calculus. Routledge, 2015. https://www.routledge.com/Numerical-Methods-for-Fractional-Calculus/Li-Zeng/p/book/9780367658793
LI C., and CAI M. Theory and Numerical Approximations of Fractional Integrals and Derivatives, Society for Industrial and Applied Mathematics. Philadelphia, 2019. https://www.semanticscholar.org/paper/Theory-and-Numerical-Approximations-of-Fractional-Li-Cai/90886eab1cc3596ca2e66b5a030b46299820f0f3
LI C., QIAN D., and CHEN Y. On Riemann-Liouville and Caputo derivatives. Discrete Dynamics in Nature and Society, 2011, Article ID 562494. https://doi.org/10.1155/2011/562494
QIAN D., LI C., AGARWAL R.P., and WONG P.J.Y. Stability analysis of fractional differential system with Riemann-Liouville derivative. Mathematical and Computer Modelling, 2010, 52(5-6): 862-874. https://doi.org/10.1016/j.mcm.2010.05.016
DENG W., LI C., and GUO Q. Analysis of fractional differential equations with multi-orders. Fractals, 2007, 15(2): 173-182. https://doi.org/10.1142/S0218348X07003472
ODIBAT Z.M. Analytic study on linear systems of fractional differential equations. Computers and Mathematics with Applications, 2010, 59(3): 1171-1183. https://doi.org/10.1016/j.camwa.2009.06.035
DENG W., LI C., and LÜ J. Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics, 2007, 48(4): 409-416.
BONILLA B., RIVERO M., and TRUJILLO J.J. On systems of linear fractional differential equations with constant coefficients. Applied Mathematics and Computation, 2007, 187(1): 68-78. https://doi.org/10.1016/j.amc.2006.08.104
LI C., and ZHANG F. Stability analysis of fractional differential systems with order lying in (1,2). Advances in Difference Equations, 2011, 59(5): 1594-1609. https://doi.org/10.1155/2011/213485
RADWAN A.G., SOLIMAN A.M., ELWAKIL A.S., and SEDEEK A. On the stability of linear systems with fractional order elements. Chaos, Solitons and Fractals, 2009, 40(5): 2317-2328. https://doi.org/10.1016/j.chaos.2007.10.033
QIAN D., and LI C. Stability analysis of the fractional differential systems with Miller-Ross sequential derivative, In: Proceedings of the World Congress on Intelligent Control and Automation (WCICA), 2010: 213-219. https://doi.org/10.1109/WCICA.2010.5555024
QIN Z., WU R., and LU Y. Stability analysis of fractional order systems with the Riemann-Liouville derivative. Systems Science and Control Engineering, 2014, 2(1): 727-731. https://doi.org/10.1080/21642583.2013.877857
BRANDIBUR O., GARRAPPA R., and KASLIK E. Stability of systems of fractional order differential equations with Caputo derivatives. Mathematics, 2021, 9(8), 914. https://doi.org/10.3390/math9080914
MOMANI S., and HADID S. Lyapunov stability solutions of fractional integrodifferential equations. International Journal of Mathematics and Mathematical Sciences, 2004, 47: 2503-2507.
LONGGE Z., JUNMIN L., and GUO-PEI C. Extension of Lyapunov second method by fractional calculus. Pure and Applied Mathematics, 2005, 21: 291-294.
TARASOV V.E. Fractional Stability. Springer, 2007. https://doi.org/10.48550/arXiv.0711.2117
LI Y., CHEN Y. Q., and PODLUBNY I. Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica, 2009, 45(8): 1965-1969. https://doi.org/10.1016/j.automatica.2009.04.003
LI Y., CHEN Y.Q., and PODLUBNY I. Stability of fractional order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Computers and Mathematics with Applications, 2010, 59(5): 1810-1821. https://doi.org/10.1016/j.camwa.2009.08.019
HASSAN K. Nonlinear Systems, 3rd ed. Prentice Hall, New Jersey, 2002.
VIANA M., and ESPINAR J.M. Differential Equations: A Dynamical Systems Approach to Theory and Practice. American Mathematical Society, 2021.
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