Asymptotic behavior of solutions to some classes of multi-order fractional cooperative systems
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- by La Van Thinh and Hoang The Tuan;
- Proc. Amer. Math. Soc. 153 (2025), 1559-1574
- DOI: https://doi.org/10.1090/proc/17119
- Published electronically: February 18, 2025
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Abstract:
This paper is devoted to the study of the asymptotic behavior of solutions to multi-order fractional cooperative systems. First, we demonstrate the boundedness of solutions to fractional-order systems under certain conditions imposed on the vector field. We then prove the global attractivity and the convergence rate of solutions to such systems (in the case when the orders of fractional derivatives are equal, the convergence rate of solutions is sharp and optimal). To our knowledge, these kinds of results are new contributions to the qualitative theory of multi-order fractional positive systems and they seem to have been unknown before in the literature. As a consequence of this result, we obtain the convergence of solutions toward a nontrivial equilibrium point in an ecosystem model (a particular class of fractional-order Kolmogorov systems). Finally, some numerical examples are also provided to illustrate the obtained theoretical results.References
- Dirk Aeyels and Patrick De Leenheer, Extension of the Perron-Frobenius theorem to homogeneous systems, SIAM J. Control Optim. 41 (2002), no. 2, 563–582. MR 1920271, DOI 10.1137/S0363012900361178
- E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models, J. Math. Anal. Appl. 325 (2007), no. 1, 542–553. MR 2273544, DOI 10.1016/j.jmaa.2006.01.087
- D. Bǎleanu and A. M. Lopes, Handbook of fractional calculus with applications: applications in engineering, life and social sciences, Part A, De Gruyter, Berlin, Boston, 2019.
- D. Bǎleanu and A. M. Lopes, Handbook of fractional calculus with applications: applications in engineering, life and social sciences, Part B, De Gruyter, Berlin, Boston, 2019.
- L. Benvenuti, L. Farina, and B. D. O. Anderson, The positive side of filters: a summary, IEEE Circ. Sys. Magazine 1 (2001), no. 3, pp. 32–36.
- F. Blanchini, P. Colaneri, and M. E. Valcher, Switched linear positive systems, Foundations and Trends in Systems and Control, 2 (2015), no. 2, pp. 101–273.
- E. Carson and C. Cobelli, Modelling methodology for physiology and medicine, Academic Press, San Diego, 2001.
- Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund, and Hoang The Tuan, Linearized asymptotic stability for fractional differential equations, Electron. J. Qual. Theory Differ. Equ. , posted on (2016), Paper No. 39, 13. MR 3513975, DOI 10.14232/ejqtde.2016.1.39
- N. D. Cong, H. T. Tuan, and H. Trinh, On asymptotic properties of solutions to fractional differential equations, J. Math. Anal. Appl. 484 (2020), no. 2, 123759, 24. MR 4041539, DOI 10.1016/j.jmaa.2019.123759
- Pamela G. Coxson and Helene Shapiro, Positive input reachability and controllability of positive systems, Linear Algebra Appl. 94 (1987), 35–53. MR 902065, DOI 10.1016/0024-3795(87)90076-0
- Kai Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, vol. 2004, Springer-Verlag, Berlin, 2010. An application-oriented exposition using differential operators of Caputo type. MR 2680847, DOI 10.1007/978-3-642-14574-2
- A. A. Elsadany and A. E. Matouk, Dynamical behaviors of fractional-order Lotka-Volterra predator-prey model and its discretization, J. Appl. Math. Comput. 49 (2015), no. 1-2, 269–283. MR 3393779, DOI 10.1007/s12190-014-0838-6
- Hamid Reza Feyzmahdavian, Themistoklis Charalambous, and Mikael Johansson, Exponential stability of homogeneous positive systems of degree one with time-varying delays, IEEE Trans. Automat. Control 59 (2014), no. 6, 1594–1599. MR 3225234, DOI 10.1109/TAC.2013.2292739
- Wassim M. Haddad, VijaySekhar Chellaboina, and Qing Hui, Nonnegative and compartmental dynamical systems, Princeton University Press, Princeton, NJ, 2010. MR 2655815, DOI 10.1515/9781400832248
- Esteban Hernandez-Vargas, Patrizio Colaneri, Richard Middleton, and Franco Blanchini, Discrete-time control for switched positive systems with application to mitigating viral escape, Internat. J. Robust Nonlinear Control 21 (2011), no. 10, 1093–1111. MR 2839841, DOI 10.1002/rnc.1628
- Morris W. Hirsch, Systems of differential equations which are competitive or cooperative. I. Limit sets, SIAM J. Math. Anal. 13 (1982), no. 2, 167–179. MR 647119, DOI 10.1137/0513013
- Patrick De Leenheer and Dirk Aeyels, Stability properties of equilibria of classes of cooperative systems, IEEE Trans. Automat. Control 46 (2001), no. 12, 1996–2001. MR 1878229, DOI 10.1109/9.975508
- Oliver Mason and Mark Verwoerd, Observations on the stability properties of cooperative systems, Systems Control Lett. 58 (2009), no. 6, 461–467. MR 2518095, DOI 10.1016/j.sysconle.2009.02.009
- Y. Moreno, R. Pastor-Satorras, and A. Vespignani, Epidemic outbreaks in complex heterogeneous networks, The European Physical J. B: Condensed Matter and Complex Systems, 26 (2002), no. 4, 521–529.
- J. W. Nieuwenhuis, Some results about a Leontieff-type model, Frequency domain and state space methods for linear systems (Stockholm, 1985) North-Holland, Amsterdam, 1986, pp. 213–225. MR 924211
- I. Petráš, Handbook of fractional calculus with applications: applications in control, De Gruyter, Berlin, Boston, 2019.
- Wenxian Shen and Xiao-Qiang Zhao, Convergence in almost periodic cooperative systems with a first integral, Proc. Amer. Math. Soc. 133 (2005), no. 1, 203–212. MR 2085171, DOI 10.1090/S0002-9939-04-07556-2
- John Smillie, Competitive and cooperative tridiagonal systems of differential equations, SIAM J. Math. Anal. 15 (1984), no. 3, 530–534. MR 740693, DOI 10.1137/0515040
- Hal L. Smith, On the asymptotic behavior of a class of deterministic models of cooperating species, SIAM J. Appl. Math. 46 (1986), no. 3, 368–375. MR 841454, DOI 10.1137/0146025
- Hal L. Smith, Monotone dynamical systems, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995. An introduction to the theory of competitive and cooperative systems. MR 1319817
- V. E. Tarasov, Handbook of fractional calculus with applications: applications in physics, Part A, De Gruyter, Berlin, Boston, 2019.
- V. E. Tarasov, Handbook of fractional calculus with applications: applications in physics, Part B, De Gruyter, Berlin, Boston, 2019.
- Qiang Xiao, Zhenkun Huang, Zhigang Zeng, Tingwen Huang, and Frank L. Lewis, Stability of homogeneous positive systems with time-varying delays, Automatica J. IFAC 152 (2023), Paper No. 110965, 7. MR 4557769, DOI 10.1016/j.automatica.2023.110965
- Gennadi Vainikko, Which functions are fractionally differentiable?, Z. Anal. Anwend. 35 (2016), no. 4, 465–487. MR 3556757, DOI 10.4171/ZAA/1574
- D. Del Vecchio and R. M. Murray, Biomolecular feedback systems, Princeton University Press, Princeton, New Jersey, 2014.
Bibliographic Information
- La Van Thinh
- Affiliation: Academy of Finance, No. 58, Le Van Hien St., Duc Thang Wrd., Bac Tu Liem Dist., Hanoi, Viet Nam
- MR Author ID: 1520688
- ORCID: 0000-0002-7595-9836
- Email: lavanthinh@hvtc.edu.vn
- Hoang The Tuan
- Affiliation: Department of Mathematics, Great Bay University, Dongguan, Guangdong 523000, People’s Republic of China; and Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Hanoi, Viet Nam
- MR Author ID: 1016449
- Email: tuanht@gbu.edu.vn
- Received by editor(s): November 19, 2023
- Received by editor(s) in revised form: July 11, 2024
- Published electronically: February 18, 2025
- Additional Notes: The second author was partly supported by the Guangdong Basic and Applied Basic Research Foundation (2023A1515140016).
- Communicated by: Wenxian Shen
- © Copyright 2025 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 153 (2025), 1559-1574
- MSC (2020): Primary 34A08, 34K37, 45G05, 45M05, 45M20
- DOI: https://doi.org/10.1090/proc/17119