124
Views
2
CrossRef citations to date
0
Altmetric
Research Article

A novel second-order nonstandard finite difference method preserving dynamical properties of a general single-species model

Pages 2047-2062 | Received 15 Apr 2023, Accepted 28 Jul 2023, Published online: 18 Aug 2023

References

  • E.M. Adamu, K.C. Patidar, and A. Ramanantoanina, An unconditionally stable nonstandard finite difference method to solve a mathematical model describing Visceral Leishmaniasis, Math. Comput. Simul. 187 (2021), pp. 171–190.
  • O. Adekanye and T. Washington, Nonstandard finite difference scheme for a Tacoma narrows bridge model, Appl. Math. Model. 62 (2018), pp. 223–236.
  • R. Anguelov, Y. Dumont, J.M.-S. Lubuma, and M. Shillor, Dynamically consistent nonstandard finite difference schemes for epidemiological models, J. Comput. Appl. Math. 255 (2014), pp. 161–182.
  • R. Anguelov, P. Kama, and J.M.-S. Lubuma, On non-standard finite difference models of reaction-diffusion equations, J. Comput. Appl. Math. 175 (2005), pp. 11–29.
  • R. Anguelov and J.M.-S. Lubuma, Contributions to the mathematics of the nonstandard finite difference method and applications, Numer. Methods. Partial. Differ. Equ. 17 (2001), pp. 518–543.
  • R. Anguelov, J.M.-S. Lubuma, and M. Shillor, Topological dynamic consistency of non-standard finite difference schemes for dynamical systems, J. Differ. Equ. Appl. 17 (2011), pp. 1769–1791.
  • A.J. Arenas, G. Gonzalez-Parra, and B.M. Chen-Charpentier, Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order, Math. Comput. Simul. 121 (2016), pp. 48–63.
  • U.M. Ascher and L.R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Society for Industrial and Applied Mathematics, Philadelphia, 1998.
  • R.L. Burden and J.D. Faires, Numerical Analysis, 9th ed., Cengage Learning, 2015.
  • J. Calatayud and M. Jornet, An improvement of two nonstandard finite difference schemes for two population mathematical models, J. Differ. Equ. Appl. 27 (2021), pp. 422–430.
  • J. Calatayud and M. Jornet, On the symmetrization and composition of nonstandard finite difference schemes as an alternative to Richardson's extrapolation, J. Differ. Equ. Appl. 28 (2022), pp. 716–724.
  • B.M. Chen-Charpentier, D.T. Dimitrov, and H.V. Kojouharov, Combined nonstandard numerical methods for ODEs with polynomial right-hand sides, Math. Comput. Simul. 73 (2006), pp. 105–113.
  • D. Conte, G. Pagano, and B. Paternoster, Nonstandard finite differences numerical methods for a vegetation reaction-diffusion model, J. Comput. Appl. Math. 419 (2023), Article ID 114790.
  • K. Cooke, P. van den Driessche, and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol. 39 (1999), pp. 332–352.
  • J. Cresson and F. Pierret, Non standard finite difference scheme preserving dynamical properties, J. Comput. Appl. Math. 303 (2016), pp. 15–30.
  • J. Cresson and A. Szafrańska, Discrete and continuous fractional persistence problems – the positivity property and applications, Commun. Nonlinear Sci. Numer. Simul. 44 (2017), pp. 424–448.
  • Q.A. Dang and M.T. Hoang, Complete global stability of a metapopulation model and its dynamically consistent discrete models, Qual. Theory Dyn. Syst. 18 (2019), pp. 461–475.
  • Q.A. Dang and M.T. Hoang, Nonstandard finite difference schemes for a general predator-prey system, J. Comput. Sci. 36 (2019), Article ID 101015.
  • Q.A. Dang and M.T. Hoang, Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems, Int. J. Comput. Math. 97 (2020), pp. 2036–2054.
  • Q.A. Dang and M.T. Hoang, Positivity and global stability preserving NSFD schemes for a mixing propagation model of computer viruses, J. Comput. Appl. Math. 374 (2020), Article ID 112753.
  • D.T. Dimitrov and H.V. Kojouharov, Nonstandard finite-difference schemes for general two-dimensional autonomous dynamical systems, Appl. Math. Lett. 18 (2005), pp. 769–774.
  • M. Ehrhardt and R.E. Mickens, A nonstandard finite difference scheme for convection-diffusion equations having constant coefficients, Appl. Math. Comput. 219 (2013), pp. 6591–6604.
  • S. Elaydi, An Introduction to Difference Equations, Springer, New York, NY, 2005.
  • H. Fatoorehchi and M. Ehrhardt, Numerical and semi-numerical solutions of a modified Thévenin model for calculating terminal voltage of battery cells, J. Energy Storage 145 (2022), Article ID 103746.
  • Y. Gao and W. Zhao, Stability Analysis for the Fractional-Order Single-Species Model with the Dispersal, 29th Chinese Control And Decision Conference (CCDC), 2017, pp. 7822–7826. https://doi.org/10.1109/CCDC.2017.7978613
  • S.M. Garba, A.B. Gumel, A.S. Hassan, and J.M.-S. Lubuma, Switching from exact scheme to nonstandard finite difference scheme for linear delay differential equation, Appl. Math. Comput. 258 (2015), pp. 388–403.
  • G. Gonzalez-Parra, A.J. Arenas, and B.M. Chen-Charpentier, Combination of nonstandard schemes and Richardson's extrapolation to improve the numerical solution of population models, Math. Comput. Model. 52 (2010), pp. 1030–1036.
  • M. Gupta, J.M. Slezak, F. Alalhareth, S. Roy, and H.V. Kojouharov, Second-order nonstandard explicit Euler method, AIP Conf. Proc. 2302 (2020), Article ID 110003.
  • M.T. Hoang, A novel second-order nonstandard finite difference method for solving one-dimensional autonomous dynamical systems, Commun. Nonlinear Sci. Numer. Simul. 114 (2022), Article ID 106654.
  • M.T. Hoang, Dynamically consistent nonstandard finite difference schemes for a virus-patch dynamic model, J. Appl. Math. Comput. 68 (2022), pp. 3397–3423.
  • M.T. Hoang, Global asymptotic stability of a general fractional-order single-species model, Bol. Soc. Mat. Mex. 28 (2022), Article number: 2. https://doi.org/10.1007/s40590-021-00396-2
  • M.T. Hoang, Reliable approximations for a hepatitis B virus model by nonstandard numerical schemes, Math. Comput. Simul. 193 (2022), pp. 32–56.
  • M.T. Hoang, A class of second-order and dynamically consistent nonstandard finite difference schemes for nonlinear Volterra's population growth model, Comput. Appl. Math. 42 (2023), Article number: 85. https://doi.org/10.1007/s40314-023-02230-z
  • M.T. Hoang, Dynamical analysis of a generalized hepatitis B epidemic model and its dynamically consistent discrete model, Math. Comput. Simul. 205 (2023), pp. 291–314.
  • Z. Jiang and W. Zhang, Bifurcation analysis in single-species population model with delay, Sci. China Math. 53 (2010), pp. 1475–1481.
  • D.C. Joyce, Survey of extrapolation processes in numerical analysis, SIAM Rev. 13 (1971), pp. 435–490.
  • H.K. Khalil, Nonlinear Systems, 3rd ed., Prentice Hall, 2002.
  • H.V. Kojouharov, S. Roy, M. Gupta, F. Alalhareth, and J.M. Slezak, A second-order modified nonstandard theta method for one-dimensional autonomous differential equations, Appl. Math. Lett.112 (2021), Article ID 106775.
  • M.C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science 197 (1977), pp. 287–289.
  • J. Martin-Vaquero, A.M. del Rey, A.H. Encinas, J.D.H. Guillen, A. Queiruga-Dios, and G.R. Sanchez, Higher-order nonstandard finite difference schemes for a MSEIR model for a malware propagation, J. Comput. Appl. Math. 317 (2017), pp. 146–156.
  • J. Martin-Vaquero, A. Queiruga-Dios, A.M. del Rey, A.H. Encinas, J.D.H. Guillen, and G.R. Sanchez, Variable step length algorithms with high-order extrapolated non-standard finite difference schemes for a SEIR model, J. Comput. Appl. Math. 330 (2018), pp. 848–854.
  • R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, 1993.
  • R.E. Mickens, Applications of Nonstandard Finite Difference Schemes, World Scientific, 2000.
  • R.E. Mickens, Nonstandard finite difference schemes for differential equations, J. Differ. Equ. Appl. 8 (2002), pp. 823–847.
  • R.E. Mickens, Advances in the Applications of Nonstandard Finite Difference Schemes, World Scientific, 2005.
  • R.E. Mickens, Dynamic consistency: A fundamental principle for constructing nonstandard finite difference schemes for differential equations, J. Differ. Equ. Appl. 11 (2005), pp. 645–653.
  • R.E. Mickens, Nonstandard Finite Difference Schemes: Methodology and Applications, World Scientific, 2020.
  • R.E. Mickens and A.B. Gumel, Construction and analysis of a non-standard finite difference scheme for the Burgers-Fisher equation, J. Sound. Vib. 257 (2002), pp. 791–797.
  • R.E. Mickens and I.H. Herron, Approximate rational solutions to the Thomas-Fermi equation based on dynamic consistency, Appl. Math. Lett. 116 (2021), Article ID 106994.
  • R.E. Mickens and T.M. Washington, NSFD discretizations of interacting population models satisfying conservation laws, Comput. Math. Appl. 66 (2013), pp. 2307–2316.
  • R.M. Nisbet and W.S.C. Gurney, Modelling Fluctuating Populations, John Wiley and Sons, New York, 1982.
  • S.A. Pasha, Y. Nawaz, and M.S. Arif, On the nonstandard finite difference method for reaction-diffusion models, Chaos Solit. Fractals 166 (2023), Article ID 112929.
  • K.C. Patidar, On the use of nonstandard finite difference methods, J. Differ. Equ. Appl. 11 (2005), pp. 735–758.
  • K.C. Patidar, Nonstandard finite difference methods: Recent trends and further developments, J. Differ. Equ. Appl. 22 (2016), pp. 817–849.
  • L.F. Richardson and J.A. Gaunt, The deferred approach to the limit, Philos. Trans. R. Soc. London226A (1927), pp. 299–361.
  • Z. Sun, J. Lv, and X. Zou, Dynamical analysis on two stochastic single-species models, Appl. Math. Lett. 99 (2020), Article ID 105982.
  • S. Treibert, H. Brunner, and M. Ehrhardt, A nonstandard finite difference scheme for the SVICDR model to predict COVID-19 dynamics, Math. Biosci. Eng. 19 (2022), pp. 1213–1238.
  • J.X. Velasco-Hernández, A model for Chagas disease involving transmission by vectors and blood transfusion, Theor. Popul. Biol. 46 (1994), pp. 1–31.
  • D.T. Wood, D.T. Dimitrov, and H.V. Kojouharov, A nonstandard finite difference method for n-dimensional productive-destructive systems, J. Differ. Equ. Appl. 21 (2015), pp. 240–254.
  • D.T. Wood and H.V. Kojouharov, A class of nonstandard numerical methods for autonomous dynamical systems, Appl. Math. Lett. 50 (2015), pp. 78–82.
  • D.T. Wood, H.V. Kojouharov, and D.T. Dimitrov, Universal approaches to approximate biological systems with nonstandard finite difference methods, Math. Comput. Simul. 133 (2017), pp. 337–350.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.