Abstract
In this paper, we construct nonstandard finite difference (NSFD) schemes that preserve essential qualitative properties of a Logistics model with feedback control using the Mickens’ methodology. These properties of the model include positivity of solutions and stability of equilibrium points. Dynamical properties of the proposed NSFD schemes are rigorously investigated. The results show that the constructed nonstandard schemes preserve the essential properties of the continuous model for all finite step sizes. Consequently, we obtain NSFD schemes which are dynamically consistent with the continuous model. Especially, the constructed NSFD schemes become exact finite difference ones in some special cases. Besides, some numerical simulations are performed to validate the theoretical results as well as the advantages of the proposed NSFD schemes. The numerical simulations also indicate that the NSFD schemes are effective and appropriate to solve the continuous model. Meanwhile, the standard finite difference schemes such as the Euler scheme, the second order and fourth order Runge–Kutta schemes can generate numerical solutions that are completely different from the solutions of the continuous model and therefore, they fail to correctly reflect the correct behavior of the continuous model.
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Adekanye, O., Washington, T.: Nonstandard finite difference scheme for a tacoma narrows bridge model. Appl. Math. Model. (2018). https://doi.org/10.1016/j.apm.2018.05.027
Allen, J.L.S.: An Introduction to Mathematical Biology. Pearson, London (2006)
Anguelov, R., Lubuma, J.M.-S.: Nonstandard finite difference method by nonlocal approximations. Math. Comput. Simul. 61, 465–475 (2003)
Anguelov, R., Dumont, Y., Lubuma, J.M.-S., Shillor, M.: Dynamically consistent nonstandard finite difference schemes for epidemiological models. J. Comput. Appl. Math. 255, 161–182 (2014)
Anguelov, R., Kama, P., Lubuma, J.M.-S.: On non-standard finite difference models of reaction–diffusion equations. J. Comput. Appl. Math. 175, 11–29 (2005)
Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential–Algebraic Equations. Society for Industrial and Applied Mathematics Philadelphia, Philadelphia (1998)
Chapwanya, M., Lubuma, J.M.-S., Mickens, R.E.: Positivity–preserving nonstandard finite difference schemes for cross–diffusion equations in biosciences. Comput. Math. Appl. 68, 1071–1082 (2014)
Chapwanya, M., Lubuma, J.M.-S., Mickens, R.E.: Nonstandard finite difference schemes for Michaelis–Menten type reaction–diffusion equations. Numer. Methods Partial Differ. Equ. 29(1), 337–360 (2013)
Chapwanya, M., Lubuma, J.M.-S., Mickens, R.E.: From enzyme kinetics to epidemiological models with Michaelis-Menton contact rate: design of nonstandard finite difference schemes. Comput. Math. Appl. 64, 201–213 (2012)
Chapwanya, M., Jejeniwa, O.A., Appadu, A.R., Lubuma, J.M.-S.: An explicit nonstandard finite difference scheme for the FitzHugh–Nagumo equations. Int. J. Comput. Math. (2019). https://doi.org/10.1080/00207160.2018.1546849
Chen, L., Sun, J.: Global stability of an SI epidemic model with feedback controls. Appl. Math. Lett. 28, 53–55 (2014)
Chen, L., Chen, F.: Global stability of a Leslie–Gower predator–prey model with feedback controls. Appl. Math. Lett. 22, 1330–1334 (2009)
Dang, Q.A., Hoang, M.T., Trejos, D.Y., Valverde, J.C.: Feedback control variables to restrain the Babesiosis disease. Math. Methods Appl. Sci. (2019). https://doi.org/10.1002/mma.5877
Dang, Q.A., Hoang, M.T.: Dynamically consistent discrete metapopulation model. J. Differ. Equ. Appl. 22, 1325–1329 (2016)
Dang, Q.A., Hoang, M.T.: Lyapunov direct method for investigating stability of nonstandard finite difference schemes for metapopulation models. J. Differ. Equ. Appl. 24, 32–47 (2018)
Dang, Q.A., Hoang, M.T.: Complete global stability of a metapopulation model and its dynamically consistent discrete models. Qual. Theory Dyn. Syst. 18, 461–475 (2019)
Dang, Q.A., Hoang, M.T.: Exact finite difference schemes for three-dimensional linear systems with constant coefficients. Vietnam J. Math. 46, 471–492 (2018)
Dang, Q.A., Hoang, M.T.: Nonstandard finite difference schemes for a general predator–prey system. J. Comput. Sci. 36, 101015 (2019)
Dang, Q.A., Hoang, M.T., Dang, Q.L.: Nonstandard finite difference schemes for solving a modified epidemiological model for computer viruses. J. Comput. Sci. Cybern. 32(2), 171–185 (2018). https://doi.org/10.15625/1813-9663/32/2/13078
Dang, Q.A., Hoang, M.T.: Positive and elementary stable explicit nonstandard Runge–Kutta methods for a class of autonomous dynamical systems. Int. J. Comput. Math. (2019). https://doi.org/10.1080/00207160.2019.1677895
Dimitrov, D.T., Kojouharov, H.V.: Stability–preserving finite–difference methods for general multi-dimensional autonomous dynamical systems. Int. J. Numer. Anal. Model. 4(2), 280–290 (2007)
Dimitrov, D.T., Kojouharov, H.V.: Nonstandard finite difference schemes for general two-dimensional autonomous dynamical systems. Appl. Math. Lett. 18, 769–774 (2005)
Dimitrov, D.T., Kojouharov, H.V.: Positive and elementary stable nonstandard numerical methods with applications to predator–prey models. J. Comput. Appl. Math. 189, 98–108 (2006)
Dimitrov, D.T., Kojouharov, H.V.: Nonstandard finite–difference methods for predator–prey models with general functional response. Math. Comput. Simul. 78, 1–11 (2008)
Egbelowo, O., Harley, C., Jacobs, B.: Nonstandard finite difference method applied to a linear pharmacokinetics model. Bioengineering 4, 40 (2017)
Egbelowo, O.: Nonlinear elimination of drugs in one-compartment pharmacokinetic models: nonstandard finite difference approach for various routes of administration. Math. Comput. Appl. 23(2), 27 (2018)
Egbelowo, O.F.: Nonstandard finite difference approach for solving 3-compartment pharmacokinetic models. Int. J. Numer. Methods Biomed. Eng. 34, e3114 (2018). https://doi.org/10.1002/cnm.3114
Elaydi, S.: An Introduction to Difference Equations. Springer, Berlin (2005)
Fan, Y.H., Wang, L.L.: Global asymptotical stability of a Logistic model with feedback control. Nonlinear Anal. Real World Appl. 11, 2686–2697 (2010)
Gopalsamy, K., Weng, P.X.: Feedback regulation of Logistic growth. Int. J. Math. Sci. 16, l77–l92 (1993)
Hoang, M.T., Nagy, A.M.: Uniform asymptotic stability of a Logistic model with feedback control of fractional order and nonstandard finite difference schemes. Chaos Solitons Fractals 123, 24–34 (2019)
Hoang, M.T., Egbelowo, O.F.: Nonstandard finite difference schemes for solving an SIS epidemic model with standard incidence. Rend. Circolo Mat. Palermo Ser. (2019). https://doi.org/10.1007/s12215-019-00436-x
Korpusik, A.: A nonstandard finite difference scheme for a basic model of cellular immune response to viral infection. Commun. Nonlinear Sci. Numer. Simul. 43, 369–384 (2017)
Li, H.L., Zhang, L., Teng, Z., Jiang, Y.L., Muhammadhaji, A.: Global stability of an SI epidemic model with feedback controls in a patchy environment. Appl. Math. Comput. 321, 372–384 (2018)
Lin, Q.: Stability analysis of a single species logistic model with Allee effect and feedback control. Adv. Differ. Equ. 2018, 190 (2018). https://doi.org/10.1186/s13662-018-1647-2
Liao, L.S.: Feedback regulation of a Logistic growth with variable coefficients. J. Math. Anal. Appl. 259(2), 489–500 (2001)
Mickens, R.E.: Applications of Nonstandard Finite Difference Schemes. World Scientific, Singapore (2000)
Mickens, R.E.: Nonstandard Finite Difference Models of Differential Equations. World Scientific, Singapore (1993)
Mickens, R.E.: Nonstandard finite difference schemes for differential equations. J. Differ. Equ. Appl. 8, 823–847 (2012)
Mickens, R.E.: Advances in the Applications of Nonstandard Finite Difference Schemes. World Scientific, Singapore (2006)
Mickens, R.E.: A nonstandard finite–difference scheme for the Lotka–Volterra system. Appl. Numer. Math. 45, 309–314 (2003)
Patidar, K.C.: Nonstandard finite difference methods: recent trends and further developments. J. Differ. Equ. Appl. 22, 817–849 (2016)
Roeger, L.-I.W.: Dynamically Consistent Discrete-Time Lotka–Volterra Competition Models. Discrete and Continuous Dynamical Systems. In: Proceedings of the 7th AIMS International Conference, Arlington, Supplement, pp. 650-658 (2009)
Wood, D.T., Kojouharov, H.V.: A class of nonstandard numerical methods for autonomous dynamical systems. Appl. Math. Lett. 50, 78–82 (2016)
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The authors thank the anonymous referees for useful comments that led to a great improvement of the paper.
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Hoang, M.T., Egbelowo, O.F. Numerical dynamics of nonstandard finite difference schemes for a Logistics model with feedback control. Ann Univ Ferrara 66, 51–65 (2020). https://doi.org/10.1007/s11565-020-00338-2
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DOI: https://doi.org/10.1007/s11565-020-00338-2
Keywords
- Logistics model
- Positivity
- Stability
- Nonstandard finite difference schemes
- Exact finite difference schemes