Nonstandard finite difference schemes for a general predator–prey system

Dang, Quang A; Hoang, Manh Tuan

doi:10.1016/j.jocs.2019.07.002

Cited by 35 publications

(20 citation statements)

References 36 publications

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“…The similar comments can be found in previous works (see, for example, [1, 4, 13-16, 22, 27, 33-36, 40, 41]). The numerical approximations generated by NSFD scheme (10) in Example 3 with t 2 ½0; 100 and h ¼ 2 From Fig. 5, we see that NSFD scheme (10) exactly preserves the essential properties of the continuous model.…”

Section: Numerical Simulationsmentioning

confidence: 92%

“…(ii) Consider NSFD scheme (10) and suppose that R 0 [ 1. Let /ðhÞ be a function satisfying where the matrix A is given by (16).…”

Section: Theoremmentioning

confidence: 99%

“…Fig. 1 The solutions of model (1) generated by NSFD scheme (10) In this case, R 0 ¼ 0:7529\1 and hence, the equilibrium point E 0 ¼ ð0:8; 0; 2:4Þ is globally asymptotically stable.…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…Appendix 1: The basic reproduction number of NFSD scheme (10) We compute the basic reproduction number for NSFD scheme (10) by the nextgeneration matrix approach proposed in [3].…”

mentioning

confidence: 99%

“…Note that the first and second equations of system (10) do not include the component R. Hence, it is sufficient to consider the following sub-system: Appendix 2: An extension of the Lyapunov stability theorem for discrete-time systems…”

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confidence: 99%

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On the global asymptotic stability of a hepatitis B epidemic model and its solutions by nonstandard numerical schemes

Hoang

Egbelowo

2020

Bol. Soc. Mat. Mex.

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Very recently, a hepatitis B epidemic model with saturated incidence rate has been proposed and analyzed in Khan et al. (Chaos Solitons Fractals 124:1-9, 2019). The local asymptotic stability of the disease endemic equilibrium (DEE) point of model has been established theoretically but its global asymptotic stability has not been studied. In this paper, we present a mathematically rigorous analysis for the global asymptotic stability of the DEE point of the model. More precisely, we prove that if the DEE point exists, then it is not only locally asymptotically stable but also globally asymptotically stable. Furthermore, we present an alternative proof for the global stability of the disease-free equilibrium point. The main result is that we obtain the complete global stability of the hepatitis B virus model. Besides, we construct nonstandard finite difference (NSFD) schemes preserving the essential qualitative properties of the continuous model. These properties include the positivity of solutions and the stability of the model. Dynamical properties of the proposed schemes are rigorously investigated by mathematical analyses and numerical simulations. Finally, numerical simulations are performed to confirm the validity of the theoretical results as well as the advantages of the NSFD schemes. Numerical simulations also indicate that the NSFD schemes are appropriate and effective to solve the continuous model. Meanwhile, the standard finite difference schemes such as the Euler scheme, the classical fourth-order Runge-Kutta scheme cannot preserve the essential properties of the continuous model. Consequently, they can generate numerical approximations which are completely different from the solutions of the model.

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Section: Numerical Simulationsmentioning

confidence: 92%

“…(ii) Consider NSFD scheme (10) and suppose that R 0 [ 1. Let /ðhÞ be a function satisfying where the matrix A is given by (16).…”

Section: Theoremmentioning

confidence: 99%

“…Fig. 1 The solutions of model (1) generated by NSFD scheme (10) In this case, R 0 ¼ 0:7529\1 and hence, the equilibrium point E 0 ¼ ð0:8; 0; 2:4Þ is globally asymptotically stable.…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…Appendix 1: The basic reproduction number of NFSD scheme (10) We compute the basic reproduction number for NSFD scheme (10) by the nextgeneration matrix approach proposed in [3].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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