A boundedness and monotonicity preserving method for a generalized population model. Issue 9 (2nd October 2020)

• Record Type: Journal Article
• Authors: Qin, W. D.; Ma, Qiang; Man, Z. Y.; Ding, X. H.
• Is Part Of: Journal of difference equations and applications. Volume 26:Issue 9/10(2020)
• Page Start: 1347
• Page End: 1368
• Publication Date: 2020-10-02
• Subjects: Reaction–diffusion equation -- positivity -- boundedness -- monotonicity -- stability -- convergence; 65Q10 -- 65M06 -- 65Q30 -- 35K55; Difference equations -- Periodicals; 515.625
• DOI: 10.1080/10236198.2020.1832088
• Languages: English
• ISSNs: 1023-6198

Abstract

Abstract : In this work, a nonstandard finite difference (NSFD) method is proposed to approximate the solutions of a nonlinear reaction–diffusion equation which appears in population dynamics. It is well known that the model under study has some travelling-wave solutions, which are positive, bounded and monotone in both space and time. First, a robust NSFD method is presented for the diffusion-free case of original equation. Then, combined with the NSFD method for the diffusion-free equation, an NSFD method is constructed for the full reaction–diffusion equation. It is shown that, under certain conditions on the denominator function of the time-step size, the proposed method is capable of preserving the positivity, boundedness and the spatial and temporal monotonicity of these travelling-wave solutions. Moreover, the nonlinear stability and convergence of this method are also analysed. Finally, some numerical simulations are provided to verify the validity of our analytical results.