A boundedness and monotonicity preserving method for a generalized population model

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.

Keywords:

• Reaction–diffusion equation
• positivity
• boundedness
• monotonicity
• stability
• convergence

AMS Classifications:

• 65Q10
• 65M06
• 65Q30
• 35K55

The authors would like to express their appreciation to the editors and the anonymous referees for their many valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11701124] and the Natural Science Foundation of Shandong Province of China [grant number ZR2017PA006].