A boundedness and monotonicity preserving method for a generalized population model

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.