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.

中文翻译：

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一种广义种群模型的有界和单调性保持方法

在这项工作中，提出了一种非标准有限差分 (NSFD) 方法来逼近出现在种群动力学中的非线性反应扩散方程的解。众所周知，所研究的模型有一些行波解，它们在空间和时间上都是正的、有界的和单调的。首先，针对原始方程的无扩散情况，提出了一种鲁棒的NSFD方法。然后，结合无扩散方程的NSFD方法，构建了全反应-扩散方程的NSFD方法。结果表明，在时间步长分母函数的一定条件下，所提出的方法能够保持这些行波解的正性、有界性和时空单调性。而且，还分析了该方法的非线性稳定性和收敛性。最后，提供了一些数值模拟来验证我们的分析结果的有效性。