Abstract This paper provides the essential mathematical basis for computational studies of space fractional reaction–diffusion systems, from biological and numerical analysis perspectives. We adopt linear stability analysis to derive conditions on the choice of parameters that lead to biologically meaningful equilibria. The stability analysis has a lot of implications for understanding the various spatiotemporal and chaotic behaviors of the species in the spatial domain. For the solution of the full reaction–diffusion system modelled by the fractional partial differential equations, we introduced the Fourier transform method to discretize in space and advance the resulting system of ordinary differential equation in time with the fourth-order exponential time differencing scheme. Results of numerical experiments are presented.

中文翻译：

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分数反应扩散系统建模捕食者-猎物相互作用的数学和计算研究

摘要 本文从生物学和数值分析的角度为空间分数反应扩散系统的计算研究提供了基本的数学基础。我们采用线性稳定性分析来推导出导致具有生物学意义的平衡的参数选择条件。稳定性分析对于理解物种在空间域中的各种时空和混沌行为有很多意义。对于由分数阶偏微分方程建模的全反应-扩散系统的解，我们引入了傅立叶变换方法在空间上离散化，并使用四阶指数时间差分格式在时间上推进常微分方程的结果系统。给出了数值实验的结果。