This paper focuses on the design and analysis of an efficient numerical method based on the novel implicit finite difference scheme for the solution of the dynamics of reaction-diffusion models. The work replaces the integer first order derivative in time with the Caputo fractional derivative operator. The dynamics of activator-inhibitor as encountered in chemistry, physics and engineering processes, and predator-prey models are two cases addresses in this study. In order to provide a good guidelines on the correct choice of parameters for the numerical simulation of full fractional reaction-diffusion system, its linear stability analysis is also examined. The resulting scheme is applied to solve cross-diffusion problem in two-dimensions. In the experimental results, a number of spatiotemporal and chaotic patterns that are related to Turing pattern are observed. It was discovered in the simulation experiments that the species predator-prey model distribute in almost same fashion, while that of the activator-inhibitor dynamics behaved differently regardless of the value of fractional order chosen.

本文着重设计和分析一种基于新型隐式有限差分格式的有效数值方法，用于求解反应扩散模型的动力学问题。该作品用Caputo分数导数运算符及时替换了整数一阶导数。在化学，物理和工程过程中以及在捕食者-猎物模型中遇到的活化剂-抑制剂动力学是本研究的两个案例。为了对全分数反应扩散系统的数值模拟提供正确选择参数的良好指导，还对其线性稳定性分析进行了检查。所得方案用于解决二维交叉扩散问题。在实验结果中 观察到许多与图灵模式有关的时空和混沌模式。在模拟实验中发现，物种捕食-被捕食模型的分布几乎相同，而无论选择何种分数阶，激活-抑制剂动力学的行为都不同。