The evolutionary dynamics of cross-reaction–diffusion equations of predator–prey type are investigated in the sense of fractional operator. In the models, we replace the classical time and spatial derivatives with the Caputo–Fabrizio and Riesz fractional derivatives, respectively. The nature of the resulting problem (is nonlinear, nonlocal, and nonsingular) do not either admit a closed form solution, while in most cases the analytical solution is too involved to be useful. As a result, there is need to provide a reliable numerical scheme that can approximate these derivatives in time and space. Hence, we formulate an approximation scheme with second-order convergence rate for the time-Caputo–Fabrizio fractional operator of order 0 < α ≤ 1 and L₁ formula for the Riesz fractional derivative of order 1 < β ≤ 2 in space. As a case study, we consider two examples of strongly coupled cross fractional reaction–diffusion systems describing the interaction between two individual species that prey on the other one. We examine the system for stability analysis and establish the condition for the occurrence of Turing instability. The complexity of the dynamics of time–space cross fractional reaction–diffusion systems is theoretically studied and numerically in one and two dimensions for some instances of fractional orders.

中文翻译：

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具有 Caputo-Fabrizio 和 Riesz 算子的交叉反应-扩散系统的分析和数值模拟

在分数算子的意义上研究了捕食者-猎物型交叉反应扩散方程的演化动力学。在模型中，我们分别用 Caputo–Fabrizio 和 Riesz 分数阶导数替换了经典的时间和空间导数。由此产生的问题的性质（非线性、非局部和非奇异）不允许封闭形式的解决方案，而在大多数情况下，分析解决方案过于复杂而无用。因此，需要提供一种可靠的数值方案，可以在时间和空间上逼近这些导数。因此，我们为 0 < α  ≤ 1 和L ₁的时间-Caputo–Fabrizio 分数阶算子制定了一个具有二阶收敛速度的近似方案  空间中1 < β ≤ 2阶 Riesz 分数阶导数的公式 。作为案例研究，我们考虑了两个强耦合交叉分数反应-扩散系统的例子，描述了两个捕食另一个物种的物种之间的相互作用。我们检查系统进行稳定性分析，并建立图灵不稳定性发生的条件。时空交叉分数阶反应扩散系统动力学的复杂性在理论上和数值上在一维和二维上进行了一些分数阶实例的研究。