In this paper, we investigate the spatial dynamics of a class of spatial fractional predator-prey systems with time delay and Allee effect. Firstly, Hopf bifurcation and Turing bifurcation conditions are obtained by using stability theory and bifurcation theory. Then, the abundant dynamic behaviors of the system are demonstrated by numerical simulation. Finally, numerical results show that time delay, Allee effect, and fractional diffusion can affect the formation 3D Turing patterns such as spherical, tube, spherical and tube coexisting and the constitution of 3D spiral wave patterns.

中文翻译：

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具有时滞和阿利效应的分数捕食系统空间动力学

在本文中，我们研究了一类具有时间延迟和 Allee 效应的空间分数捕食者 - 猎物系统的空间动力学。首先，利用稳定性理论和分岔理论得到Hopf分岔和图灵分岔条件。然后，通过数值模拟证明了系统丰富的动态行为。最后，数值结果表明，时间延迟、阿利效应和分数扩散会影响球面、管状、球面和管状共存等3D图灵图案的形成和3D螺旋波图案的构成。