In this paper, we consider a diffusive predator-prey system with spatial memory and predator-taxis. Since in this system, the memory delay appears in the diffusion term, and the diffusion term is nonlinear, the classical normal form of Hopf bifurcation in the reaction-diffusion system with delay can't be applied to this system. Thus, in this paper, we first derive an algorithm for calculating the normal form of Hopf bifurcation in this system. Then in order to illustrate the effectiveness of our newly developed algorithm, we consider the diffusive Holling-Tanner model with spatial memory and predator-taxis. The stability and Hopf bifurcation analysis of this model are investigated, and the direction and stability of Hopf bifurcation periodic solution are also researched by using our newly developed algorithm for calculating the normal form of Hopf bifurcation. At last, we carry out some numerical simulations, two stable spatially inhomogeneous periodic solutions corresponding to the mode-1 and mode-2 Hopf bifurcations are found, which verifies our theoretical analysis results.
翻译：在本文中, 我们考虑的是带有空间内存和掠食税的悬浮掠食动物系统。 由于在这个系统中, 记忆延迟出现在扩散术语中, 扩散术语为非线性, 反应扩散系统中典型的Hopf 双壁化形式不能被延迟应用到这个系统中。 因此, 在本文中, 我们首先得出一个算法来计算这个系统中Hopf 双壁化的正常形式。 然后为了说明我们新开发的算法的有效性, 我们考虑的是带有空间内存和掠食税的 diffive Holling- Tanner 模型。 对这个模型的稳定性和Hopf 双壁化分析进行了调查, 并且利用我们新开发的算法来计算Hopf 双壁化的正常形式。 最后, 我们进行了一些数字模拟, 发现了两种与模式-1 和 模式-2 Hopf 双壁化相应的稳定空间不均匀周期解决方案, 从而证实了我们的理论分析结果 。
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