A diffusive predator-prey model with nonlocal prey competition and the homogeneous Neumann boundary conditions is considered, to explore the effects of nonlocal reaction term. Firstly, conditions of the occurrence of Hopf, Turing, Turing-Turing and double zero bifurcations, are established. Then, several concise formulas of computing normal form at a double zero singularity for partial functional differential equations, are provided. Next, via analyzing normal form derived by utilizing these formulas, we find that diffusive predator-prey system admits interesting spatiotemporal dynamics near the double zero singularity, like tristable phenomenon that a stable spatially inhomogeneous periodic solution with the shape of $ \cos\omega_0 t\cos\frac{x}{l}- $like which is unstable in model without nonlocal competition and also greatly different from these with the shape of $ \cos\omega_0 t+\cos\frac{x}{l}- $like resulting from Turing-Hopf bifurcation, coexists with a pair of spatially inhomogeneous steady states with the shape of $ \cos\frac{x}{l}- $like. At last, numerical simulations are shown to support theory analysis. These investigations indicate that nonlocal reaction term could stabilize spatially inhomogeneous periodic solutions with the shape of $ \cos\omega_0 t\cos\frac{kx}{l}- $like for reaction-diffusion systems subject to the homogeneous Neumann boundary conditions.

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具有非局部猎物竞争的扩散捕食者—食饵模型的双零奇异性和时空分布

考虑具有非局部猎物竞争和齐次诺伊曼边界条件的扩散捕食者—食饵模型，以探讨非局部反应项的影响。首先，建立了Hopf，Turing，Turing-Turing和双零分叉的发生条件。然后，针对部分泛函微分方程，提供了几个在双零奇点计算范式的简洁公式。接下来，通过分析利用这些公式得出的正态形式，我们发现扩散的捕食者－食饵系统在双零奇点附近接受了有趣的时空动力学，例如三稳态现象，即形状稳定的空间非均匀周期解，其形状为\\ cos \ omega_0 t \ cos \ frac {x} {l}-$ like不稳定在没有非局部竞争的模型中，并且也与这些模型有很大的不同，例如$ \ cos \ omega_0 t + \ cos \ frac {x} {l}-由Turing-Hopf分叉产生的$，与一对空间非均匀稳态共存形状为$ \ cos \ frac {x} {l}-$ like。最后，显示了数值模拟以支持理论分析。这些研究表明，对于局部均质Neumann边界条件下的反应扩散系统，非局部反应项可以稳定空间非均匀周期解，形状为\\ cos \ omega_0 t \ cos \ frac {kx} {l}-$。