In this paper, we are mainly concerned with the effect of nonlocal diffusion and dispersal spread on bifurcations of a general activator-inhibitor system in which the activator has a nonlocal dispersal. We find that spatially inhomogeneous patterns always exist if the dispersal rate of the activator is sufficiently small, while a larger dispersal spread and an increase of the activator diffusion inhibit the formation of spatial patterns. Compared with the "spatial averaging" nonlocal dispersal model, our model admits a larger parameter region supporting pattern formations, which is also true if compared with the local reaction-diffusion one when the dispersal spread is small. We also study the existence of nonconstant positive steady states through bifurcation theory and find that there could exist finite or infinite steady state bifurcation points of the inhibitor diffusion constant. As an example of our results, we study a water-biomass model with nonlocal dispersal of plants and show that the water and plant distributions could be inphase and antiphase.

中文翻译：

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具有非局部扩散的一般活化剂-抑制剂模型的分叉分析

在本文中，我们主要关注非局部扩散和分散扩散对其中活化剂具有非局部分散的一般活化剂-抑制剂系统的分叉的影响。我们发现，如果活化剂的分散速率足够小，则空间不均匀的图案将始终存在，而较大的分散扩散和活化剂扩散的增加​​会抑制空间图案的形成。与“空间平均”非局部扩散模型相比，我们的模型允许较大的参数区域来支持图案形成，如果与散布分布较小的局部反应扩散模型相比，这也是正确的。我们还通过分叉理论研究了非恒定正稳态的存在，发现抑制剂扩散常数可能存在有限或无限的稳态分叉点。作为结果的一个示例，我们研究了具有植物非局部分散性的水-生物量模型，并表明水和植物的分布可能是同相和反相的。