We extend a spatially non-local cross-diffusion model of aggregation between multiple species with directed motion toward resource gradients to include many species and more general kinds of dispersal. We first consider diffusive instabilities, determining that for directed motion along fecundity gradients, the model permits the Turing instability leading to colony formation and persistence provided there are three or more interacting species. We also prove that such patterning is not possible in the model under the Turing mechanism for two species under directed motion along fecundity gradients, confirming earlier findings in the literature. However, when the directed motion is not along fecundity gradients, for instance, if foraging or migration is sub-optimal relative to fecundity gradients, we find that very different colony structures can emerge. This generalization also permits colony formation for two interacting species. In the advection-dominated case, aggregation patterns are more broad and global in nature, due to the inherent non-local nature of the advection which permits directed motion over greater distances, whereas in the diffusion-dominated case, more highly localized patterns and colonies develop, owing to the localized nature of random diffusion. We also consider the interplay between Turing patterning and spatial heterogeneity in resources. We find that for small spatial variations, there will be a combination of Turing patterns and patterning due to spatial forcing from the resources, whereas for large resource variations, spatial or spatiotemporal patterning can be modified greatly from what is predicted on homogeneous domains. For each of these emergent behaviors, we outline the theoretical mechanism leading to colony formation and then provide numerical simulations to illustrate the results. We also discuss implications this model has for studies of directed motion in different ecological settings.

中文翻译：

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人口动态的非局部交叉扩散模型 I：紧急时空模式

我们扩展了多个物种之间聚集的空间非局部交叉扩散模型，该模型具有朝向资源梯度的定向运动，以包括许多物种和更一般的扩散类型。我们首先考虑扩散不稳定性，确定对于沿着繁殖力梯度的定向运动，该模型允许图灵不稳定性导致集落形成和持久性，前提是存在三个或更多相互作用的物种。我们还证明，在图灵机制下的模型中，对于沿着繁殖力梯度的定向运动的两个物种，这种模式是不可能的，这证实了文献中的早期发现。然而，当定向运动不沿着繁殖力梯度时，例如，如果觅食或迁徙相对于繁殖力梯度不是最佳的，我们发现可以出现非常不同的群体结构。这种概括还允许两个相互作用的物种形成菌落。在以对流为主的情况下，聚集模式在本质上更广泛和全局，因为对流固有的非局部性质允许在更远的距离上进行定向运动，而在以扩散为主的情况下，聚集模式更加高度局部化发展，由于随机扩散的局部性质。我们还考虑了图灵模式和资源空间异质性之间的相互作用。我们发现，对于小的空间变化，由于来自资源的空间强迫，将存在图灵模式和模式的组合，而对于大的资源变化，空间或时空模式可以根据在同质域上的预测进行很大的修改。对于这些紧急行为中的每一个，我们概述了导致菌落形成的理论机制，然后提供数值模拟来说明结果。我们还讨论了该模型对不同生态环境中定向运动研究的影响。