In nature, different species compete among themselves for common resources and favorable habitat. Therefore, it becomes really important to determine the key factors in maintaining the bio-diversity. Also, some competing species follow cyclic competition in real world where the competitive dominance is characterized by a cyclic ordering. In this paper, we study the formation of a wide variety of spatiotemporal patterns including stationary, periodic, quasi-periodic and chaotic population distributions for a diffusive Lotka–Volterra type three-species cyclic competition model with two different types of cyclic ordering. For both types of cyclic ordering, the temporal dynamics of the corresponding non-spatial system show the extinction of two species through global bifurcations such as homoclinic and heteroclinic bifurcations. For the spatial system, we show that the existence of Turing patterns is possible for a particular cyclic ordering, while it is not the case for the other cyclic ordering through both the analytical and numerical methods. Further, we illustrate an interesting scenario of short-range invasion as opposed to the usual invasion phenomenon over the entire habitat. Also, our study reveals that both the stationary and dynamic population distributions can coexist in different parts of a habitat. Finally, we extend the spatial system by incorporating nonlocal intra-specific competition terms for all the three competing species. Our study shows that the introduction of nonlocality in intra-specific competitions stabilizes the system dynamics by transforming a dynamic population distribution to stationary. Surprisingly, this nonlocality-induced stationary pattern formation leads to the extinction of one species and hence, gives rise to the loss of bio-diversity for intermediate ranges of nonlocality. However, the bio-diversity can be restored for sufficiently large extent of nonlocality.

在自然界中，不同的物种相互竞争共同的资源和有利的栖息地。因此，确定维持生物多样性的关键因素变得非常重要。此外，一些竞争物种在现实世界中遵循循环竞争，其中竞争优势以循环排序为特征。在本文中，我们研究了具有两种不同类型循环排序的扩散 Lotka-Volterra 型三物种循环竞争模型的各种时空模式的形成，包括平稳、周期性、准周期性和混沌种群分布。对于这两种类型的循环排序，相应的非空间系统的时间动态表明两个物种通过全局分叉如同宿分叉和异宿分叉而灭绝。对于空间系统，我们通过解析和数值方法证明了图灵模式的存在对于特定的循环排序是可能的，而对于其他循环排序则不是这种情况。此外，我们展示了一个有趣的短程入侵场景，而不是整个栖息地的常见入侵现象。此外，我们的研究表明，静态和动态人口分布可以在栖息地的不同部分共存。最后，我们通过为所有三个竞争物种合并非本地种内竞争术语来扩展空间系统。我们的研究表明，在种内竞争中引入非定域性通过将动态人口分布转换为静态来稳定系统动态。出奇，这种非定域性引起的静止模式形成导致一个物种的灭绝，从而导致非定域性中间范围的生物多样性丧失。然而，生物多样性可以在足够大的非定域性范围内恢复。