Mathematical modeling and mechanisms of pattern formation in ecological systems: a review

How populations distribute in both space and time is one of the key issues in ecological systems, which can characterize the relationship between populations, space–time structure and evolution law. Consequently, pattern dynamics in ecosystems has been widely investigated including their causes and ecological functions. In order to systematically understand the interactions in ecosystems, we summarize the related results in pattern formation of ecological systems. Based on mathematical modeling and analysis, we show the mechanisms of different patterns including feedback, scale-dependent, phase separation, nonlocal effects, time delay and spatial heterogeneity. This work offers assistance for better understanding the complexity of ecosystems and provides new insights for self-organizations evolution and ecosystem protection. We hope that our results may be applied in other related fields such as epidemiology, medical science, atmospheric science and so on.