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Nestedness is a property of bipartite complex networks that has been shown to characterize the peculiar structure of biological and economical networks. In a nested network, a node of low degree has its neighborhood included in the neighborhood of nodes of higher degree. Emergence of nestedness is commonly due to two different schemes: i) mutualistic behavior of nodes, where nodes of each class have an advantage in associating with each other, such as plant pollination or seed dispersal networks; ii) geographic distribution of species, captured in a so-called biogeographic network where species represent one class and geographical areas the other one. Nestedness has useful applications on real-world networks such as node ranking and link prediction. Motivated by analogies with biological networks, we study the nestedness property of the public Internet peering ecosystem, an important part of the Internet where autonomous systems (ASes) exchange traffic at Internet eXchange Points (IXPs). We propose two representations of this ecosystem using a bipartite graph derived from PeeringDB data. The first graph captures the AS [is member of] IXP relationship which is reminiscent of the mutualistic networks. The second graph groups IXPs into countries, and we define the AS [is present at] country relationship to mimic a biogeographic network. We statistically confirm the nestedness property of both graphs, which has never been observed before in Internet topology data. From this unique observation, we show that we can use node metrics to extract new key ASes and make efficient prediction of newly created links over a two-year period.