What are the shapes of diversity with respect the shapes of the city? How those shapes can inform about the ecological fitness and guide the enhancement of a city? Can we detect sentinels for the ''wing of change''? Eco-climatically sensitive species carry information of envionmental change in their distribution, fitness, and preferential gradients of the habitat suitability that is largely linked to the hydrological stress. Conversely, environmental features outside of these species' niche convey information of potential extreme climate to mitigate. Here we propose a model, as a proof of concept and paradigm, to extract the multivariable functional networks and their attraction basins carrying the signature of compounding hydroclimatic pressure on species fitness. Specifically, we consider butterflies and their habitat suitability (HS) to infer maximum suitability gradients that are meaningful of potential species flows with the least resistance within an ecosystem and sentinels of climate stress. These flows are compared to the distribution of urban parks to identify their ecological attractiveness for species, observed and potential connectivity, and park potential to reduce hydroclimatic impacts. Results are shown for the city of Shenzhen that is one the most quickly developing and green cities per population size worldwide. Ecoflow networks are much more extended, scale-free, and clustering for low HS gradient areas populated by large reservoirs that act as sources of ecological corridors ''draining'' into urban parks. Parks have higher HS and gradients, and small-world connectivity that does not overlap with hydrological networks. We find diverging patterns of abundance and diversity increasing and decreasing with HS where the latter is largely determined by temperature and precipitation of the coldest quarter, and seasonality, which are critical fetaures of the hydrologic cycle that is worrisomely changing worldwide. Interestingly a U-shape pattern is found between abudance and diversity, similarly to the one in natural ecosystems. Additionally, both abundance and diversity are mildly associated with park area according to a power function, unrelated to longitude but related to the degree of urbanization/park centrality (due to isolation in rich parks) counterintuitively. Ecological patterns such as the Preston plot and abundance rank can reflect the stationarity of ecological communities (and their divergence) but cannot be informative about the community connectivity and fitness. The maximum HS gradients are meaningful of potential species flows and eco-change. Those, within their attraction basins, can inform about the collective function of ecological communities, like parks in cities. More importantly, flows as distributed stress-response of species are the blueprint for the design of ecotone networks along hydrologic corridors, regulating temperature and evapotranspiration, connecting communities and mitigating climate-driven eco-shifts locally and butterfly effects at larger scales.