Many systems in chemistry, biology, finance, and social sciences present emerging features that are not easy to guess from the elementary interactions of their microscopic individual components. In the past, the macroscopic behavior of such systems was modeled by assuming that the collective dynamics of microscopic components can be effectively described collectively by equations acting on spatially continuous density distributions. It turns out that, to the contrary, taking into account the actual individual͞ discrete character of the microscopic components of these systems is crucial for explaining their macroscopic behavior. In fact, we find that in conditions in which the continuum approach would predict the extinction of all of the population (respectively the vanishing of the invested capital or the concentration of a chemical substance, etc.), the microscopic granularity insures the emergence of macroscopic localized subpopulations with collective adaptive properties that allow their survival and development. In particular it is found that in two dimensions ''life'' (the localized proliferating phase) always prevails.I n addition to physics, an increasing range of sciences (chemistry, biology, ecology, finance, urban and social planning) have moved in the last century to quantitative mathematical methods.Along with the obvious benefits, it turns out that the traditional differential equations approach has brought some fallacy into the study of such sciences.We present here a very simple generic model that contains proliferating (and dying) individuals, and we show that in reality it behaves very differently than its representation in terms of continuum density distributions. In conditions in which the continuum equations predict the population extinction, the individuals self-organize in spatio-temporally localized adaptive patches, which ensure their survival and development.This phenomenon admits multiple interpretations in various fields:If the individuals are interpreted as interacting molecules, the resulting chemical system emerges spatial patches of high density that evolve adaptively in a way similar with the first selfsustaining systems that might have anticipated living cells.If the individuals are the carriers of specific genotypes represented in the genetic space, the patches can be identified with species, which rather than becoming extinct, evolve between various genomes (locations in the genetic space) by abandoning regions of low viability in favor of more viable regions. This adaptive speciation behavior emerges despite the total randomness we assume for the individuals motions in the genetic space (mutations).Interpreted as financial traders, the individuals develop a ''herding'' behavior despite the fact that we do not introduce communication or interaction between them. The model leads to the flourishing of markets, which the continuum analysis would doom to extinction.All these phenomena have in common the emergence of large, macroscopic structures from apparently uniform background (...