Stochastic growth models are very common in real life owing to their ability to capture the underlying mechanisms. This thesis considers three of such models. Each model can be seen as describing the evolution in time of a complex population of interacting ``particles": competing types of individuals in the first model, nodes in a dynamic network in the second, and species in an ecosystem in the third. A common feature of these models is that the population size grows in time and is represented by a transient (generalized) birth and death Markov process. This dissertation studies asymptotic structure of the ``particles landscape" which is represented in these three models by, respectively, type structure of the population, graph of interconnections, and the empirical distribution of species fitness.