Lotka-Volterra systems serve as a still-valuable paradigm for modeling the population dynamics of many interacting species. Motivated by numerics and hints on multi-stability in the biological literature, W. Hong and I study competitive exclusion principles for stable communities of species, and existence criteria for very many stable states. Some results relate closely to evolutionary game theory. Hong also uses Lojasiewicz inequalities to study long-time convergence for a related selection-mutation PDE and a concentration-dispersion model whose fixed points also describe nonlinear waves.