Extinction of multiple populations and a team of Die-out Lyapunov functions

The extinction of species is a major problem of concern with a large literature. We investigate a differential equations model for population interactions with the goal of determining when several species (i.e., coordinates of a bounded solution) must die out or ``go extinct'' and must do so exponentially fast. Typically each coordinate represents the population density of a different species. For our main tool, we create what we call ``die-out'' Lyapunov functions. A given system may have several or many such functions, each of which is a function of a different set of coordinates. That die-out function implies that one of the species in its subset must die out exponentially fast -- for almost every choice of coefficients of the system. We create a ``team'' of die-out functions that work together to show that $k$ species must die, where $k$ is determined separately. Secondly, we present a ``trophic'' condition for generalized Lotka-Volterra systems that guarantees that there is a trapping region that is globally attracting. That implies that all solutions are bounded.