Abstract
Two type of mathematical population models will be studied in this dissertation. One is the three-level food chain, by adding a third trophic level to a predator-prey system. The other one is competing predators system, competition for a single prey by two competitors. LaSalle's extension theorem of Lyapunov stability theory is the main tool. The mathematical results are intended to determine the eventual behavior of solutions of the relevant differential equations. We extend the Lyapunov functions, construted by A. Ardito and P. Ricciardi for predator-prey systems, to the three-level food chain and competing predators systems. Sufficient conditions are obtained for the global stability of equilibrium which respresents the extinction of top predator in three-level food chains. Moreover, by using the extended Lyapunov functions, we give a complete description of the globally asymptotic behaviors of the surviving predator and its prey in the competing predators systems.