The Lyapunov functionals for delay Lotka-Volterra-type models

By means of Lyapunov functional, we have succeeded in establishing the global attractivity of the positive equilibrium of n-species Lotka-Volterra systems modeled by equations of the "pure-delay type" with both finite and infinite delays involved. As a corollary, we show that if the delay system is dissipative and the coefficient matrix is VL-stable, then the global attractivity of the unique positive equilibrium is maintained provided the delays are small. Estimates on the allowable sizes of delays are indicated. Applications to 2-species prey-predator models and n-species food chains and competition models are included.