Degenerate Lyapunov functionals of a well-known prey-predator model with discrete delays

It is commonly believed that, as far as stabilities are concerned, 'small d elays are negligible in some modelling processes'. However, to have an affi rmative answer for this common belief is still an open problem for many non linear equations. In this paper, the classical Lotka-Volterra prey-predator equation with discrete delays (x) over dot(t) = x(t)[r(1) - x(t - tau(1)) - ay(t - tau(2))], (y) over dot (t) = y(t)[-r(2) + bx(t - tau(3))], is considered, and, by using degenerate Lyapunov functionals method, an aff irmative answer to this open problem on both local and global stabilities o f the prey-predator delay equations is given. It is shown that degenerate L yapunov functional method is a powerful tool for studying the stability of such nonlinear delay systems. A detailed and explicit procedure of construc ting such functionals is provided. Furthermore, some explicit estimates on the allowable sizes of the delays are obtained.