TY - JOUR
T1 - Degenerate Lyapunov functionals of a well-known prey-predator model with discrete delays
AU - He, Xue Zhong
PY - 1999
Y1 - 1999
N2 - It is commonly believed that, as far as stabilities are concerned, 'small delays are negligible in some modelling processes'. However, to have an affirmative answer for this common belief is still an open problem for many nonlinear equations. In this paper, the classical Lotka-Volterra prey-predator equation with discrete delays ẋ(t) = x(t)[r1 - x(t - τ1) - ay(t - τ2)], ẏ(t) = y(t)[-r2 + bx(t - τ3)], is considered, and, by using degenerate Lyapunov functionals method, an affirmative answer to this open problem on both local and global stabilities of the prey-predator delay equations is given. It is shown that degenerate Lyapunov functional method is a powerful tool for studying the stability of such nonlinear delay systems. A detailed and explicit procedure of constructing such functionals is provided. Furthermore, some explicit estimates on the allowable sizes of the delays are obtained.
AB - It is commonly believed that, as far as stabilities are concerned, 'small delays are negligible in some modelling processes'. However, to have an affirmative answer for this common belief is still an open problem for many nonlinear equations. In this paper, the classical Lotka-Volterra prey-predator equation with discrete delays ẋ(t) = x(t)[r1 - x(t - τ1) - ay(t - τ2)], ẏ(t) = y(t)[-r2 + bx(t - τ3)], is considered, and, by using degenerate Lyapunov functionals method, an affirmative answer to this open problem on both local and global stabilities of the prey-predator delay equations is given. It is shown that degenerate Lyapunov functional method is a powerful tool for studying the stability of such nonlinear delay systems. A detailed and explicit procedure of constructing such functionals is provided. Furthermore, some explicit estimates on the allowable sizes of the delays are obtained.
UR - http://www.scopus.com/inward/record.url?scp=33747135058&partnerID=8YFLogxK
U2 - 10.1017/s0308210500013123
DO - 10.1017/s0308210500013123
M3 - Article
AN - SCOPUS:33747135058
SN - 0308-2105
VL - 129
SP - 755
EP - 771
JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics
JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics
IS - 4
ER -