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

Abstract

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 delaysis 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. However, to answer this in the affirmative is still a difficult problem in general. It is well known that, for the classical autonomous Lotka-Volterra prey-predator system,with bi, a {j > 0 (i,j = 1, 2), if it has a positive equilibrium N* = (AT*, AT^), then it must be globally asymptotically stable. In order to account for time-lag effects in this model, Cushing [4,5] considered the following prey-predator integrodifferential 'Present address:

Persistence and global stability in a delayed predator-prey system with Holling-type functional response

“…In this section, we provide conditions under which the positive equilibrium E* of system (1.1) is globally asymptotically stable. The method of proof is to construct a suitable Lyapunov functional for system (1.1) by borrowing the technique used in [14,15]. It is immediate that if the conditions for the global stability of the positive equilibrium E*(x*, x 2 ) are explicitly independent of x* and x 2 , then E*(x*, x 2 ) is in fact unique.…”

“…We note that time delays of one type or another have been incorporated into biological models by many researchers; we refer to the monographs of Cushing [5], Gopalsamy [10], Kuang [25] and MacDonald [28] for general delayed biological systems and to Beretta and Kuang [4], Gopalsamy [8,9], Hastings [13], He [14,15], May [30], Ruan [34], Wang and Ma [37], Xu and Yang [39], and the references cited therein for studies on delayed biological systems. In general, delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause a stable equilibrium to become unstable and cause the population to fluctuate.…”

Persistence and global stability in a delayed predator–prey system with Michaelis–Menten type functional response

“…The method of proof is to construct a suitable Lyapunov functional for system (1.1) by borrowing the technique used in [14,15] On substituting (4.1) into (1.1), we derive…”