Global Attractivity and Extinction for Lotka-Volterra systems with infinite delay and feedback controls

摘要

The paper deals with a multiple species Lotka-Volterra model with infinitedistributed delays and feedback controls, for which we assume a weak form ofdiagonal dominance of the instantaneous negative intra-specific terms over theinfinite delay effect in both the population variables and controls. Generalsufficient conditions for the existence and attractivity of a saturatedequilibrium are established. When the saturated equilibrium is on the boundaryof $\R^n_+$, sharper criteria for the extinction of all or part of thepopulations are given. While the literature usually treats the case ofcompetitive systems only, here no restrictions on the signs of the intra- andinter-specific delayed terms are imposed. Moreover, our technique does notrequire the construction of Lyapunov functionals.