Boundedness and asymptotic stability for delayed equations of logistic type

Abstract

For a scalar Lotka–Volterra-type delay equation ẋ(t) = b(t)x(t)[1 − L(xt)], where L: C([−r, 0];R) → R is a bounded linear operator and b a positive continuous function, sufficient conditions are established for the boundedness of positive solutions and for the global stability of the positive equilibrium, when it exists. Special attention is given to the global behaviour of solutions for the case of L a positive linear operator. The approach used for this situation is applied to address the global asymptotic stability of delayed logistic models in the more general form ẋ(t) = b(t)x(t)[a(t) − L(t, xt)], with L(t, ·) being linear and positive.