Stability of the delay logistic equation of population dynamics

Abstract

The nonlinear logistic equation dy/dt=/spl epsiv/y(t) (1-/spl Sigma//sub k=0/ /sup n/b/sub k/y(t-/spl tau//sub k/), /spl epsiv/>0, b/sub k/, /spl tau//spl isin/(0;/spl infin/) (0/spl les/k/spl les/n) is discussed. The local stability of the nonzero stationary solution of this equation depends on the stability of linear equation dx/dt=-/spl Sigma//sub k=1/ /sup n/ a/sub k/x(t-/spl tau//sub k/), where a/sub k/=/spl epsiv/b/sub k///spl Sigma//sub j=0/ /sup n/b/sub j/ (0/spl les/k/spl les/n). It is shown that the condition /spl Sigma//sub k=1/ /sup n/a/sub k//spl tau//sub k/ 0, k=0,l, ..., n), then the stationary solution y/spl equiv/1//spl Sigma//sub k=0/ /sup n/b/sub k/ of logistic equation is stable with respect to small perturbations when the sequence (b/sub k/) is nonnegative and convex.