Stability of the delay logistic equation of population dynamics

The nonlinear logistic equation dy/dt=εy(t) (1-Σ_k=0 ⁿb_ky(t-τ_k), ε>0, b_k, τ∈(0;∞) (0≤k≤n) is discussed. The local stability of the nonzero stationary solution of this equation depends on the stability of linear equation dx/dt=-Σ_k=1 ⁿ a_kx(t-τ_k), where a_k=εb_k/Σ_j=0 ⁿb_j (0≤k≤n). It is shown that the condition Σ_k=1 ⁿa_kτ_k<π/2 is sufficient for zero solution stability of linear equation. We prove, that there is no restriction above on the value Σ_k=1 ⁿa_kτ_k which is necessary for the stability of linear equation. It disproves one of the propositions of K. Gopalsamy. It is shown that, if all the delays τ_k are multiples of one of them: τ_k=kτ(τ>0, k=0,l, ..., n), then the stationary solution y≡1/Σ_k=0 ⁿb_k of logistic equation is stable with respect to small perturbations when the sequence (b_k) is nonnegative and convex.