A global stability criterion for scalar functional differential equations

Abstract

We consider scalar delay differential equations $x^{\prime }(t)=-\delta x(t)+f(t,x_t)(\ast )$ with nonlinear f satisfying a sort of negative feedback condition combined with a boundedness condition. The well known Mackey-Glass type equations, equations satisfying the Yorke condition, equations with maxima are kept within our considerations. Here, we establish a criterion for the global asymptotical stability of a unique steady state to $(\ast )$. As an example, we study Nicholson's blowflies equation, where our computations support Smith's conjecture about the equivalence of global and local asymptotical stability in this population model.

28 pages, 2 figures. Final version, to appear in the SIAM Journal on Mathematical Analysis