Mackey-Glass type delay differential equations near the boundary of absolute stability

For equations $x'(t) = -x(t) + \zeta f(x(t-h)), x \in \R, f'(0)= -1, \zeta > 0,$ with $C^3$-nonlinearity $f$ which has negative Schwarzian derivative and satisfies $xf(x) < 0$ for $x\not=0$, we prove convergence of all solutions to zero when both $\zeta -1 >0$ and $h(\zeta-1)^{1/8}$ are less than some constant (independent on $h,\zeta$). This result gives additional insight to the conjecture about the equivalence between local and global asymptotical stabilities in the Mackey-Glass type delay differential equations.Comment: 16 pages, 1 figure, accepted for publication in the Journal of Mathematical Analysis and Application

Global stability in a regulated logistic growth model

Trofimchuk, S. Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile.We investigate global stability of the regulated logistic growth model (RLC) n'(t) = rn(t)(1-n(t-h)/K-cu(t)), u'(t) = -au(t)+bn(t-h). It was proposed by Gopalsamy and Weng [1, 2] and studied recently in [4, 5, 6, 9]. Compared with the previous results, our stability condition is of different kind and has the asymptotical form. Namely, we prove that for the fixed parameters K and mu = bcK/a (which determine the levels of steady states in the delayed logistic equation n'(t) rn(t)(1 - n(t - h)/K) and in RLG) and for every hr < root 2 the regulated logistic growth model is globally stable if we take the dissipation parameter a sufficiently large. On the other hand, studying the local stability of the positive steady state, we observe the improvement of stability for the small values of a: in this case, the inequality rh < pi(1 + mu)/2 guaranties such a stabilit

Yorke and Wright 3/2-stability theorems from a unified point of view

We consider a family of scalar delay differential equations $x'(t)=f(t,x_t)$, with a nonlinearity $f$ satisfying a negative feedback condition combined with a boundedness condition. We present a global stability criterion for this family, which in particular unifies the celebrated 3/2-conditions given for the Yorke and the Wright type equations. We illustrate our results with some applications.Comment: 10 pages, accepted for publication in the Expanded Volume of DCDS, devoted to the fourth international conference on Dynamical Systems and Differential Equations, held at UNC at Wilmington, May 2002. Minor changes from the previous versio

Pushed traveling fronts in monostable equations with monotone delayed reaction

We study the existence and uniqueness of wavefronts to the scalar reaction-diffusion equations $u_{t}(t,x) = \Delta u(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone delayed reaction term $g: \R_+ \to \R_+$ and $h >0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x) \leq g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a similar result for $h>0$. We prove the existence of the minimal speed of propagation, the uniqueness of wavefronts (up to a translation) and describe their asymptotics at $-\infty$. We also present a new uniqueness result for a class of nonlocal lattice equations.Comment: 17 pages, submitte

ON THE MINIMAL SPEED OF FRONT PROPAGATION IN A MODEL OF THE BELOUSOV-ZHABOTINSKY REACTION

Univ Talca, Inst Matemat & Fis, Talca, Chile. Trofimchuk, S (Trofimchuk, Sergei)In this paper, we answer the question about the existence of the minimal speed of front propagation in a delayed version of the Murray model of the Belousov-Zhabotinsky (BZ) chemical reaction. It is assumed that the key parameter r of this model satisfies 0 < r <= 1 that makes it formally monostable. By proving that the set of all admissible speeds of propagation has the form [c(*), +infinity), we show here that the BZ system with r is an element of (0, 1] is actually of the monostable type (in general, c(*) is not linearly determined). We also establish the monotonicity of wavefronts and present the principal terms of their asymptotic expansions at infinity (in the critical case r = 1 inclusive)

Global stability in discrete population models with delayed-density dependence

Trofimchuk, S.Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, ChileWe address the global stability issue for some discrete population models with delayed-density dependence. Applying a new approach based on the concept of the generalized Yorke conditions, we establish several criteria for the convergence of all solutions to the unique positive steady state. Our results support the conjecture stated by Levin and May in 1976 affirming that the local asymptotic stability of the equilibrium of some delay difference equations (including Ricker’s and Pielou’s equations) implies its global stability. We also discuss the robustness of the obtained results with respect to perturbations of the model