Global asymptotic behavior in chemostat-type competition models with delay

The asymptotic behavior of solutions of a chemostat-type model in which two species compete for a limiting nutrient supplied at a constant rate is considered. The model incorporates a general nutrient uptake function and two distributed delays. The first delay models the fact that the nutrient is partially recycled after the death of the biomass by bacterial decomposition and the second indicates that the growth of the species depends on the past concentration of the nutrient. Furthermore, it is assumed that there is interspecific competition between the two species as well as intraspecific competition within each species. Conditions for boundedness of solutions and existence of nonnegative equilibria are given. By constructing appropriate Liapunov-like functionals, some sufficient conditions for global attractivity of the positive equilibrium is obtained. The combined effects of the two different delays are studied. The main results of Freedman and Xu [H.I. Freedman, Y. Xu, Models of competition in the chemostat with instantaneous and delayed nutrient recycling, J. Math. Biol. 31 (1993) 513–527] and Ruan and He [S. Ruan, X.-Z. He, Global stability in chemostat-type competition models with nutrient recycling, SIAM J. Appl. Math. 58 (1) (1998) 170–192] are improved and extended.     

Oscillations and asymptotic behavior of first-order neutral delay differential equations

Consider the neutral delay differential equation [display math001] In this paper we are concerned with the asymptotic behavior and the oscillatory nature of solutions of Eq. (1).     

具有可变时滞的非线性非自治差分方程的线性化全局渐近稳定性及其应用

证明了在一定条件下,具有可变时滞的非线性非自治差分方程的全局渐近稳定性可由某种线性差分方程的渐近稳定性确定,给出了这类差分方程全局渐近稳定的充分条件.作为实例,获得了具有可变时滞的离散型非自治广义Log istic方程的全局吸收性判别准则.     

The asymptotic behavior of a class of nonlinear delay difference equations

The asymptotic behavior of difference equations of type 0, \end{equation*}">is studied, where and each are continuous real functions with decreasing and increasing. Results include sufficient conditions for permanence, oscillations and global attractivity.

     

On oscillatory and asymptotic behavior of fourth order nonlinear neutral delay dynamic equations with positive and negative coefficients

In this paper, oscillatory and asymptotic properties of solutions of nonlinear fourth order neutral dynamic equations of the form $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = 0(H)$ and $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = f(t),(NH)$ are studied on a time scale $\mathbb{T}$ under the assumption that $\int\limits_{t_0 }^\infty {\tfrac{t} {{r(t)}}\Delta t = \infty } $ and for various ranges of p(t). In addition, sufficient conditions are obtained for the existence of bounded positive solutions of the equation (NH) by using Krasnosel’skii’s fixed point theorem.     

Uniform asymptotic stability for perturbed neutral delay differential equations

One-dimensional perturbed neutral delay differential equations of the form (x(t)−P(t,x(t−τ)))′=f(t,x_t)+g(t,x_t) are considered assuming that f satisfies −v(t)M(φ)?f(t,φ)?v(t)M(−φ), where M(φ)=max{0,max_s∈[−r,0]φ(s)}. A typical result is the following: if ‖g(t,φ)‖?w(t)‖φ‖ and , then the zero solution is uniformly asymptotically stable providing that the zero solution of the corresponding equation without perturbation (x(t)−P(t,x(t−τ)))′=f(t,x_t) is uniformly asymptotically stable. Some known results associated with this equation are extended and improved.     

Global existence and asymptotic behavior of solutions for nonlinear parabolic equations on unbounded domains

We study the existence and the asymptotic behavior of positive solutions for the parabolic equation on D×(0,∞), where is a some unbounded domain in and V belongs to a new parabolic class J^∞ of singular potentials generalizing the well-known parabolic Kato class at infinity P^∞ introduced recently by Zhang. We also show that the choice of this class is essentially optimal.     

Global asymptotic behavior of positive solutions for exponential form difference equations with three parameters

In this paper, we study a class of second order difference equations with three paremeters. With positive initial values, the asymptotic behavior of positive solutions are investigated.     

非线性时滞差分议程的全局渐近稳定性

In this paper,a sufficient condition for the global asymptotic stability of the solutions of the following nonlinear delay difference equation is obtained,xn 1=xn xn-1xn-2 a/xmxm-1 xn-2 a,n=0,1…,where a∈(0,∞) and the initial values x-2,x-1,x0∈(0,∞).As a special case,a conjecture by Ladas is confirmed.     

Oscillation and asymptotic behavior of bounded solutions of functional differential equations with delay

Conditions on a(t), g(t), and f(t) have been found under which the bounded nonoscillatory solutions of the equation y⁽ⁿ⁾(t) ? a(t) y(g(t)) = f(t) approach zero. For the even order equation y⁽²ⁿ⁾(t) ? a(t) y(g(t)) = f(t) the delay is shown to be causing the oscillatory behavior.     

Global structure and asymptotic behavior of weak solutions to flood wave equations

The present paper concerns with the global structure and asymptotic behavior of the discontinuous solutions to flood wave equations. By solving a free boundary problem, we first obtain the global structure and large time behavior of the weak solutions containing two shock waves. For the Cauchy problem with a class of initial data, we use Glimm scheme to obtain a uniform BV estimate both with respect to time and the relaxation parameter. This yields the global existence of BV solution and convergence to the equilibrium equation as the relaxation parameter tends to 0.     

Numerical radius for the asymptotic stability of delay differential equations

In this note, we make use of the numerical radius to investigate the asymptotic stability of linear systems of delay differential equations. We present examples to illustrate our assumption is weaker than the classical ones.     

On dynamic iteration for delay differential equations

In this paper we study dynamic iteration techniques for systems of nonlinear delay differential equations. After pointing out a close connection to the truncated infinite embedding, as proposed by Feldstein, Iserles, and Levin, we give a proof of the superlinear convergence of the simple dynamic iteration scheme. Then we propose a more general scheme that in addition allows for a decoupling of the equations into disjoint subsystems, just like what we are used to from dynamic iteration schemes for ODEs. This scheme is also shown to converge superlinearly.