Xingfu Zou

Global attractivity in delayed Hopfield neural network models

Two different approaches are employed to investigate the global attractivity of delayed Hopfield neural network models. Without assuming the monotonicity and differentiability of the activation functions, Liapunov functionals and functions (combined with the Razumikhin technique) are constructed and employed to establish sufficient conditions for global asymptotic stability independent of the delays. In the case of monotone and smooth activation functions, the theory of monotone dynamical systems is applied to obtain criteria for global attractivity of the delayed model. Such criteria depend on the magnitude of delays and show that self-inhibitory connections can contribute to the global convergence.

Traveling Wave Fronts of Reaction-Diffusion Systems with Delay

This paper deals with the existence of traveling wave front solutions of reaction-diffusion systems with delay. A monotone iteration scheme is established for the corresponding wave system. If the reaction term satisfies the so-called quasimonotonicity condition, it is shown that the iteration converges to a solution of the wave system, provided that the initial function for the iteration is chosen to be an upper solution and is from the profile set. For systems with certain nonquasimonotone reaction terms, a convergence result is also obtained by further restricting the initial functions of the iteration and using a non-standard ordering of the profile set. Applications are made to the delayed Fishery–KPP equation with a nonmonotone delayed reaction term and to the delayed system of the Belousov–Zhabotinskii reaction model.

Harmless delays in Cohen-Grossberg neural networks

Without assuming monotonicity and differentiability of the activation functions and any symmetry of interconnections, we establish some sufficient conditions for the globally asymptotic stability of a unique equilibrium for the Cohen–Grossberg neural network with multiple delays. Lyapunov functionals and functions combined with the Razumikhin technique are employed. The criteria are all independent of the magnitudes of the delays, and thus the delays under these conditions are harmless.

A reaction-diffusion model for a single species with age structure. I Travelling wavefronts on unbounded domains

In this paper, we derive the equation for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment. We show that if the mature death and diffusion rates are age independent, then the total mature population is governed by a reaction-diffusion equation with time delay and non-local effect. We also consider the existence, uniqueness and positivity of solution to the initial-value problem for this type of equation. Moreover, we establish the existence of a travelling–wave front for the special case when the birth function is the one which appears in the well-known Nicholsons blowflies equation and we consider the dependence of the minimal wave speed on the mobility of the immature population.

Traveling waves for the diffusive Nicholson's blowflies equation

We consider traveling wave front solutions for the diffusive Nicholsons blowflies equation on the real line. The existence of such solutions is proved using the technique developed by J. Wu and X. Zou (J. Dyn. Differ. Equations 13 (3) (2001)). Some numerical simulation using the iteration formula of Wu and Zou [7] is also provided.

Impact of delays in cell infection and virus production on HIV-1 dynamics

Analysed is a mathematical model for HIV-1 infection with two delays accounting, respectively, for (i) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and (ii) a virus production period for new virions to be produced within and released from the infected cells. For this model, the basic reproduction number is identified and its threshold property is discussed: the uninfected steady state is proved to be globally asymptotically stable if and unstable if . In the latter case, an infected steady state occurs and is proved to be locally asymptotically stable. The formula for shows that increasing either of the two delays will decrease . This may suggest a new direction for new drugs-drugs that can prolong the latent peri od and/or slow down the virus production process.

Stable periodic solutions in a discrete periodic logistic equation

Abstract In this paper, we consider a discrete logistic equation x ( n +1)= x ( n ) exp r ( n ) 1 − x ( n ) K ( n ) where {r(n)} and {K(n)} are positive ω-periodic sequences. Sufficient conditions are obtained for the existence of a positive and globally asymptotically stable ω-periodic solution. Counterexamples are given to illustrate that the conclusions in [1] are incorrect.

Stabilization role of inhibitory self-connections in a delayed neural network

Abstract In a delayed Hopfield neural network that is strongly connected with non-inhibitory interconnections, fast and inhibitory self-connections lead to global convergence to a unique equilibrium of the network. By applying monotone dynamical systems theory and an embedding technique, we prove that this conclusion remains true without the requirement of strong connectivity or non-inhibitory interconnections.

Dynamics of a non-autonomous ratio-dependent predator{prey system

We investigate a non-autonomous ratio-dependent predator{prey system, whose autonomous versions have been analysed by several authors. For the general non-autonomous case, we address such properties as positive invariance, permanence, non-persistence and the globally asymptotic stability for the system. For the periodic and almost-periodic cases, we obtain conditions for existence, uniqueness and stability of a positive periodic solution, and a positive almost-periodic solution, respectively.

Existence of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method

The monotone iteration method is employed to establish the existence of traveling wave fronts in delayed reaction-diffusion systems with monostable nonlinearities.