Pingzhou Liu

Discrete Competitive and Cooperative Models of Lotka–Volterra Type

The dynamics of discrete Lotka–Volterra system of two species is investigated. It is shown that the proposed discrete models for competitive and cooperative systems possess “dynamical consistency” with their continuous counterparts.

Periodic solutions of difference equations

We give an overview of results on the existence of periodic solutions of difference equations that have been obtained in the last two decades. The survey covers both ordinary and Volterra difference systems. Some extensions and generalizations of known result are also presented.

Trend of Income-related Inequality of Child Oral Health in Australia

It is important that we monitor socio-economic inequality in health. Inequality in child oral health has been expected to widen because of widening socio-economic inequality. This study aimed to evaluate trends in income-related inequality in caries experience of Australian children. Cross-sectional studies in 1992/93 and 2002/03 collected data on deciduous caries experience of 5- to 10-year-olds and permanent caries experience of 6- to 12-year-olds. Household composition and income was used to calculate quartiles of equivalized income. Slope Index of Inequality (SII), Concentration Index (CI), and regression-based rate ratios were used to quantify income-related inequality and to evaluate trends. Income-related inequality in caries experience was evident regardless of time and dentition. The three indicators of inequality indicate a significant increase in income-related inequality in child deciduous caries experience during the decade. The income inequality in permanent caries experience did not change significantly. Income inequalities increased in deciduous teeth, but not in permanent teeth, among Australian children.

Discrete-Time Analogs for a Class of Continuous-Time Recurrent Neural Networks

This paper is concerned with the problem of local and global asymptotic stability for a class of discrete-time recurrent neural networks, which provide discrete-time analogs to their continuous-time counterparts, i.e., continuous-time recurrent neural networks with distributed delay. Some stability criteria, which include some existing results as their special cases, are derived. A discussion about the dynamical consistence of discrete-time neural networks versus their continuous-time counterparts is provided. An unconventional finite difference method is proposed and an example is also given to show the effectiveness of the method.

On stability of recurrent neural networks-an approach from volterra integro-differential equations

The uniform asymptotic stability of recurrent neural networks (RNNs) with distributed delay is analyzed by comparing RNNs to linear Volterra integro-differential systems under Lipschitz continuity of activation functions. The stability criteria obtained have unified and extended many existing results on RNNs.

Almost Periodic Solutions of Nonautonomous Linear Difference Equations

Explicit almost periodic solutions are obtained for a class of almost periodic linear difference equations. The stability characteristics of the almost periodic solution are investigated. The results are applied to a nonautonomous hyperbolic difference equation modelling the dynamics of a single species population in temporally varying environments.

A discrete model of competition

Sufficient conditions are derived for the global asymptotic stability of the positive equilibrium of the coupled system of difference equations with distributed delays modelling the dynamics of two competing species with discrete generations. The conditions obtained are the same as in the continuous case which are sufficient for the globally asymptotic stability of the positive equilibrium.

Hyperbolic logistic difference equation with infinitely many delays

A difference equation with infinitely many delays was considered, which gives a ‘best’ approximation of logistic integrodifferential equation. Some biological and evolutionary consequences are briefly discussed.

Stability analysis of recurrent neural networks - a Volterra integro-differential equation approach

The stability of a special class of nonlinear Volterra integro-differential systems are analyzed by comparing them to linear Volterra integro-differential systems. The results are used to determine the stability properties of recurrent neural networks with distributed delays, including constant discrete delays as a special case. The obtained stability criteria have unified and extended many existing results on recurrent neural networks.