A note on nonautonomous logistic equation with random perturbation

AbstractThis paper discusses a randomized nonautonomous logistic equation dN(t)=N(t)[(a(t)−b(t)N(t))dt+α(t)dB(t)], where B(t) is 1-dimensional standard Brownian motion. We show that E[1N(t)] has a unique positive T-periodic solution E[1Np(t)] provided a(t), b(t), and α(t) are continuous T-periodic functions, a(t)>0, b(t)>0 and ∫0T[a(s)−α2(s)]ds>0

A note on a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation

AbstractIn this paper, we show there is a stationary distribution of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation and it has ergodic property

Information criterion-based clustering with order-restricted candidate profiles in short time-course microarray experiments

<p>Abstract</p> <p>Background</p> <p>Time-course microarray experiments produce vector gene expression profiles across a series of time points. Clustering genes based on these profiles is important in discovering functional related and co-regulated genes. Early developed clustering algorithms do not take advantage of the ordering in a time-course study, explicit use of which should allow more sensitive detection of genes that display a consistent pattern over time. Peddada <it>et al</it>. <abbrgrp><abbr bid="B1">1</abbr></abbrgrp> proposed a clustering algorithm that can incorporate the temporal ordering using order-restricted statistical inference. This algorithm is, however, very time-consuming and hence inapplicable to most microarray experiments that contain a large number of genes. Its computational burden also imposes difficulty to assess the clustering reliability, which is a very important measure when clustering noisy microarray data.</p> <p>Results</p> <p>We propose a computationally efficient information criterion-based clustering algorithm, called ORICC, that also takes account of the ordering in time-course microarray experiments by embedding the order-restricted inference into a model selection framework. Genes are assigned to the profile which they best match determined by a newly proposed information criterion for order-restricted inference. In addition, we also developed a bootstrap procedure to assess ORICC's clustering reliability for every gene. Simulation studies show that the ORICC method is robust, always gives better clustering accuracy than Peddada's method and saves hundreds of times computational time. Under some scenarios, its accuracy is also better than some other existing clustering methods for short time-course microarray data, such as STEM <abbrgrp><abbr bid="B2">2</abbr></abbrgrp> and Wang <it>et al</it>. <abbrgrp><abbr bid="B3">3</abbr></abbrgrp>. It is also computationally much faster than Wang <it>et al</it>. <abbrgrp><abbr bid="B3">3</abbr></abbrgrp>.</p> <p>Conclusion</p> <p>Our ORICC algorithm, which takes advantage of the temporal ordering in time-course microarray experiments, provides good clustering accuracy and is meanwhile much faster than Peddada's method. Moreover, the clustering reliability for each gene can also be assessed, which is unavailable in Peddada's method. In a real data example, the ORICC algorithm identifies new and interesting genes that previous analyses failed to reveal.</p

The Asymptotic Behavior of a Stochastic Predator-Prey System with Holling II Functional Response

We discuss a stochastic predator-prey system with Holling II functional response. First, we show that this system has a unique positive solution as this is essential in any population dynamics model. Then, we deduce the conditions that there is a stationary distribution of the system, which implies that the system is permanent. At last, we give the conditions for the system that is going to be extinct

Global stability of two-group SIR model with random perturbation

AbstractIn this paper, we discuss the two-group SIR model introduced by Guo, Li and Shuai [H.B. Guo, M.Y. Li, Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q. 14 (2006) 259–284], allowing random fluctuation around the endemic equilibrium. We prove the endemic equilibrium of the model with random perturbation is stochastic asymptotically stable in the large. In addition, the stability condition is obtained by the construction of Lyapunov function. Finally, numerical simulations are presented to illustrate our mathematical findings

Minimax confidence bound of the normal mean under an asymmetric loss function

Confidence bound, LINEX loss function, normal mean, Bayes risk, minimaxity, admissibility,

Semiparametric estimation of regression functions in autoregressive models

This paper proposes a semiparametric method for an autoregressive model by combining a parametric regression estimator with a nonparametric adjustment. The regression has a parametric framework. After the parameter is estimated through a general parametric method, the obtained regression function is adjusted by a nonparametric factor, and the nonparametric factor is obtained through a natural consideration of the local L2-fitting criterion. Some asymptotic and simulation results for the semiparametric method are discussed.