Analysis of a nonlinear diffusion problem With Michaelis-Menten kinetics by an integral equation method

A mathematical model for oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics is analyzed by means of an intergral equation method. It is shown that an integral equation formulation can be used to obtain a numerical solution associated with this boundary and initial value problem. Through an illustrative numerical calculation we are able to obtain an accurate solution for both the steady and transient problems. Finally, a comparison is made with the numerical solution of McElwain and the variational solution of Anderson and Arthurs for the steady state and Lin's result concerning the unsteady state.     

Mathematical analysis of a delayed stage-structured predator–prey model with impulsive diffusion between two predators territories

In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete, but in reality, many species diffuse only during a single period, and diffusion often occurs in regular pulses. Further, in forest habitats, the highest-level predator species are restricted to a specific territory, but prey can impulsively move between territories. Therefore, in this paper, we consider a delayed stage-structured predator–prey model with impulsively diffusive prey between two patches; in the model, patches represent the territories of two different predator populations. Here, we analytically obtain the global attractivity condition of predator-extinction periodic solutions for the system by using the concepts of Hui and Chen (2005); a numerical simulation is also included to illustrate this result. Further, we establish permanence conditions for the coexistence of the species using the theory of impulsive delayed differential equations. Finally, we explore the possibilities of the permanence of the system by using the growth rates of immature predators and the impulse period as critical parameters, and we also obtain the parameters’ threshold limits using numerical experimentation.     

一类考虑收获的时滞捕食系统的稳定性与Hopf分支研究

给出了一类考虑收获的时滞捕食系统的局部稳定性判断,并由规范型理论和中心流形定理推导出了Hopf分支的方向、稳定性等条件,最后给出了两个数值模拟例子验证了结论的正确性.     

食饵具有周期强迫的捕食模型

建立并研究了一类具有周期强迫和脉冲扰动的捕食模型,通过理论分析和数值模拟,得到了食饵灭绝周期解全局渐近稳定和系统持久的充分条件,利用分支理论证明了边界周期解附近会分支出正周期解.     

Complementary variational principles for diffusion problems with Michaelis-Menten kinetics

The theory of complementary variational principles is used to obtain maximum and minimum principles for diffusion problems with Michaelis-Menten kinetics. In an illustrative calculation we obtain an extremely accurate variational solution in good agreement with the numerical solution of McElwain (1978).     

Dynamic analysis of lactic acid fermentation in membrane bioreactor

In this paper, a mathematical model for the lactic acid fermentation in membrane bioreactor is investigated. This novel theoretical framework could result in an objective criterion on how to control the substrate concentration in order to keep a sustainable and steady output of lactic acid. Firstly, continuous input substrate is taken. The existence and local stability of two equilibria are studied. According to Poincaré-Bendixson Theorem, we obtain the conditions for the globally asymptotical stability of the equilibrium. Secondly, impulsive input substrate is also considered. Using Floquet's theorem and small-amplitude perturbation, we obtain the biomass-free periodic solution is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations.     

Mathematical model and numerical simulation of the cell growth in scaffolds

A scaffold is a three-dimensional matrix that provides a structural base to fill tissue lesion and provides cells with a suitable environment for proliferation and differentiation. Cell-seeded scaffolds can be implanted immediately or be cultured in vitro for a period of time before implantation. To obtain uniform cell growth throughout the entire volume of the scaffolds, an optimal strategy on cell seeding into scaffolds is important. We propose an efficient and accurate numerical scheme for a mathematical model to predict the growth and distribution of cells in scaffolds. The proposed numerical algorithm is a hybrid method which uses both finite difference approximations and analytic closed-form solutions. The effects of each parameter in the mathematical model are numerically investigated. Moreover, we propose an optimization algorithm which finds the best set of model parameters that minimize a discrete l ₂ error between numerical and experimental data. Using the mathematical model and its efficient and accurate numerical simulations, we could interpret experimental results and identify dominating mechanisms.     

Imitation dynamics of vaccine decision-making behaviours based on the game theory

Based on game theory, we propose an age-structured model to investigate the imitation dynamics of vaccine uptake. We first obtain the existence and local stability of equilibria. We show that Hopf bifurcation can occur. We also establish the global stability of the boundary equilibria and persistence of the disease. The theoretical results are supported by numerical simulations.     

A dynamic numerical method for models of renal tubules

We show that an explicit method for solving hyperbolic partial differential equations can be applied to a model of a renal tubule to obtain both dynamic and steady-state solutions. Appropriate implementation of this method eliminates numerical instability arising from reversal of intratubular flow direction. To obtain second-order convergence in space and time, we employ the recently developed ENO (Essentially Non-Oscillatory) methodology. We present examples of computed flows and concentration profiles in representative model contexts. Finally, we indicate briefly how model tubules may be coupled to construct large-scale simulations of the renal counterflow system.     

具有社会行为的HIV传播模型

考虑到HIV传播的社会行为的影响,建立了一个HIV传播的偏微分方程系统.利用预测校正方法我们给出了系统的数值解法,由此说明人的行为在HIV传播中的重要作用.     

根表面养分吸收通量和根围溶质浓度的近似解析解

该文用Nye-Tinker-Barber模型来研究植物根系表面的养分吸收通量和根围溶质浓度的近似解析解。将根围区域分为远场区域和近场区域, 在远场用相似变量, 在近场用尺度变换, 将远场解在根表面展开并与近场解进行待定函数的匹配, 从而获得对流扩散方程根表面通量和浓度的一阶近似解析解, 该解能够简化到扩散方程的解的形式。对氮、钾、硫、磷、镁、钙的养分吸收通量和氮、钾的浓度分别进行数值模拟, 比较模型的数值解、Roose的近似解析解和该文的近似解析解。结果表明: 在扩散方程中, 6种元素通量的解析解与Roose解析解相近, 但均高于数值解, 钾和磷的通量在短时间内迅速衰减; 钾和氮浓度的全局近似解析解与Roose解析解接近, 并与数值解的变化趋势一致。在对流扩散方程中, 除氮外的5种元素通量的近似解较Roose的解析解更接近于数值解, 且没有奇性。     

Some exact solutions of a generalized Fisher equation related to the problem of biological invasion

The problem of biological invasion in a model single-species community is considered, the spatiotemporal dynamics of the system being described by a modified Fisher equation. For a special case, we obtain an exact solution describing self-similar growth of the initially inhabited domain. By comparison with numerical solutions, we show that this exact solution may be applicable to describe an early stage of a biological invasion preceding the propagation of the stationary travelling wave. Also, the exact solution is applied to the problem of critical aggregation to derive sufficient conditions of population extinction. Finally, we show that the solution we obtain is in agreement with some data from field observations.     

An intervened Poisson distribution and its medical application

Among probability distributions that are used to describe a chance mechanism whose observational apparatus becomes active only when at least one event occurs is the zero-truncated Poisson distribution (ZTPD). A modified version of the ZTPD, which we call an intervened Poisson distribution (IPD), is discussed in this paper. We give a genesis of IPD and obtain its statistical properties. A numerical example is included to illustrate the results.     

Investigation on Optical Properties of Ag–Au Alloy Nanoparticles

The outstanding chemical stability of Au and intense localized surface plasmon resonance of Ag make it possible to obtain a nanostructure with a good balance of good chemical stability and optical response. In this paper, we investigated the relationship between optical properties and the composition and size of Ag–Au alloy nanoparticle with numerical calculation by applying experimental data. Simplified empirical formulas are proposed through numerical simulation. The properties of extinction efficiency and the relative contribution of scattering and absorption efficiency to the extinction efficiency have been researched in detail. The calculated result and experimental data has been compared, and good agreement is obtained. Our work contributes greatly to catalysis application of Au–Ag alloy NPs in specific regions.     

Bio-economics of a renewable resource in a seasonally varying environment

In this paper we study the bio-economics of a renewable resource with governing dynamics described by two distinct growth functions (viz., logistic and Gompertz growth functions) in a seasonally varying environment. Seasonality is introduced into the system by taking the involved ecological parameters to be periodic. In this work, we establish a procedure to obtain the optimal path and compute the optimal effort policy which maximizes the net revenue to the harvester for a fairly general optimal control problem and apply this procedure to the considered models to derive some important conclusions. These problems are solved on the infinite horizon. We find that, for both the models, the optimal harvest policy and the corresponding optimal path are periodic after a finite time. We also obtain optimal solution, a suboptimal harvesting policy and the corresponding suboptimal approach path to reach this optimal solution. The key results are illustrated using numerical simulations and we compare the revenues to the harvester along the optimal and suboptimal paths. The general procedure developed in this work, for obtaining the optimal effort policy and the optimal path, has wider applicability.     

Smart platform for the analysis of cupula deformation caused by otoconia presence within SCCs

In this paper, we first briefly describe the mechanical model of cupula deformation with the appropriate analytical solution. Then, we present the numerical solution of the same problem and compare it with the analytical one. Besides, we provide another numerical solution based on the Finite Element Method procedure, in an effort to encompass a more realistic approach to the problem such as considering the real geometry of the SCCs and the obstruction of the fluid flow during head movement due to the presence of otoconia. As a result, we obtain fifty solutions for a head rotation angle in a range from 0° to 120°, taking into account that such a manoeuvre lasts exactly 3?seconds. In the end, we present a mobile client–server application for visualising the finite element solutions in a way that is convenient for the clinical practice.     

一类带Monod增长率及脉冲状态反馈控制的微生物杀虫剂模型的周期解

研究了一类带Monod增长率及脉冲状态反馈控制的微生物杀虫剂模型．证明了无脉冲系统的负向全局渐近稳定性及带有脉冲状态反馈控制系统具有阶一周期解,并且给出阶一周期解存在和稳定的充分条件．数值模拟验证了理论结果．     

Bayesian Nonparametric Estimation of the Probability of Discovering New Species

We consider the problem of evaluating the probability of discoveringa certain number of new species in a new sample of populationunits, conditional on the number of species recorded in a basicsample. We use a Bayesian nonparametric approach. The differentspecies proportions are assumed to be random and the observationsfrom the population exchangeable. We provide a Bayesian estimator,under quadratic loss, for the probability of discovering newspecies which can be compared with well-known frequentist estimators.The results we obtain are illustrated through a numerical exampleand an application to a genomic dataset concerning the discoveryof new genes by sequencing additional single-read sequencesof cdna fragments.     

延迟遗传调控网络稳定性及分支

本文主要研究了延迟遗传调控网络的局部稳定性和该网络的Hopf分支存在条件．延迟遗传调控网络是无穷维系统,此类系统在平衡点线性化后的特征方程为超越方程。通过对此超越方程进行研究,得到了系统系数不同时的系统稳定的条件及相关结论,又进一步说明了此系统的Hopf分支存在条件．最后,举一个例子进行了数值仿真验证了所得到的结论．     

Optical Nanofocusing on Tapered Metallic Waveguides

Tapered metal wires show a remarkable ability to ‘squeeze’ the lateral extent of a propagating surface-plasmon-polariton mode as it travels toward the tip of the taper. The transformation can be continued well below the diffraction limit to terminate at a nanoscale apex where intense near-fields are created. We perform the first full numerical simulations to investigate and quantify this phenomenon. We find optimal angles for maximal tip-field enhancement on conical wires by considering absorption, scattering to radiation and reflection. The optimal parameters we obtain contradict the conditions for adiabatic tapering, thereby advocating the use of numerical simulations. Despite the influence of losses, nanofocusing is still highly efficient for a broad range of practical metals, visible wavelengths and taper geometries. Diverse nano-optic applications can benefit directly and significantly from the results.