Mathematical modeling of church growth

The possibility of using mathematics to model church growth is investigated using ideas from population modeling. It is proposed that a major mechanism of growth is through contact between religious enthusiasts and unbelievers, where the enthusiasts are only enthusiastic for a limited period. After that period they remain church members but less effective in recruitment. This leads to the general epidemic model which is applied to a variety of church growth situations. Results show that even a simple model like this can help understand the way in which churches grow, particularly in times of religious revival.     

Diffusion with a Discontinuous Potential: a Non-Linear Semigroup Approach

This paper studies existence of mild solution to a sharp cut off model for contact driven tumor growth. Analysis is based on application of the Crandall-Liggett theorem for ω-quasi-contractive semigroups on the Banach space L~1(?). Furthermore,numerical computations are provided which compare the sharp cut off model with the tumor growth model of Perthame, Quirós, and Vázquez [13].     

A model for glioma growth

Glioblastoma Multiforme (GBM) is the most invasive form of primary brain tumor. We propose a mathematical model that describes such tumor growth and allows us to describe two different mechanisms of cell invasion: diffusion (random motion) and chemotaxis (directed motion along the gradient of the chemoattractant concentration). The results are in a quantitative agreement with recent in vitro experiments. It was observed in experiments that the outer invasive zone grows faster than the inner proliferative region. We argue that this feature indicates transient behavior, and that the growth velocities tend to the same constant value for larger times. A longer‐time experiment is needed to verify this hypothesis and to choose between the two basic mechanisms for tumor growth. © 2005 Wiley Periodicals, Inc. Complexity 11: 53–57, 2005     

A microbial continuous culture system with diffusion and diversified growth

A reaction-diffusion model is presented to describe the microbial continuous culture with diversified growth. The existence of nonnegative solutions and attractors for the system is obtained, the stability of steady states and the steady state bifurcation are studied under three growth conditions. In the case of no growth inhibition or only product inhibition, the system admits one positive constant steady state which is stable; in the case of growth inhibition only by substrate, the system can have two positive constant steady states, explicit conditions of the stability and the steady state bifurcation are also determined. In addition, numerical simulations are given to exhibit the theoretical results.     

New epitaxial thin‐film models and numerical approximation

This paper concerns new continuum phenomenological model for epitaxial thin‐film growth with three different forms of the Ehrlich–Schwoebel current. Two of these forms were first proposed by Politi and Villain 1996 and then studied by Evans, Thiel, and Bartelt 2006. The other one is completely new. Energy structure and properties of the new model are studied. Following the techniques used in Li and Liu 2003, we present rigorous analysis of the well‐posedness, regularity, and time stability for the new model. We also studied both the global and the local behavior of the surface roughness in the growth process. By using a convex–concave time‐splitting scheme, one can naturally build unconditionally stable semi‐implicit numerical discretizations with linear implicit parts, which is much easier to implement than conventional models requiring nonlinear implicit parts. Despite this fundamental difference in the model, numerical experiments show that the nonlinear morphological instability of the new model agrees well with results of other models published before which indicates that the new model correctly captures the essential morphological states in the thin‐film growth process. Copyright © 2017 John Wiley & Sons, Ltd.     

Spike patterns in a reaction–diffusion ODE model with Turing instability

We explore a mechanism of pattern formation arising in processes described by a system of a single reaction–diffusion equation coupled with ordinary differential equations. Such systems of equations arise from the modeling of interactions between cellular processes and diffusing growth factors. We focus on the model of early carcinogenesis proposed by Marciniak‐Czochra and Kimmel, which is an example of a wider class of pattern formation models with an autocatalytic non‐diffusing component. We present a numerical study showing emergence of periodic and irregular spike patterns because of diffusion‐driven instability. To control the accuracy of simulations, we develop a numerical code on the basis of the finite‐element method and adaptive mesh grid. Simulations, supplemented by numerical analysis, indicate a novel pattern formation phenomenon on the basis of the emergence of nonstationary structures tending asymptotically to a sum of Dirac deltas. Copyright © 2013 John Wiley & Sons, Ltd.     

Analysis of electronic models for solar cells including energy resolved defect densities

We introduce an electronic model for solar cells including energy resolved defect densities. The resulting drift–diffusion model corresponds to a generalized van Roosbroeck system with additional source terms coupled with ordinary differential equations containing space and energy as parameters for all defect densities. The system has to be considered in heterostructures and with mixed boundary conditions from device simulation. We give a weak formulation of the problem. If the boundary data and the sources are compatible with thermodynamic equilibrium, the free energy along solutions decays monotonously. In other cases, it may be increasing, but we estimate its growth. We establish boundedness and uniqueness results and prove the existence of a weak solution. This is carried out by considering a regularized problem, showing its solvability and the boundedness of its solutions independent of the regularization level. Copyright © 2011 John Wiley & Sons, Ltd.     

三个不同领域实际问题相结合的教学方法探讨

通过建立肿瘤增长动力学模型,研究肿瘤增长动态规律时发现:肿瘤增长的全过程与龚珀兹增长曲线描述的经济增长过程完全相符;进一步给出了龚珀兹增长曲线与描述种群的增长与调节的逻辑斯蒂增长曲线完全相似.将上述三个属性完全不同的实际应用问题打包传授给学生的授课方法是有别于孤立地介绍三个具体问题的,学生的学习效果也会不同.     

Positive computational modelling of the dynamics of active and inert biomass with extracellular polymeric substances

Departing from a complex system of nonlinear partial differential equations that models the growth dynamics of biological films, we provide a finite-difference model to approximate its solutions. The variables of interest are measured in absolute scales, whence the need of preserving the positivity of the solutions is a mathematical constraint that must be observed. In this work, we provide a numerical discretization of our mathematical model which is capable of preserving the non-negative character of approximations under suitable conditions on the model and computational parameters. As opposed to the nonlinear model which motivates this report, our numerical technique is a linear method which, under suitable circumstances, may be represented by an M-matrix. The fact that our method is a positivity-preserving scheme is established using the inverse-positive properties of these matrices. Computer simulations corroborate the validity of the theoretical findings.     

A Lyapunov functional for a triangular reaction–diffusion system with nonlinearities of exponential growth

The aim of this work is to study the global existence of solutions to a triangular system of reaction–diffusion equations, which describes epidemiological or chemical situations. On the basis of the construction of a suitable Lyapunov functional, we show that for any initial data, classical global solutions exist even when the nonlinearities are of exponential growth. Copyright © 2012 John Wiley & Sons, Ltd.     

A STRUCTURALLY BASED ANALYTIC MODEL OF GROWTH AND BIOMASS DYNAMICS IN SINGLE SPECIES STANDS OF CONIFERS

A theoretically based analytic model of plant growth in single species conifer communities based on the species fully occupying a site and fully using the site resources is introduced. Model derivations result in a single equation simultaneously describes changes over both, different site conditions (or resources available), and over time for each variable for each species. Leaf area or biomass, or a related plant community measurement, such as site class, can be used as an indicator of available site resources. Relationships over time (years) are determined by the interaction between a stable foliage biomass in balance with site resources, and by the increase in the total heterotrophic biomass of the stand with increasing tree size. This structurally based, analytic model describes the relationships between plant growth and each species’ functional depth for foliage, its mature crown size, and stand dynamics, including the self‐thinning. Stand table data for seven conifer species are used for verification of the model. Results closely duplicate those data for each variable and species. Assumptions used provide a basis for interpreting variations within and between the species. Better understanding of the relationships between the MacArthur consumer resource model, the Chapman–Richards growth functions, the metabolic theory of ecology, and stand development resulted.     

Optimal control and long-run dynamics for a spatial economic growth model with physical capital accumulation and pollution diffusion

In this work we analyze the large-time behavior of a spatially structured economic growth model coupling physical capital accumulation and pollution diffusion. This model extends other results in the literature along different directions. Alongside the classical Cobb–Douglas production function, a convex–concave production function is considered. We add a negative feedback to the production function in order to describe the (negative) influence of pollution on output, and therefore on capital accumulation. We also present an optimal control problem for the above model.     

一个一般的两部门增长模型

本文提出一个一般的资本驱动型两部门经济增长模型 ,它是对Uzawa和Lucas的模型的一个改进 .通过此模型 ,我们发现经济在最一般的情况时将会出现停滞状态 ,并讨论和估计系统的动态性质     

On the growth of filamentary structures in planar media

We analyse a mathematical model for the growth of thin filaments into a two dimensional medium. More exactly, we focus on a certain reaction/diffusion system, describing the interaction between three chemicals (an activator, an inhibitor and a growth factor), and including a fourth cell variable characterising irreversible incorporation to a filament. Such a model has been shown numerically to generate structures shaped like nets. We perform an asymptotical analysis of the behaviour of solutions, in the case when the system has parameters very large and very small, thereby allowing the onset of different time and space scales. In particular, we describe the motion of the tip of a filament, and the changes in the relevant chemical species nearby. Copyright © 2004 John Wiley & Sons, Ltd.     

Vanishing viscosities and error estimate for a Cahn–Hilliard type phase field system related to tumor growth

In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn–Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in Colli et al. (2015), letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.     

An Improved Solow—Swan Model

§1.　IntroductionThetheoryofeconomicgrowthisanimportanttopicinMacroeconomics.Afterador-mancynearlytwodecades,thestudyofeconomicgrowthbecamevigorousagaininthemid-1980s.TheSolow-Swanmodel[1]isoneofthemostusedinthisfield.Thismodelhasarousedmanyresearchersinterestintherecentyears.N.G.Manki,etal(1992)[2]appliedthismodeltoexaminewhetheritisconsistentwiththeinternationalvariationinthestandardofliving.W.Easterly(1993)[3]gavearevisedSolow-SwanmodelbyintroducingtheCESproductionfunction.Inthepaper…     

Linear and nonlinear stability thresholds for a diffusive model of pioneer and climax species interaction

In this paper a reaction–diffusion model describing two interacting pioneer and climax species is considered. The role of diffusivity and forcing (stocking or harvesting of the species) on the nonlinear stability of a coexistence equilibrium is analysed. The study is performed in the context of a new approach to nonlinear L²‐stability based on the analysis of stability of the zero solution of a suitable linear system of ordinary differential equations. Theorems concerning the effect of forcing and diffusivity on the dynamics are established and stability–instability thresholds for the system are obtained. An example to illustrate the practical use of the results is also provided. Copyright © 2008 John Wiley & Sons, Ltd.     

A General Model of Church Growth and Decline

An earlier model of church growth (Hayward, 1999 Hayward , J. ( 1999 ). Mathematical modeling of church growth . Journal of Mathematical Sociology , 23 ( 4 ): 255 – 292 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) is extended to include long-term effects due to births, deaths and reversion from the church. It is proposed that only a subset of the church, the enthusiasts, is involved in the recruitment process, and only for a limited period of time after their conversion. It is found that the church reaches equilibrium in its proportion of society according to the potential of these enthusiasts to reproduce themselves, and the losses from the church. If this reproduction potential is below a threshold that depends on losses, then extinction occurs. If it is above a higher threshold, then the church sees rapid revival growth. The model is applied to a number of church denominations to examine their prospects for survival or revial growth. Generally, declining churches do so because their reproduction potential is inadequate, rather than due to excessive losses.     

A fourth-order parabolic equation modeling epitaxial thin film growth

We study the continuum model for epitaxial thin film growth from Phys. D 132 (1999) 520-542, which is known to simulate experimentally observed dynamics very well. We show existence, uniqueness and regularity of solutions in an appropriate function space, and we characterize the existence of nontrivial equilibria in terms of the size of the underlying domain. In an investigation of asymptotical behavior, we give a weak assumption under which the ω-limit set of the dynamical system consists only of steady states. In the one-dimensional setting we can characterize the set of steady states and determine its unique asymptotically stable element. The article closes with some illustrative numerical examples.     

基于Improved-Romer模型的经济一体化对区域经济增长影响的分析

以Romer模型为基础,通过构造一个两区域模型,研究了经济一体化对区域经济增长的影响,证明了实行经济一体化后每个区域的经济增长率均能得到显著提高,并阐明了经济一体化促进各区域经济增长的途径主要有两条:一是规模效应,即经济一体化使得各区域中间产品和最终产品的市场规模得到扩大而获得的规模收益;二是技术扩散效应,即经济一体化...