Mode-doubling and tripling in reaction-diffusion patterns on growing domains: A piecewise linear model

 Reaction-diffusion equations are ubiquitous as models of biological pattern formation. In a recent paper [4] we have shown that incorporation of domain growth in a reaction-diffusion model generates a sequence of quasi-steady patterns and can provide a mechanism for increased reliability of pattern selection. In this paper we analyse the model to examine the transitions between patterns in the sequence. Introducing a piecewise linear approximation we find closed form approximate solutions for steady-state patterns by exploiting a small parameter, the ratio of diffusivities, in a singular perturbation expansion. We consider the existence of these steady-state solutions as a parameter related to the domain length is varied and predict the point at which the solution ceases to exist, which we identify with the onset of transition between patterns for the sequence generated on the growing domain. Applying these results to the model in one spatial dimension we are able to predict the mechanism and timing of transitions between quasi-steady patterns in the sequence. We also highlight a novel sequence behaviour, mode-tripling, which is a consequence of a symmetry in the reaction term of the reaction-diffusion system. Received: 19 December 2000 / Revised version: 24 May 2001 / Published online: 7 December 2001     

A tractable deterministic model with realistic latent period for an epidemic in a linear habitat

 This paper describes a model for the spatial spread of an epidemic in which the latent period has a Gamma distribution with arbitrary mean and variance while the infected and uninfected may have different diffusion coefficients. Using a variant of the “linear chain trick” we derive a simple algorithm for the propagation speed of the infection front. By way of verification, we show that, with suitably matched parameters, the algorithm agrees closely with speeds obtained numerically from the corresponding model in which the latent period is a constant. Received: 6 March 2001 / Revised version: 15 July 2001 / Published online: 8 February 2002     

Modeling visuospatial perception in neglect patients

 The spatial distortion hypothesis is one of several theories that explain certain aspects of neglect in patients with right parietal lesions. To determine whether a distorted representation of space can account for the performance of neglect patients in different visuospatial tasks, we asked 26 neglect patients to: (1) bisect horizontal lines and (2) to compare the width of two horizontally aligned bars. A simple mathematical model compatible with the idea of a stationary distortion of represented space in egocentric coordinates explained the results of the line-bisection task. A second model that had basically the same structure and was compatible with the idea of a distorted egocentric representation based on a dynamic remapping of space approximated the size-comparison data. These results support the view that abnormalities observed in the line-bisection and size-comparison tasks are due to a distorted internal representation of the external world. Certain findings suggest that this distortion could be based on a dynamic mapping of space determined by the distribution of visuospatial attention. Received: 14 June 1999 / Accepted in revised form: 30 May 2001     

The evolution of dispersal rates in a heterogeneous time-periodic environment

A reaction-diffusion model for the evolution of dispersal rates is considered in which there is both spatial heterogeneity and temporal periodicity. The model is restricted to two phenotypes because of technical difficulties, but a wide range of mathematical techniques and computational effort are needed to obtain useful answers. We find that the question of selection is a great deal richer than in the autonomous case, where the phenotype with the lowest diffusion is selected for. In the current model either the lower or higher diffuser rate may be selected, or there may be coexistence of phenotypes. The paper raises several open questions and suggests in particular that a mutation-selection multi-phenotypic model would repay study. Received: 17 April 2000 / Revised version: 2 May 2001 / Published online: 12 October 2001     

Unravelling the Turing bifurcation using spatially varying diffusion coefficients

. The Turing bifurcation is the basic bifurcation generating spatial pattern, and lies at the heart of almost all mathematical models for patterning in biology and chemistry. In this paper the authors determine the structure of this bifurcation for two coupled reaction diffusion equations on a two-dimensional square spatial domain when the diffusion coefficients have a small explicit variation in space across the domain. In the case of homogeneous diffusivities, the Turing bifurcation is highly degenerate. Using a two variable perturbation method, the authors show that the small explicit spatial inhomogeneity splits the bifurcation into two separate primary and two separate secondary bifurcations, with all solution branches distinct. This splitting of the bifurcation is more effective than that given by making the domain slightly rectangular, and shows clearly the structure of the Turing bifurcation and the way in which the! var ious solution branches collapse together as the spatial variation is reduced. The authors determine the stability of the solution branches, which indicates that several new phenomena are introduced by the spatial variation, including stable subcritical striped patterns, and the possibility that stable stripes lose stability supercritically to give stable spotted patterns.. Received: 10 January 1996/Revised version: 3 July 1996     

Dynamic volume-averaged model of heat and mass transport within a compost biofilter: I. Model development

Successful, long-term operation of a biofilter system depends on maintaining a suitable biofilm environment within a porous medium reactor. In this article a mathematical study was conducted of the spatial and temporal changes of biofilter performance due to interphase heat and mass transport. The method of volume averaging was used to spatially smooth the three-phase (solid, liquid, and gas) conservation equations over the biofilter domain. The packing medium was assumed to be inert, removing the solid phase mass continuity equation from the system. The finite element method was used to integrate the resulting nonlinear-coupled partial differential equations, tracking eight state variables: temperature, water vapor, dry air, liquid water, biofilm, gas and liquid phase organic pollutant, and nutrient densities, through time and space. A multiphase, gas and liquid flow model was adapted to the biofilter model from previous studies of unsaturated groundwater flow. Newton's method accelerated by an LU direct solver was used to iterate the model for solutions. Effects of packing media on performance were investigated to illustrate the utility of the model. The moisture dynamics and nutrient cycling are presented in Part II of this article.     

High-resolution structure of ybfF from Escherichia coli K12: a unique substrate-binding crevice generated by domain arrangement

Esterases are one of the most common enzymes and are involved in diverse cellular functions. ybfF protein from Escherichia coli (Ec_ybfF) belongs to the esterase family for the large substrates, palmitoyl coenzyme A and malonyl coenzyme A, which are important cellular intermediates for energy conversion and biomolecular synthesis. To obtain molecular information on ybfF esterase, which is found in a wide range of microorganisms, we elucidated the crystal structures of Ec_ybfF in complexes with small molecules at resolutions of 1.1 and 1.68 Å, respectively. The structure of Ec_ybfF is composed of a globular α/β hydrolase domain with a three-helical bundle cap, which is linked by a kinked helix to the α/β hydrolase domain. It contains a catalytic tetrad of Ser-His-Asp-Ser with the first Ser acting as a nucleophile. The unique spatial arrangement and orientation of the helical cap with respect to the α/β hydrolase domain form a substrate-binding crevice for large substrates. The helical cap is also directly involved in catalysis by providing a substrate anchor, viz., the conserved residues of Arg123 and Tyr208. The high-resolution structure of Ec_ybfF shows that the inserted helical bundle structure and its spatial orientation with respect to the α/β hydrolase domain are critical for creating a large inner space and constituting a specific active site, thereby providing the broad substrate spectrum toward large biomolecules.     

Three‐Phase Multiscale Modeling of a LiCoO2 Cathode: Combining the Advantages of FIB–SEM Imaging and X‐Ray Tomography

LiCoO₂ electrodes contain three phases, or domains, each having specific‐intended functions: ion‐conducting pore space, lithium‐ion‐reacting active material, and electron conducting carbon‐binder domain (CBD). Transport processes take place in all domains on different characteristic length scales: from the micrometer scale in the active material grains through to the nanopores in the carbon‐binder phase. Consequently, more than one imaging approach must be utilized to obtain a hierarchical geometric representation of the electrode. An approach incorporating information from the micro‐ and nanoscale to calculate 3D transport‐relevant properties in a large‐reconstructed active domain is presented. Advantages of focused ion beam/scanning electron microscopy imaging and X‐ray tomography combined by a spatial stochastic model, validated with an artificially produced reference structure are used. This novel approach leads to significantly different transport relevant properties compared with previous tomographic approaches: nanoporosity of the CBD leads to up to 42% additional contact area between active material and pore space and increases ionic conduction by a factor of up to 3.6. The results show that nanoporosity within the CBD cannot be neglected.     

Epileptiform activity in a neocortical network: a mathematical model

 A simple mathematical model describing the generation and propagation of epileptiform activity in a cerebral cortical network is presented. The model consists of a system of nonlinear delay differential equations. Physiological properties are taken into account as nonlinear transmission of signals at the synapse, temporal and spatial summation of incoming signals at the soma, active membrane characteristics, and dendritic and axonal propagation times. The influence of the connectivity and the temporal parameters on the oscillatory properties of the model is studied. The computer simulations are in agreement with experimental observations in cortical networks: whereas a weak excitatory or strong inhibitory synaptic connection strength produces a stationary status with short-lasting responses to external stimuli, increases in excitation or decreases in inhibition induce spontaneous and stimulus-evoked rhythmic discharges. Synaptic burst-like activity is observed only for an intermediate range of excitatory and inhibitory connection strengths and external inputs. The form and duration of the bursts can also be controlled by the temporal parameters. The results demonstrate that relatively simple mathematical equations are sufficient to model some of the network properties underlying the generation and propagation of epileptiform activity. Received: 2 October 2000 / Accepted in revised form: 4 March 2001     

Multiple Comparisons for a Large Number of Parameters

When there are many parameters of interest (finitely large or infinite), standard multiple comparison procedures for a finite number of parameters (called discrete‐domain approaches) may lead to a simultaneous confidence region (SCR) too conservative to be useful. Such cases often arise in locating disease genes, detecting changes in image data and examining shapes and patterns in growth curves; or generally, in quantifying uncertainty in an estimate of a regression function (as one entity). In these cases, procedures designed for a continuous domain must be used. Scheffe's method is a classical example of continuous‐domain approaches. It provides an SCR for a regression function when errors are iid Gaussian and the predictor space is unconstrained, i.e. the domain of interest is the q dimensional Euclidean space. In practice, however, functions defined on finite intervals or other constrained domains are often of interest and data may not be Gaussian. Thus, Scheffe's SCR becomes either too conservative or inadequate. In this paper, we introduce and survey a modern‐type continuous‐domain approach, and explore a connection between some discrete‐ and continuous‐domain multiple comparison procedures. We show that, in some cases, even for a small number of parameters, it is still better to use a continuous‐domain multiple comparison procedure. The main ideas behind the continuous‐domain procedures are shown. A new procedure for comparing a finite number of contrasts about k regression curves is developed. Relevant software is provided.     

Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model

A computational model for retinal hemodynamics accounting for ocular curvature is presented. The model combines (i) a hierarchical Darcy model for the flow through small arterioles, capillaries and small venules in the retinal tissue, where blood vessels of different size are comprised in different hierarchical levels of a porous medium; and (ii) a one-dimensional network model for the blood flow through retinal arterioles and venules of larger size. The non-planar ocular shape is included by (i) defining the hierarchical Darcy flow model on a two-dimensional curved surface embedded in the three-dimensional space; and (ii) mapping the simplified one-dimensional network model onto the curved surface. The model is solved numerically using a finite element method in which spatial domain and hierarchical levels are discretized separately. For the finite element method, we use an exterior calculus-based implementation which permits an easier treatment of non-planar domains. Numerical solutions are verified against suitably constructed analytical solutions. Numerical experiments are performed to investigate how retinal hemodynamics is influenced by the ocular shape (sphere, oblate spheroid, prolate spheroid and barrel are compared) and vascular architecture (four vascular arcs and a branching vascular tree are compared). The model predictions show that changes in ocular shape induce non-uniform alterations of blood pressure and velocity in the retina. In particular, we found that (i) the temporal region is affected the least by changes in ocular shape, and (ii) the barrel shape departs the most from the hemispherical reference geometry in terms of associated pressure and velocity distributions in the retinal microvasculature. These results support the clinical hypothesis that alterations in ocular shape, such as those occurring in myopic eyes, might be associated with pathological alterations in retinal hemodynamics.     

Environment is more relevant than spatial structure as a driver of regional variation in tropical tree community richness and composition

Background: Understanding how factors related to environment and geographical distance explain community variation allows insights about how ecological niche and neutral processes control tropical community assembly.

Aims: Quantify how variation in regional tree community richness and composition in a humid tropical forest across a mountain chain are related to niche and putative neutral processes.

Methods: We used a variation partitioning routine based on Redundancy Analysis to model tropical tree community richness and composition within three distinct elevation belts, as a function of environment and spatial structure, using data from 32 studies in the Serra do Mar Range, south-eastern Brazil.

Results: Environmental effects were greater than spatial structure effects to explain community variation in the three elevation belts. There was a trend of decreasing spatial structure effects while environmental effects remained constant from lower to higher elevations. Patterns were congruent for species richness and composition.

Conclusions: We suggest that on tropical mountains, niche-related processes are equally relevant for tropical forest community assembly at all elevations, while neutral processes become weaker towards higher elevations. Determining if this trend is a consequence of the greater heterogeneity of environmental conditions associated with higher elevations in tropical mountainous terrain remains an important area of research.     

Predator-mediated coexistence: an equilibrium interpretation

I have constructed a spatially distributed analytical model of predators, superior prey competitors, and inferior prey competitors, based on the limiting deterministic version of a simulation model by Caswell (1978). Persistence regions for the three populations are mapped in parameter space. Conceptually shrinking the system from infinite size (i.e. infinitely many spatial “cells”) to some finite size introduces demographic stochasticity, increasing the chance of extinction of one or more populations within a given time interval. But some of the finite (stochastic) system's behavior, such as any tendency to damp perturbations, can be related to the behavior of the deterministic system at the same location in parameter space.     

A review on spatial aggregation methods involving several time scales

This article is a review of spatial aggregation of variables for time continuous models. Two cases are considered. The first case corresponds to a discrete space, i.e. a set of discrete patches connected by migrations, which are assumed to be fast with respect to local interactions. The mathematical model is a set of coupled ordinary differential equations (O.D.E.). The spatial aggregation allows one to derive a global model governing the time variation of the total numbers of individuals of all patches in the long term. The second case considers a continuous space and is a set of partial differential equations (P.D.E.). In that case, we also assume that diffusion is fast in comparison with local interactions. The spatial aggregation allows us again to obtain an O.D.E. governing the total population density, which is obtained by integration all over the spatial domain, at the slow time scale. These aggregations of variables are based on time scales separation methods which have been presented largely elsewhere and we recall the main results. We illustrate the methods by examples in population dynamics and prey–predator models.     

基于微粒群-马尔科夫复合模型的生态空间预测模拟——以长株潭城市群为例

生态空间具有重要的生态功能,对生态空间进行科学预测模拟可为保护国土空间生态安全提供决策依据。利用Arc GIS及MATLAB软件,在生态空间风险评价的基础上构建了微粒群-马尔科夫复合模型,并以长株潭城市群为研究区,基于2013年土地利用现状数据,对2020年的生态空间进行了预测模拟,最后在此基础上提出了生态空间重构的基本思路。结果表明:1)微粒群-马尔科夫复合模型(PSO-Markov)构建的基本步骤为:第一步:粒子的选择与设计,以2000 m×2000 m的正方形单元作为基本粒子。第二步:粒子的初始化设定,根据生态空间风险由低到高的原则进行选择。第三步:适应度函数的建立,用生态空间的风险值来确定生态空间的空间格局。第四步:空间位置的更新,根据自身的历史最优值及粒子群的全局最优值进行速度和位置更新。2)微粒群-马尔科夫复合模型(PSO-Markov)是一种土地利用格局预测的新途径,生态空间的数量规模可以通过改进后的马尔科夫模型进行预测,生态空间的格局可以通过微粒群模型进行预测。3)微粒群-马尔科夫复合模型具有4个特点:第一、数量预测较为合理。第二、搜索范围大、较好地考虑到局部对全局的影响。第三、受问题维数变化影响小,在求解多目标问题时具有明显优势。第四、收敛时间短、运算速度快、易于实现。4)2020年,长株潭城市群的生态空间总体数量减少,其中林地和未利用地面积变化最明显,空间变化主要集中分布在西南部地区。生态空间总面积减小的主要原因是建设用地的扩张。因此,要控制城市群的人口密度,优化城市群生产—生活—生态的数量结构及空间布局,尤其要合理规划与利用城市建设用地,充分发挥水体与未利用地的生态价值,重点保护好生态源地、廊道及关键结点,构建结构合理、功能齐全的生态网络系统,提高系统的生态服务价值功能,要在规划的指导下合理调整城市群的城乡局部空间结构,保护生态环境,提高生境质量和景观多样性。这是今后一段时期面临的主要任务。     

Differences in time and space use between two sympatric Acrocephalus warblers with similar diets

Capsule: We found high diet overlap and different uses of space and time between Moustached Warblers Acrocephalus melanopogon and Reed Warblers Acrocephalus scirpaceus breeding in sympatry at a marshland in Spain.

Aims: To study the degree of diet overlap between both species, their space use on a local scale and their breeding phenologies.

Methods: We studied the breeding phenologies of the two species by standardized ringing activity. Spatial distribution was investigated by point counts. We determined diet composition from emetic samples and we collected invertebrates by standardized sweep-netting to estimate food availability.

Results: Diet and prey selection were similar among species. Conversely, spatial overlap was relatively small (<50%) and breeding phenologies were not synchronized. Both food availability and the overall abundance of the two species increased throughout the breeding season.

Conclusion: The two species are potential competitors for food and the observed differences in spatial and temporal niches may represent a way to lower competition for trophic resources: Moustached Warblers could reduce competition by breeding early, while Reed Warblers could avoid settling in areas occupied by the other species.     

Spatial scaling in a benthic population model with density-dependent disturbance.

This work investigates approaches to simplifying individual-based models in which the rate of disturbance depends on local densities. To this purpose, an individual-based model for a benthic population is developed that is both spatial and stochastic. With this model, three possible ways of approximating the dynamics of mean numbers are examined: a mean-field approximation that ignores space completely, a second-order approximation that represents spatial variation in terms of variances and covariances, and a patch-based approximation that retains information about the age structure of the patch population. Results show that space is important and that a temporal model relying on mean disturbance rates provides a poor approximation to the dynamics of mean numbers. It is possible, however, to represent relevant spatial variation with second-order moments, particularly when recruitment rates are low and/or when disturbances are large and weak. Even better approximations are obtained by retaining patch age information.     

Synaptic efficacy shapes resource limitations in working memory

Working memory (WM) is limited in its temporal length and capacity. Classic conceptions of WM capacity assume the system possesses a finite number of slots, but recent evidence suggests WM may be a continuous resource. Resource models typically assume there is no hard upper bound on the number of items that can be stored, but WM fidelity decreases with the number of items. We analyze a neural field model of multi-item WM that associates each item with the location of a bump in a finite spatial domain, considering items that span a one-dimensional continuous feature space. Our analysis relates the neural architecture of the network to accumulated errors and capacity limitations arising during the delay period of a multi-item WM task. Networks with stronger synapses support wider bumps that interact more, whereas networks with weaker synapses support narrower bumps that are more susceptible to noise perturbations. There is an optimal synaptic strength that both limits bump interaction events and the effects of noise perturbations. This optimum shifts to weaker synapses as the number of items stored in the network is increased. Our model not only provides a circuit-based explanation for WM capacity, but also speaks to how capacity relates to the arrangement of stored items in a feature space.     

Modeling alignment and movement of animals and cells

 Schools of fish and flocks of birds are examples for groups of individuals moving in a highly organized way. Individuals adapt their orientation and speed to that of their (nearest) neighbors. Adaptation of orientation can also be found on the cellular and subcellular level and is called alignment. A model for alignment and movement is derived on the basis of reaction transport equations in one space dimension. Existence of solutions is shown and long time behavior of the system is described. The effect of schooling on the risk of predation is investigated. Then the model is generalized to two space dimensions and compared to other models for alignment which do not incorporate individual movement in space. Received: 27 March 2001 / Revised version: 17 January 2002 / Published online: 23 August 2002