Pattern formation in morphogenesis

We first treat the Gierer-Meinhardt equations by linear stability analysis to determine the critical parameter, at which the homogeneous distributions of activator and inhibitor concentrations become unstable. We find two types of instabilities: one leading to spatial pattern formation and another one leading to temporal oscillations. We consider the case where two instabilities are present. Using the method of generalized Ginzburg-Landau equations introduced earlier we then analyze the nonlinear equations. As we are mainly interested in spatial pattern formation on a sphere we consider the problem under an appropriate constraint. Combining the two occurring solutions we find patterns well-known in biology, such as a gradient system and temporal oscillations.     

Homogenization techniques for population dynamics in strongly heterogeneous landscapes

An important problem in spatial ecology is to understand how population-scale patterns emerge from individual-level birth, death, and movement processes. These processes, which depend on local landscape characteristics, vary spatially and may exhibit sharp transitions through behavioural responses to habitat edges, leading to discontinuous population densities. Such systems can be modelled using reaction–diffusion equations with interface conditions that capture local behaviour at patch boundaries. In this work we develop a novel homogenization technique to approximate the large-scale dynamics of the system. We illustrate our approach, which also generalizes to multiple species, with an example of logistic growth within a periodic environment. We find that population persistence and the large-scale population carrying capacity is influenced by patch residence times that depend on patch preference, as well as movement rates in adjacent patches. The forms of the homogenized coefficients yield key theoretical insights into how large-scale dynamics arise from the small-scale features.     

Travelling wave dynamics and self-organization in a spatio-temporally structured population

We analyse spatial population dynamics showing that periodic or period-like chaotic dynamics produce self-organization structures, such as travelling waves. We suggest that self-organized patterns are associated with spatial synchrony patterns that often depend on geographical distance between subpopulations. The population dynamics also show statistical spatial autocorrelation patterns. We contrast our theoretical simulations with empirical data on annual damages in young sapling stands caused by voles. We conclude, on the basis of the periodicity, synchrony, and spatial autocorrelation patterns, and our simulation results, that vole dynamics represent travelling waves in population dynamics. We suggest that because such synchrony patterns are frequently observed in natural populations, spatial self-organization may be more common in population dynamics than reported in the literature.     

Nonparametric One‐way Analysis of Variance of Replicated Bivariate Spatial Point Patterns

A common problem in neuropathological studies is to assess the spatial patterning of cells on tissue sections and to compare spatial patterning between disorder groups. For a single cell type, the cell positions constitute a univariate point process and interest focuses on the degree of spatial aggregation. For two different cell types, the cell positions constitute a bivariate point process and the degree of spatial interaction between the cell types is of interest. We discuss the problem of analysing univariate and bivariate spatial point patterns in the one‐way design where cell patterns have been obtained for groups of subjects. A bootstrapping procedure to perform a nonparametric one‐way analysis of variance of the spatial aggregation of a univariate point process has been suggested by Diggle, Lange and Bene? (1991). We extend their replication‐based approach to allow the comparison of the spatial interaction of two cell types between groups, to include planned comparisons (contrasts) and to assess whole groups against complete spatial randomness and spatial independence. We also accommodate several replicate tissue sections per subject. An advantage of our approach is that it can be applied when processes are not stationary, a common problem in brain tissue sections since neurons are arranged in cortical layers. We illustrate our methods by applying them to a neuropathological study to investigate abnormalities in the functional relationship between neurons and astrocytes in HIV associated dementia. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Subbarrel Patterns in Somatosensory Cortical Barrels Can Emerge from Local Dynamic Instabilities

Complex spatial patterning, common in the brain as well as in other biological systems, can emerge as a result of dynamic interactions that occur locally within developing structures. In the rodent somatosensory cortex, groups of neurons called “barrels” correspond to individual whiskers on the contralateral face. Barrels themselves often contain subbarrels organized into one of a few characteristic patterns. Here we demonstrate that similar patterns can be simulated by means of local growth-promoting and growth-retarding interactions within the circular domains of single barrels. The model correctly predicts that larger barrels contain more spatially complex subbarrel patterns, suggesting that the development of barrels and of the patterns within them may be understood in terms of some relatively simple dynamic processes. We also simulate the full nonlinear equations to demonstrate the predictive value of our linear analysis. Finally, we show that the pattern formation is robust with respect to the geometry of the barrel by simulating patterns on a realistically shaped barrel domain. This work shows how simple pattern forming mechanisms can explain neural wiring both qualitatively and quantitatively even in complex and irregular domains.     

The interplay between facilitation and habitat type drives spatial vegetation patterns in global drylands

The spatial configuration of vascular vegetation has been linked to variations in land degradation and ecosystem functioning in drylands. However, most studies on spatial patterns conducted to date have focused on a single or a few study sites within a particular region, specific vegetation types, or in landscapes characterized by a certain type of spatial patterns. Therefore, little is known on the general typology and distribution of plant spatial patterns in drylands worldwide, and on the relative importance of biotic and abiotic factors as predictors of their variations across geographical regions and habitat types. We analyzed 115 dryland plant communities from all continents except Antarctica to: 1) investigate the general typology of spatial patterns, and 2) assess the relative importance of biotic (plant cover, frequency of facilitation, soil amelioration, height of the dominant species) and abiotic (aridity, rainfall seasonality and sand content) factors as predictors of spatial patterns (median patch size, shape of patch‐size distribution and regularity) across contrasting habitat types (shrublands and grasslands). Precipitation during the warmest period and sand content were particularly strong predictors of plant spatial patterns in grasslands and shrublands, respectively. Facilitation associated with power‐law like and irregular spatial patterns in both shrublands and grasslands, although it was mediated by different mechanisms (respectively soil ammelioration and percentage of facilitated species). The importance of biotic attributes as predictors of the shape of patch‐size distributions declined with aridity in both habitats, leading to the emergence of more regular patterns under the most arid conditions. Our results expand our knowledge about patch formation in drylands and the habitat‐dependency of their drivers. They also highlight different ways in which facilitation affects ecosystem structure through the formation of plant spatial patterns.     

Bistability and regular spatial patterns in arid ecosystems

A variety of patterns observed in ecosystems can be explained by resource–concentration mechanisms. A resource–concentration mechanism occurs when organisms increase the lateral flow of a resource toward them, leading to a local concentration of this resource and to its depletion from areas farther away. In resource–concentration systems, it has been proposed that certain spatial patterns could indicate proximity to discontinuous transitions where an ecosystem abruptly shifts from one stable state to another. Here, we test this hypothesis using a model of vegetation dynamics in arid ecosystems. In this model, a resource–concentration mechanism drives a positive feedback between vegetation and soil water availability. We derived the conditions leading to bistability and pattern formation. Our analysis revealed that bistability and regular pattern formation are linked in our model. This means that, when regular vegetation patterns occur, they indicate that the system is along a discontinuous transition to desertification. Yet, in real systems, only observing regular vegetation patterns without identifying the pattern-driving mechanism might not be enough to conclude that an ecosystem is along a discontinuous transition because similar patterns can emerge from different ecological mechanisms.     

The spatial correlation and interaction between manufacturing agglomeration and environmental pollution

Using statistical data from 285 cities in China, this paper studies the spatial correlation and interaction between manufacturing agglomeration and environmental pollution. Using a widely used spatial correlation index, bivariate Moran's I, we first estimate the spatial correlation between manufacturing agglomeration and environmental pollution. We show that there is significant spatial correlation between them, and distinct patterns of local spatial concentration are identified. Then, we use a spatial simultaneous equations (SSE) model to analyze the interaction between manufacturing agglomeration and environmental pollution. We show that manufacturing agglomeration aggravates environmental pollution, while environmental pollution restrains manufacturing agglomeration. In addition, manufacturing agglomeration and environmental pollution in any one city can be affected by manufacturing agglomeration and environmental pollution in surrounding cities through spatial spillover. Finally, we put forward specific suggestions based on the conclusions for more sustainable development.     

Uncovering Patterns of Inter-Urban Trip and Spatial Interaction from Social Media Check-In Data

The article revisits spatial interaction and distance decay from the perspective of human mobility patterns and spatially-embedded networks based on an empirical data set. We extract nationwide inter-urban movements in China from a check-in data set that covers half a million individuals within 370 cities to analyze the underlying patterns of trips and spatial interactions. By fitting the gravity model, we find that the observed spatial interactions are governed by a power law distance decay effect. The obtained gravity model also closely reproduces the exponential trip displacement distribution. The movement of an individual, however, may not obey the same distance decay effect, leading to an ecological fallacy. We also construct a spatial network where the edge weights denote the interaction strengths. The communities detected from the network are spatially cohesive and roughly consistent with province boundaries. We attribute this pattern to different distance decay parameters between intra-province and inter-province trips.     

空间尺度对地形异质性-物种多样性关系的影响:粒度和幅度

空间尺度是影响我们理解生态学格局和过程的关键因素。目前已有多种关于物种多样性分布格局形成机制的假说且研究者未达成共识,原因之一是空间尺度对物种多样性分布格局的环境影响因子的解释力和相对重要性有重要影响。地形异质性是物种多样性分布格局的重要影响因素。本文综述了在地形异质性-物种多样性关系的研究中,不同空间粒度和幅度对研究结果的影响,以及可能的原因。尽管已认识到地形异质性-物种多样性关系的空间尺度效应,但粒度和幅度的具体影响仍未有统一结论。当前物种多样性分布格局研究未能覆盖较完整的尺度变化梯度。未来对地形异质性-物种多样性关系的研究需要同时考虑幅度和粒度的影响。建议结合可靠的模型和统计分析方法开展多尺度格局比较分析,以进一步阐明研究尺度对地形异质性-物种多样性关系的影响以及地形异质性起主导作用的空间尺度。     

Using Moment Equations to Understand Stochastically Driven Spatial Pattern Formation in Ecological Systems

Spatial patterns in biological populations and the effect of spatial patterns on ecological interactions are central topics in mathematical ecology. Various approaches to modeling have been developed to enable us to understand spatial patterns ranging from plant distributions to plankton aggregation. We present a new approach to modeling spatial interactions by deriving approximations for the time evolution of the moments (mean and spatial covariance) of ensembles of distributions of organisms; the analysis is made possible by “moment closure,” neglecting higher-order spatial structure in the population. We use the growth and competition of plants in an explicitly spatial environment as a starting point for exploring the properties of second-order moment equations and comparing them to realizations of spatial stochastic models. We find that for a wide range of effective neighborhood sizes (each plant interacting with several to dozens of neighbors), the mean-covariance model provides a useful and analytically tractable approximation to the stochastic spatial model, and combines useful features of stochastic models and traditional reaction-diffusion-like models.     

Exploring the influence of shrubs on herbaceous communities in a Mediterranean climatic context of two spatial scales

Communities of plants determine nonrandom spatial patterns defined by the intervention of abiotic and biotic factors acting at different spatial scales. We consider the influence of shrubs as one of the most important factors (biotic) affecting these spatial patterns at microscale. The macroclimate could be considered one of the most important factors (abiotic) at regional scale. To study the role and the floristic implications of each factor on the global patterns of herbaceous communities, we have developed a stratified sampling design that integrates both micro and macroscale on a 100 Km-long transect (east–west) in western central Spain. The results suggest that macroclimate could be one of the most important factors in determining herbaceous spatial patterns. Moreover, shrubs create a microspatial environmental heterogeneity that could alter such global climate patterns, modifying the spatial affinities established among species. This implies that environmental heterogeneity related to microhabitat could play a key role in spatial patterns at broad spatial scales, and consequently in the dynamics of the distribution and establishment of herbaceous species.     

Distribution Patterns in the Native Vascular Flora of Iceland

The aim of our study was to reveal biogeographical patterns in the native vascular flora of Iceland and to define ecological factors responsible for these patterns. We analysed dataset of more than 500,000 records containing information on the occurrence of vascular plants. Analysis of ecological factors included climatic (derived from WORLDCLIM data), topographic (calculated from digital elevation model) and geological (bedrock characteristics) variables. Spherical k-means clustering and principal component analysis were used to detect biogeographical patterns and to study the factors responsible for them. We defined 10 biotic elements exhibiting different biogeographical patterns. We showed that climatic (temperature-related) and topographic variables were the most important factors contributing to the spatial patterns within the Icelandic vascular flora and that these patterns are almost completely independent of edaphic factors (bedrock type). Our study is the first one to analyse the biogeographical differentiation of the native vascular flora of Iceland.     

Distribution of Vascular Plant Species Richness Along an Elevational Gradient in the Dongling Mountains, Beijing, China

Quantifying spatial patterns of species richness and determining the processes that give rise to these patterns are core problems In blodlveralty theory. The aim of the present paper was to more accurately detect patterns of vascular species richness at different scales along altitudinal gradients in order to further our understanding of biodlverslty patterns and to facilitate studies on relationships between biodiversity and environmental factors. Species richness patterns of total vascular plants species, including trees, shrubs, and herbs, were measured along an altitudinal gradient on one transect on a shady slope in the Dongling Mountains, near Beijing,China. Direct gradient analysis, regression analysis, and geostatistics were applied to describe the spatial patterns of species richness. We found that total vascular species richness did not exhibit a linear pattern of change with altitude, although species groups with different ecological features showed strong elevational patterns different from total species richness. In addition to total vascular plants, analysis of trees, shrubs, and herbs demonstrated remarkable hierarchical structures of species richness with altitude (i.e. patchy structures at small scales and gradients at large scales). Species richness for trees and shrubs had similar spatial characteristics at different scales, but differed from herbs. These results indicated that species groups with similar ecological features exhibit similar biodlveraity patterns with altitude, and studies of biodiversity based on species groups with similar ecological properties or life forms would advance our understanding of variations in species diversity. Furthermore, the gradients or trends appeared to be due mainly to local variations in species richness means with altitude. We also found that the range of spatial scale dependencies of species richness for total vascular plants, trees, shrubs, and herbs was relatively large. Thus, to detect the relationships betweenspecies richness with environmental factors along altitudinal gradients, it was necessary to quantify the scale dependencies of environmental factors in the sampling design or when establishing non-linear models.     

Soliton behaviour in a bistable reaction diffusion model

We analyze a generic reaction-diffusion model that contains the important features of Turing systems and that has been extensively used in the past to model biological interesting patterns. This model presents various fixed points. Analysis of this model has been made in the past only in the case when there is only a single fixed point, and a phase diagram of all the possible instabilities shows that there is a place where a Turing-Hopf bifurcation occurs producing oscillating Turing patterns. In here we focus on the interesting situation of having several fixed points, particularly when one unstable point is in between two equally stable points. We show that the solutions of this bistable system are traveling front waves, or solitons. The predictions and results are tested by performing extensive numerical calculations in one and two dimensions. The dynamics of these solitons is governed by a well defined spatial scale, and collisions and interactions between solitons depend on this scale. In certain regions of parameter space the wave fronts can be stationary, forming a pattern resembling spatial chaos. The patterns in two dimensions are particularly interesting because they can present a coherent dynamics with pseudo spiral rotations that simulate the myocardial beat quite closely. We show that our simple model can produce complicated spatial patterns with many different properties, and could be used in applications in many different fields.      

An Expanded Notch-Delta Model Exhibiting Long-Range Patterning and Incorporating MicroRNA Regulation

Notch-Delta signaling is a fundamental cell-cell communication mechanism that governs the differentiation of many cell types. Most existing mathematical models of Notch-Delta signaling are based on a feedback loop between Notch and Delta leading to lateral inhibition of neighboring cells. These models result in a checkerboard spatial pattern whereby adjacent cells express opposing levels of Notch and Delta, leading to alternate cell fates. However, a growing body of biological evidence suggests that Notch-Delta signaling produces other patterns that are not checkerboard, and therefore a new model is needed. Here, we present an expanded Notch-Delta model that builds upon previous models, adding a local Notch activity gradient, which affects long-range patterning, and the activity of a regulatory microRNA. This model is motivated by our experiments in the ascidian Ciona intestinalis showing that the peripheral sensory neurons, whose specification is in part regulated by the coordinate activity of Notch-Delta signaling and the microRNA miR-124, exhibit a sparse spatial pattern whereby consecutive neurons may be spaced over a dozen cells apart. We perform rigorous stability and bifurcation analyses, and demonstrate that our model is able to accurately explain and reproduce the neuronal pattern in Ciona. Using Monte Carlo simulations of our model along with miR-124 transgene over-expression assays, we demonstrate that the activity of miR-124 can be incorporated into the Notch decay rate parameter of our model. Finally, we motivate the general applicability of our model to Notch-Delta signaling in other animals by providing evidence that microRNAs regulate Notch-Delta signaling in analogous cell types in other organisms, and by discussing evidence in other organisms of sparse spatial patterns in tissues where Notch-Delta signaling is active.     

Evolutionarily stable consumer home range size in relation to resource demography and consumer spatial organization

There is a large variation in home range size within species, yet few models relate that variation to demographic and life-history traits. We derive an approximate deterministic population dynamics model keeping track of spatial structure, via spatial moment equations, from an individual-based spatial consumer-resource model; where space-use of consumers resembles that of central place foragers. Using invasion analyses, we investigate how the evolutionarily stable home range size of the consumer depends on a number of ecological and behavioral traits of both the resource and the consumer. We show that any trait variation leading to a decreased overall resource production or an increased spatial segregation between consumer and resource acts to increase consumer home range size. In this way, we extend theoretical predictions on optimal territory size to a larger range of ecological scenarios where home ranges overlap and population dynamics feedbacks are possible. Consideration of spatial traits such as dispersal distances also generates new results: (1) consumer home range size decreases with increased resource dispersal distance, and (2) when consumer agonistic behavior is weak, more philopatric consumers have larger home ranges. Finally, our results emphasize the role of the spatial correlation between consumer and resource distributions in determining home range size, and suggest resource dispersion is less important.