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At least 19 recordsLinked to original sources
PubMed1997

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. Copyright 1997 Academic Press. Copyright 1997 Academic Press

Journal Article↗
PubMedDate not supplied

[Determination of type and spatial pattern formation of flower organs: dynamic model of development].

The mathematical model imitating floral organ spatial pattern formation (positioning) was developed. Computer experiments performed on its basis demonstrated that organ spatial pattern formation in typical crucifer flower occurred in successive order: medial sepals, carpels, lateral sepals, long stamens, petals and short stamens. The positioning was advanced in two directions, acropetally in the perianth and basipetally in the stamens and carpels. The organ type specifying and positioning take place non-simultaneously in different floral areas. The organ type specifying passed ahead of organ primordial spatial pattern formation. The modeling of flower development of several mutants demonstrated that arabidopsis genes AP2 and AG in addition to specifying floral organ types also determine the particular zones in the floral meristem for futur organ development. The AG gene controls the formation of basipetal patterning zones where the reproductive organs develop, AP2 maintains the proliferative activity in the floral meristem that form acropetal patterning zones where perianth organ develop.

Arabidopsis↗
PubMedDate not supplied

[Structure of flower in Arabidopsis thaliana: spatial pattern formation].

Morphological analysis of flowers was carried out in Arabidopsis thaliana wild type plants and agamous and apetala2 mutants. No direct substitution of organs takes place in the mutants, since the number and position of organs in them do not correspond to the structure of wild type flower. In order to explain these data, a notion of spatial pattern formation in the meristem was introduced, which preceded the processes of appearance of organ primordia and formation of organs. Zones of acropetal and basipetal spatial pattern formation in the flower of wild type plants were postulated. It was shown that the acropetal spatial pattern formation alone took place in agamous mutants and basipetal spatial pattern formation alone, in apetala2 mutants. Different variants of flower structure are interpreted as a result of changes in the volume of meristem (space) and order of spatial pattern formation (time).

AGAMOUS Protein, Arabidopsis↗
PubMed1980

Scale-invariance in reaction-diffusion models of spatial pattern formation.

We propose a reaction-diffusion model of spatial pattern formation whose solutions can exhibit scale-invariance over any desired range for suitable choices of parameters in the model. The model does not invoke preset polarity or any other ad hoc distinction between cells and provides a solution to the French flag problem without sources at the boundary. Furthermore, patterns other than the polar pattern that usually arises first in a growing one-dimensional system described by Turing's model can be obtained. Evidence is given that suggests that the model may apply in the slug stage of Dictyostelium discoideum.

Cell Differentiation↗
PubMed1981

Positional information and pattern formation.

Spatial patterns of cellular differentiation may arise from cells first being assigned a position, as in a coordinate system, and then interpreting the positional value that they have acquired. This interpretation will depend on their genetic constitution and developmental history. Different patterns may thus arise from similar positional fields. The specification of positional value may involve a positional signal, such as the concentration of a diffusible morphogen, but can also depend on how long the cells remain in a particular region, such as a progress zone. Positional values may also be acquired by direct transfer from one cell layer to another, as in directed embryonic induction. Positional value, unlike a positional signal, involves long-term memory, and can be regarded as a type of cell determination. Cells of the same differentiation class may have different positional values and may thus be non-equivalent. Evidence is presented for a signal providing positional information along the antero-posterior axis during chick limb development. This signal has properties similar to those of a diffusible morphogen.

Animals↗
PubMed1997

Spatial pattern formation in an insect host-parasitoid system

Spatial models in ecology predict that populations may form patchy distributions within continuous habitats, through strong predator-prey or host-parasitoid interactions combined with limited dispersal. Empirical support of these models is provided. Parasitoids emanating from a population outbreak of tussock moths (Orgyia vetusta) suppressed the growth of nearby experimental populations of the moth, while experimental populations farther away were able to grow. This result explains the observed localized nature of tussock moth outbreaks and illustrates how population distributions can be regulated by dynamic spatial processes.

Journal Article↗
PubMed2006

Long range dispersal and spatial pattern formation in biological invasions.

In this paper we explore the consequences of long distance dispersal in biological invasion processes through simulations using a recently developed cellular automaton model. We show that long distance dispersal generate characteristic spatial patterns with several stationary scale-invariant properties. In particular, the patterns display a main patch around the focus of spread, with a fractal border structure whose fractal dimension contains information about the main statistical properties of the dispersal mechanism. Our results are in agreement with field data of spread of invaders with long distance dispersal mechanisms.

Apocynaceae↗
PubMed1992

Cell lineages, developmental timing, and spatial pattern formation in embryos of free-living soil nematodes.

From soils of various origins we have isolated a number of nematode strains and cultured them on agar plates. We have analyzed their anatomy, reproduction, and particularly their pattern of embryogenesis. With respect to early cleavage we can define six different classes. The basic scheme of embryogenesis is similar in all strains but considerable differences were observed in detail. Embryogenesis is more than five times longer in the slowest strain than in the fastest. The following general correlation was found: The slower embryogenesis proceeds in a strain, the relatively earlier the cleavage of germline cells occurs. In the fastest strain the primordial germ cell P4 is present at the 24-cell stage, while in the slowest strain it is already generated in the 5-cell stage. We hypothesize that germline cleavages have to occur within a certain time limit to preserve germline quality. The typical reversal of cleavage polarity in the division of the germline cell P2 is absent in the slowest, on other grounds apparently more primitive strain. This results in an unusual spatial arrangement of cells transiently. However, prior to gastrulation as a consequence of compensatory cell migrations (which may indicate the necessity for cell interactions), the pattern becomes very similar to that in the other strains. We propose that a standard cellular configuration is required at the beginning of gastrulation to ensure normal further development. Early cell interactions might be necessary to achieve this standard pattern. In about half of the analyzed strains cellular structures can be marked with an antibody raised against germline-specific granules of Caenorhabditis elegans. Our results do not support the notion that the staining pattern for P granules is a useful indicator for phylogenetic relationship.

Animals↗
PubMed2002

A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves.

In this paper, we employ the novel application of a reaction-diffusion model on a growing domain to examine growth patterns of the ligaments of arcoid bivalves (marine molluscs) using realistic growth functions. Solving the equations via a novel use of the finite element method on a moving mesh, we show how a reaction-diffusion model can mimic a number of different ligament growth patterns with modest changes in the parameters. Our results imply the existence of a common mode of ligament pattern formation throughout the Arcoida. Consequently, arcoids that share a particular pattern cannot be assumed, on this basis alone, to share an immediate common ancestry. Strikingly different patterns within the set can easily be generated by the same developmental program. We further show how the model can be used to make quantitatively testable predictions with biological implications.

Animals↗
PubMed1996

Spatial pattern formation during aggregation of the slime mould Dictyostelium discoideum.

Stream formation and spiral wave behaviour during the aggregation of Dictyostelium discoideum (Dd) are studied in a model based on the Martiel-Goldbeter equations for cAMP relay, combined with chemotactic motion of Dd cells. The results show that stream formation occurs if the turnover rate of intracellular cAMP is increased. This increase in the turnover rate of cAMP[in] leads to a dependence of the speed of the cAMP wave on the cell density. We propose that this dependence of wave speed on cell density is the underlying mechanism for stream formation. Besides stream formation, increasing the turnover rate of cAMP[in] also results in a spiral wave period that decreases during aggregation, a phenomenon that is commonly observed in situ. Furthermore, the dependence of wave speed on cell density is measured empirically. The speed of the cAMP wave is found to decrease as the wave travels from high to low cell density. This indicates that in situ, wave speed does depend on cell density.

Animals↗
PubMed2005

Formation of spatially patterned colloidal photonic crystals through the control of capillary forces and template recognition.

We report the formation of microscopic patterns of substrate-supported, 3D planar colloidal crystals using physical confinement in conjunction with surfaces displaying predetermined binary patterns of hydropholicity. The formation process involves a primary self-assembly wherein nano- and microscale colloids order into a photonic fcc lattice via capillary interactions followed by a secondary template-induced crystal cleavage step. Following this method, arbitrary arrays of pattern elements, which preserve structural and orientational properties of the parent crystal, can be easily obtained.

Journal Article↗
PubMed2026

Pattern Formation in a Spatial Public Goods Dilemma due to Diffusive or Directed Motion.

The costly provision of public goods serves as a model problem for the evolution of cooperative behavior, presenting a social dilemma between the collective benefits of shared resources and the individual incentive to free-ride in resource production. The spatial structure of populations can also impact cooperation over public goods, as diffusion of public goods and intentional motion of individuals towards regions with greater resources can interact with population and public goods dynamics to produce heterogeneous patterns in the spatial distribution of strategies and resources. In this paper, we build off a model introduced by Young and Belmonte for the reaction dynamics of interacting individuals and an explicit public good, deriving a system of PDEs that describes the spatial profiles of strategies and the public good in the presence of both diffusive motion of individuals and resources and chemotaxis-like directed motion of individuals in response to gradients in the concentration of public goods. Through linear stability analysis, we show that spatial patterns in strategic and public goods profiles can emerge due to either Turing instability with high defector diffusivity or a directed-motion instability through strong sensitivity of cooperators towards increasing resource concentration. We further explore the emergent spatial patterns with a mix of weakly nonlinear stability analysis and numerical simulation, showing that, for a wide range of reaction parameters, diffusion-driven instability appears to increase cooperation and public goods across the spatial domain, while directed motion of cooperators towards public goods tends to decrease cooperation and environmental quality across the environment.

Models, Biological↗
PubMed2003

The formation of spatial patterns in social insects: from simple behaviours to complex structures.

Many of the collective activities performed by social insects result in the formation of complex spatio-temporal patterns. Without centralized control, workers are able to work together and collectively tackle tasks far beyond the abilities of any one individual. The resulting patterns produced by a colony are not explicitly coded at the individual level, but rather emerge from nonlinear interactions between individuals or between individuals and their environment. We present a few selected examples to illustrate some of the basic mechanisms used by social insects, such as templates, stigmergy and self-organization. These mechanisms can be used in combination to organize pattern formation at the colony level.

Adaptation, Physiological↗
PubMed2001

Fronts, Waves, and Stationary Patterns in Electrochemical Systems In the appendix we list the minimal equations (prototype equations) that describe the phenomena of self-organization discussed in the individual sections.

Oscillatory behavior has been observed for almost all electrochemical reactions in a certain, although sometimes small, range of external parameters. Only in the past ten years has it been possible, however, to find a common explanation for the occurrence of these temporal self-organization phenomena of chemically completely different electrochemical reactions. The breakthrough was achieved because new methods and concepts, which had been developed in nonlinear dynamics to describe the spontaneous formation of order in various disciplines, could be applied. This development in turn was only possible because the underlying laws are universal at a certain abstract level. Oscillations are only one possible manifestation of nonlinear behavior. Examples of other features that are often closely associated with temporal instabilities are spatial structures and waves. Initiated by the theoretical progress and the development of new experimental techniques, spatial pattern formation in electrochemical systems has been targeted for investigations in the past few years. Based on these investigations, it can be predicted under which conditions temporal or spatial pattern formation can be expected. Furthermore, the possibility of predicting the occurrence of instabilities indicates that it might be feasible to exploit nonlinear effects to increase, for example, the yield of electrocatalytic reactions. Here we discuss physicochemical mechanisms that lead to pattern formation in electrochemical systems. At the same time, we stress the generic principles that are responsible for self-structuring processes in many chemical and biological systems.

Journal Article↗
PubMed1982

Pattern formation in morphogenesis. Analytical treatment of the Gierer-Meinhardt model on a sphere.

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.

Animals↗