Stripes,spots, or reversed spots in two-dimensional Turing systems

Two-dimensional Turing models can generate stationary striped patterns or spotted patterns, and are used to explain the body pattern formation of animals. We studied the effects of the choice of reaction terms on pattern selection, i.e., which pattern is likely to be formed. We examined in detail a model with linear reaction terms and additional constraint terms that confine two variables within a finite range. In the one-dimensional model, a periodic stationary pattern can be formed only when the activator level is constrained both from below and from above. In the two-dimensional model, the relative distance of the equilibrium level of the activator between the upper and lower limitations determines the pattern selection. Striped patterns are produced when the equilibrium is equally distant from the upper and the lower limitations, but spotted patterns are produced when the equilibrium is clearly closer to one than to the other of two limitations. We then examined models with nonlinear reaction terms, including both activator-inhibitor and activator-depletion substrate type models; we attempted to explain the pattern selection of these nonlinear models based on the results of linear models with constraints. The distribution of the activator level is skewed positively and negatively for spotted patterns and reversed spotted patterns, respectively. In contrast, the skew of the distribution of the activator level was close to zero in the case of striped patterns. This observation provides a heuristic argument of how the location of the equilibrium between the constraints leads to pattern selection.     

Photosynthetic rates in relation to leaf phosphorus content in pioneer versus climax tropical rainforest trees

In Guyana dense rainforest occurs on intensely weathered acid soils, low in soil phosphorus. To investigate whether low P availability limits photosynthesis of trees growing on these soils more than N does, leaf P and N content, and their relationship with the photosynthetic capacity (A _sat, mol CO₂ m^-2 s^-1) were studied for nine pioneer and climax tree species in a range of light climates. Light environment was described using hemispherical photographs. For both pioneer and climax species, leaf P content (r ²=0.71 and 0.23, respectively) is a more important determinant of A _sat than leaf N content (r ²=0.54 and 0.12, respectively). Pioneer species have a higher leaf P and N content than climax species. At similar P or N content, pioneers have a higher A _sat than climax species. The saplings studied had a relatively high A _sat, considering their low P concentration (15–30 mol P g^-1). All species studied had a constant leaf P and N concentration and photosynthetic capacity across light climates, because specific leaf mass (g m^-2) increased similarly with light availability. This acclimation to a change in light environment makes a possible limitation of A _sat by P or N independent of light environment.     

Modeling the Role of Diffusion Coefficients on Turing Instability in a Reaction-diffusion Prey-predator System

The paper is concerned with the effect of variable dispersal rates on Turing instability of a non-Lotka-Volterra reaction-diffusion system. In ecological applications, the dispersal rates of different species tends to oscillate in time. This oscillation is modeled by temporal variation in the diffusion coefficient with large as well as small periodicity. The case of large periodicity is analyzed using the theory of Floquet multipliers and that of the small periodicity by using Hill's equation. The effect of such variation on the resulting Turing space is studied. A comparative analysis of the Turing spaces with constant diffusivity and variable diffusivities is performed. Numerical simulations are carried out to support analytical findings.     

Transient dynamics and pattern formation: reactivity is necessary for Turing instabilities.

The theory of spatial pattern formation via Turing bifurcations - wherein an equilibrium of a nonlinear system is asymptotically stable in the absence of dispersal but unstable in the presence of dispersal - plays an important role in biology, chemistry and physics. It is an asymptotic theory, concerned with the long-term behavior of perturbations. In contrast, the concept of reactivity describes the short-term transient behavior of perturbations to an asymptotically stable equilibrium. In this article we show that there is a connection between these two seemingly disparate concepts. In particular, we show that reactivity is necessary for Turing instability in multispecies systems of reaction-diffusion equations, integrodifference equations, coupled map lattices, and systems of ordinary differential equations.     

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     

Weakly nonlinear stability analyses of one-dimensional Turing pattern formation in activator-inhibitor/immobilizer model systems

The development of one-dimensional Turing patterns characteristic of the chlorite-iodide-malonic acid/starch reaction as well as analogous Brussellator/immobilizer and Schnackenberg/immobilizer model systems is investigated by means of a weakly nonlinear stability analysis applied to the appropriately scaled governing equations. Then the theoretical predictions deduced from these pattern formation studies are compared with experimental evidence relevant to the Turing diffusive instabilities under examination in order to explain more fully the transition to such stationary symmetry-breaking spatial structures when the temperature or pool species concentrations vary.     

Pattern formation in generalized Turing systems

Turing's model of pattern formation has been extensively studied analytically and numerically, and there is recent experimental evidence that it may apply in certain chemical systems. The model is based on the assumption that all reacting species obey the same type of boundary condition pointwise on the boundary. We call these scalar boundary conditions. Here we study mixed or nonscalar boundary conditions, under which different species satisfy different boundary conditions at any point on the boundary, and show that qualitatively new phenomena arise in this case. For example, we show that there may be multiple solutions at arbitrarily small lengths under mixed boundary conditions, whereas the solution is unique under homogeneous scalar boundary conditions. Moreover, even when the same solution exists under scalar and mixed boundary conditions, its stability may be different in the two cases. We also show that mixed boundary conditions can reduce the sensitivity of patterns to domain changes.Supported in part by NIH Grant # GM29123     

Traveling wave solutions of a reaction diffusion model for competing pioneer and climax species

Presented is a reaction-diffusion model for the interaction of pioneer and climax species. For certain parameters the system exhibits bistability and traveling wave solutions. Specifically, we show that when the climax species diffuses at a slow rate there are traveling wave solutions which correspond to extinction waves of either the pioneer or climax species. A leading order analysis is used in the one-dimensional spatial case to estimate the wave speed sign that determines which species becomes extinct. Results of these analyses are then compared to numerical simulations of wave front propagation for the model on one and two-dimensional spatial domains. A simple mechanism for harvesting is also introduced.     

Propagation of Turing patterns in a plankton model

The paper is devoted to a reaction–diffusion system of equations describing phytoplankton and zooplankton distributions. Linear stability analysis of the model is carried out. Turing and Hopf stability boundaries are found. Emergence of two-dimensional spatial structures is illustrated by numerical simulations. Travelling waves between various stationary solutions are investigated. Transitions between homogeneous in space stationary solutions and Turing structures are studied.     

中国东北地带性顶极植被类型及其预测判别模型——动态地植物学说的继承与发展(Ⅰ)

本文根据刘慎谔的动态地植物学说原理,采用主成分分析法(PCA)和多组判别分析法(MGDA)研究了东北地区地带性顶极植被类型与气候指标的关系:1)以东北地区210个气象站的7个气象因子为变量,组成210×7矩阵,用PCA方法进行分类和排序,其结果明显划分7个地带性顶极植被类型,2)影响地带性顶极植被类型的形成和分布的气候因子,主要是热量和水分条件以及二者的组合状况;3)确定了7个地带性顶极植被类型的水热指数分布范围和特征;4)采用MGDA法建立了判别函数模型,预测了东北地区地带性顶极植被类型,其准确率可达90.32％。     

A power function for forest structure and regeneration pattern of pioneer and climax species in patch mosaic forests

The DBH-class distribution in natural deciduous broad-leaved forests was elucidated with a power function. A power function (y=ax ^b, y: stem density, x: represents DBH class, a and b: constants) fits the distribution better than an exponential function (y = a exp bx). The parameter b in the power function is approximately –2. This means that the natural forests studied have a patch-mosaic structure and that tree cohorts regenerate from gaps. Parameter a implies the number of juveniles, and b means size-dependent mortality. The value of –2 for parameter b means that when trees in a given DBH class double their DBH, the density of the size class should decrease by one-fourth. This phenomenon results from self-thinning and is caused by horizontal space competition among trees, called the `tile model'. The parameter describing DBH-class distribution for a forest with self-thinning patches should be approximately –2. I call this the `–2 power law' for DBH-class distribution. In a typical natural forest dominated by deciduous broadleaf tree species, trees are recognized as pioneer or climax species by the parameters describing their regeneration patterns. When I applied the power functional model to the DBH-class distribution of each dominant species, in pioneer species parameter a was high and b was less than –2 (markedly less than zero), suggesting that there are many juveniles, but mortality is high. On the other hand, in climax species parameter a was low value and the value of b was larger (negative, but closer to zero), suggesting that there are not many juveniles, but mortality is low. A power-function analysis of DBH-class distribution can be used to clarify the patch mosaic structure of a forest, and to clarify the regeneration pattern of pioneer and climax species by applying the function for each species.     

Temperature-mediated morphology changes during metamorphic climax in the African clawed frog, Xenopus laevis

(1) Investigations of the effect of temperature on body size are largely limited to the larval phase, with our understanding of the effect of temperature during metamorphic climax entirely restricted to the insects.

(2) Environmental temperature was manipulated only during metamorphosis in the aquatic amphibian Xenopus laevis.

(3) Lower temperatures during metamorphosis resulted in individuals with greater mass, head width and snout–vent length on the completion of metamorphosis.

(4) This suggests that temperatures experienced during the relatively short metamorphic phase will play an important part in determining the temperature–size relationship in amphibians.

Chemical morphogenesis: Turing patterns in an experimental chemical system

Patterns resulting from the sole interplay between reaction and diffusion are probably involved in certain stages of morphogenesis in biological systems, as initially proposed by Alan Turing. Self-organization phenomena of this type can only develop in nonlinear systems (i.e. involving positive and negative feedback loops) maintained far from equilibrium. We present Turing patterns experimentally observed in a chemical system. An oscillating chemical reaction, the CIMA reaction, is operated in an open spatial reactor designed in order to obtain a pure reaction-diffusion system. The two types of Turing patterns observed, hexagonal arrays of spots and parallel stripes, are characterized by an intrinsic wavelength. We identify the origin of the necessary difference of diffusivity between activator and inhibitor. We also describe a pattern growth mechanism by spot splitting that recalls cell division.     

Differential responses of antioxidant enzymes in pioneer and late-successional tropical tree species grown under sun and shade conditions

To investigate the ability of pioneer and late-successional species to adapt to a strong light environment in a reforestation area, we examined the activities of antioxidant enzymes in relation to photosystem II, chlorophyll a fluorescence and photosynthetic pigment concentration for eight tropical tree species grown under 100% (sun) and 10% (shade) sunlight irradiation. The pioneer (early-succession) species (PS) were Cecropia pachystachya, Croton urucurana, Croton floribundus and Schinus terebinthifolius. The non-pioneer (late succession) species (LS) were Hymenaea courbaril L. var. stilbocarpa, Esenbeckia leiocarpa, Cariniana legalis and Tabebuia roseo-alba. We observed a greater decline in the ratio of variable to maximum chlorophyll a fluorescence (F_v/F_m) under full sunlight irradiation in the late-successional species than in the pioneer species. The LS species most sensitive to high irradiance were C. legalis and H. courbaril. In LS species, chlorophyll a, chlorophyll b and total chlorophyll concentrations were higher in the shade-grown plants than in plants that developed under full sunlight, but in the PS species C. floribundus and C. pachystachya, we did not observe significant changes in chlorophyll content when grown in the two contrasting environments. The carotenoids/total chlorophyll ratio increased significantly when plants developed under high-sunlight irradiation, but this response was not observed in the PS species S.terebinthifolius and C. pachystachya. The improved performance of the pioneer species in high sunlight was accompanied by an increase in superoxide dismutase (SOD, EC 1.15.1.1) activity, though no light-dependent increase in the activity of ascorbate peroxidase (APX, EC 1.11.1.11) was observed. The activity of catalase (CAT, EC 1.11.1.6) was reduced by high irradiation in both pioneer and late-successional species. Our results show that pioneer species perform better under high-sunlight irradiation than late-successional species, as indicated by increased SOD activity and a higher F_v/F_m ratio. C. legalis was the LS species most susceptible to photoinhibition under full sunlight conditions. These results suggest that pioneer plants have more potential tolerance to photo-oxidative damage than late-successional species associated with the higher SOD activity found in pioneer species. Reduced photoinhibition in pioneer species probably results from their higher photosynthetic capacities, as has been observed in a previous survey carried out by our group.     

Oscillations in a model of repression with external control

A mathematical model for control by repression by an extracellular substance is developed, including diffusion and time delays. The model examines how active transport of a nutrient can produce either oscillatory or stable responses depending on a variety of parameters, such as diffusivity, cell size, or nutrient concentration. The system of equations for the mathematical model is reduced to a system of delay differential equations and linear Volterra equations. After linearizing these equations and forming the limiting Volterra equations, the resulting linear system no longer has any spatial dependence. Local stability analysis of the radially symmetric model shows that the system of equations can undergo Hopf bifurcations for certain parameter values, while other ranges of the parameters guarantee asymptotic stability. One numerical study shows that the model can exhibit intracellular biochemical oscillations with increasing extracellular concentrations of the nutrient, which suggests a possible trigger mechanism for morphogenesis.The work of this author was supported in part by NSF grants DMS-8603787 and DMS-8807360The work of this author was supported under the REU program of NSF by grant DMS-8807360The work of this author was supported under the REU program of NSF by grant DMS-8807360     

Turing instabilities in general systems

We present necessary and sufficient conditions on the stability matrix of a general n(≥2)-dimensional reaction-diffusion system which guarantee that its uniform steady state can undergo a Turing bifurcation. The necessary (kinetic) condition, requiring that the system be composed of an unstable (or activator) and a stable (or inhibitor) subsystem, and the sufficient condition of sufficiently rapid inhibitor diffusion relative to the activator subsystem are established in three theorems which form the core of our results. Given the possibility that the unstable (activator) subsystem involves several species (dimensions), we present a classification of the analytically deduced Turing bifurcations into p (1 ≤p≤ (n− 1)) different classes. For n = 3 dimensions we illustrate numerically that two types of steady Turing pattern arise in one spatial dimension in a generic reaction-diffusion system. The results confirm the validity of an earlier conjecture [12] and they also characterise the class of so-called strongly stable matrices for which only necessary conditions have been known before [23, 24]. One of the main consequences of the present work is that biological morphogens, which have so far been expected to be single chemical species [1–9], may instead be composed of two or more interacting species forming an unstable subsystem. Received: 21 September 1999 / Revised version: 21 June 2000 / Published online: 24 November 2000     

The role of trans-membrane signal transduction in turing-type cellular pattern formation

The Turing mechanism (Phil. Trans. R. Soc. B 237 (1952) 37) for the production of a broken spatial symmetry in an initially homogeneous system of reacting and diffusing substances has attracted much interest as a potential model for certain aspects of morphogenesis (Models of Biological Pattern Formation, Academic Press, London, 1982; Nature 376 (1995) 765) such as pre-patterning in the embryo. The two features necessary for the formation of Turing patterns are short-range autocatalysis and long-range inhibition (Kybernetik 12 (1972) 30) which usually only occur when the diffusion rate of the inhibitor is significantly greater than that of the activator. This observation has sometimes been used to cast doubt on applicability of the Turing mechanism to cellular patterning since many messenger molecules that diffuse between cells do so at more-or-less similar rates. Here we show that Turing-type patterns will be able to robustly form under a wide variety of realistic physiological conditions though plausible mechanisms of intra-cellular chemical communication without relying on differences in diffusion rates. In the mechanism we propose, reactions occur within cells. Signal transduction leads to the production of messenger molecules, which diffuse between cells at approximately equal rates, coupling the reactions occurring in different cells. These mechanisms also suggest how this process can be controlled in a rather precise way by the genetic machinery of the cell.     

Local vs. non-local interactions in population dynamics

In this work we examine two models of single-species dynamics which incorporate non-local effects. The emphasis is on the ability of these models to generate stable patterns. Global behavior of the bifurcating branches is also investigated.