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.     

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     

Labyrinthine Turing pattern formation in the cerebral cortex

I propose that the labyrinthine patterns of the cortices of mammalian brains may be formed by a Turing instability of interacting axonal guidance species acting together with the mechanical strain imposed by the interconnecting axons.     

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.     

Labyrinthine versus straight-striped patterns generated by two-dimensional Turing systems

Striped patterns are often observed on fish skin. Such patterns have been accounted for by reaction-diffusion (RD) Turing-type models, in which two substances can spontaneously form a spatially heterogeneous pattern in a homogeneous field. Among the striped patterns generated by Turing-type models, some are "straight-striped patterns," with many stripes running in parallel, while others are "labyrinthine patterns," in which the stripes often change direction, merge with each other, and frequently branch out. RD models differ in terms of their tendency to generate either labyrinthine or straight-striped patterns. Here, we studied the conditions under which either a labyrinthine or straight-striped pattern would emerge. First, we defined an index for stripe clearness, Sh. Straight-striped patterns (large Sh) are formed if only a narrow range of spatial periods corresponds to an unstable mode. Labyrinthine patterns (small Sh) are formed when a wide range of spatial periods is unstable. More specifically, labyrinthine patterns are formed when the maximum spatial period of unstable modes is more than twice that of the minimum spatial period of unstable modes; otherwise, straight-striped patterns are formed. We then examined RD models with nonlinear reaction terms, including both activator-inhibitor and substrate-depletion models, and we demonstrated that the same conclusions hold with respect to the conditions required for labyrinthine versus straight-striped patterns.     

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.     

The description of spatial pattern using two-dimensional spectral analysis

Two-dimensional spectral analysis is a general interrogative technique for describing spatial patterns. Not only is it able to detect all possible scales of pattern which can be present in the data but it is also sensitive to directional components. Four functions are described: the autocorrelation function; the periodogram; and, the R- and Θ-spectra which respectively summarize the periodogram in terms of scale and directional components of pattern. The use of these functions is illustrated by their application to a simple wave pattern, a wave pattern with added noise, and patterns simulating competition and invasion processes.     

Dynamics of pattern formation in biomimetic systems

This paper is an attempt to conceptualize pattern formation in self-organizing systems and, in particular, to understand how structures, oscillations or waves arise in a steady and homogenous environment, a phenomenon called symmetry breaking. The route followed to develop these ideas was to couple chemical oscillations produced by Belousov-Zhabotinsky reaction with confined reaction environments, the latter being an essential requirement for any process of Life. Special focus was placed on systems showing organic or lipidic compartments, which represent more reliable biomimetic matrices.     

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     

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.     

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).     

One-dimensional daisyworld: spatial interactions and pattern formation

The zero-dimensional daisyworld model of Watson and Lovelock (1983) demonstrates that life can unconsciously regulate a global environment. Here that model is extended to one dimension, incorporating a distribution of incoming solar radiation and diffusion of heat consistent with a spherical planet. Global regulatory properties of the original model are retained. The daisy populations are initially restricted to hospitable regions of the surface but exert both global and local feedback to increase this habitable area, eventually colonizing the whole surface. The introduction of heat diffusion destabilizes the coexistence equilibrium of the two daisy types. In response, a striped pattern consisting of blocks of all black or all white daisies emerges. There are two mechanisms behind this pattern formation. Both are connected to the stability of the system and an overview of the mathematics involved is presented. Numerical experiments show that this pattern is globally determined. Perturbations in one region have an impact over the whole surface but the regulatory properties of the system are not compromised by transient perturbations. The relevance of these results to the Earth and the wider climate modelling field is discussed.     

Stabilization through spatial pattern formation in metapopulations with long-range dispersal

Many studies of metapopulation models assume that spatially extended populations occupy a network of identical habitat patches, each coupled to its nearest neighbouring patches by density-independent dispersal. Much previous work has focused on the temporal stability of spatially homogeneous equilibrium states of the metapopulation, and one of the main predictions of such models is that the stability of equilibrium states in the local patches in the absence of migration determines the stability of spatially homogeneous equilibrium states of the whole metapopulation when migration is added. Here, we present classes of examples in which deviations from the usual assumptions lead to different predictions. In particular, heterogeneity in local habitat quality in combination with long-range dispersal can induce a stable equilibrium for the metapopulation dynamics, even when within-patch processes would produce very complex behaviour in each patch in the absence of migration. Thus, when spatially homogeneous equilibria become unstable, the system can often shift to a different, spatially inhomogeneous steady state. This new global equilibrium is characterized by a standing spatial wave of population abundances. Such standing spatial waves can also be observed in metapopulations consisting of identical habitat patches, i.e. without heterogeneity in patch quality, provided that dispersal is density dependent. Spatial pattern formation after destabilization of spatially homogeneous equilibrium states is well known in reaction–diffusion systems and has been observed in various ecological models. However, these models typically require the presence of at least two species, e.g. a predator and a prey. Our results imply that stabilization through spatial pattern formation can also occur in single-species models. However, the opposite effect of destabilization can also occur: if dispersal is short range, and if there is heterogeneity in patch quality, then the metapopulation dynamics can be chaotic despite the patches having stable equilibrium dynamics when isolated. We conclude that more general metapopulation models than those commonly studied are necessary to fully understand how spatial structure can affect spatial and temporal variation in population abundance.     

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.     

On modelling pattern formation by activator-inhibitor systems

Journal of Mathematical Biology - The formation of spatially patterned structures in biological organisms has been modelled in recent years by various mechanisms, including pairs of...     

Habitat choice in predator-prey systems: spatial instability due to interacting adaptive movements

The role of habitat choice behavior in the dynamics of predator-prey systems is explored using simple mathematical models. The models assume a three-species food chain in which each population is distributed across two or more habitats. The predator and prey adjust their locations dynamically to maximize individual per capita growth, while the prey's resource has a low rate of random movement. The two consumer species have Type II functional responses. For many parameter sets, the populations cycle, with predator and prey chasing each other back and forth between habitats. The cycles are driven by the aggregation of prey, which is advantageous because the predator's saturating functional response induces a short-term positive density dependence in prey fitness. The advantage of aggregation in a patch is only temporary because resources are depleted and predators move to or reproduce faster in the habitat with the largest number of prey, perpetuating the cycle. Such spatial cycling can stabilize population densities and qualitatively change the responses of population densities to environmental perturbations. These models show that the coupled processes of moving to habitats with higher fitness in predator and prey may often fail to produce ideal free distributions across habitats.     

Type specification and spatial pattern formation of floral organs: A dynamic development model

A mathematical model simulating spatial pattern formation (positioning) of floral organs is proposed. Computer experiment with this model demonstrated the following sequence of spatial pattern formation in a typical cruciferous flower: medial sepals, carpels, lateral sepals, long stamens, petals, and short stamens. The positioning was acropetal for the perianth organs and basipetal for the stamens and carpels. Organ type specification and positioning proceed non-simultaneously in different floral parts and organ type specification goes ahead of organ spatial pattern formation. Computer simulation of flower development in several mutants demonstrated that the AG and AP2 genes determine both organ type specification and formation of the zones for future organ development. The function of the AG gene is to determine the basipetal patterning zones for the development of the reproductive organs, while the AP2 gene maintains proliferative activity of the meristem establishing the acropetal patterning zone for the development of the perianth organs.