Singular perturbation analysis of cAMP signalling in Dictyostelium discoideum aggregates

In this paper, we use singular perturbation methods to study the structure of travelling waves for some reaction-diffusion models obtained from the Martiel-Goldbeter and Goldbeter-Segel's models of cAMP signalling in Dictyostelium discoideum. As a consequence, we derive analytic formulae for quantities like wave speed, maximum concentration and other magnitudes in terms of the different biochemical constants that appear in the model.     

Geometry and spatial patterns in Polysphondylium pallidum

The formation of secondary sori in whorls of Polysphondylium pallidum provides an attractive model system for the study of symmetry breaking during morphogenesis. Tip-specific antibodies that permit detection of very early stages in this patterning process are available. We have found that the patterns of tip-specific antigen expression vary considerably depending on the size, shape, and developmental stage of the whorl. All of these patterns, however, are well explained by patterning models that rely on short-range autocatalysis and long-range inhibition, as exemplified by reaction-diffusion theories. In the context of reaction-diffusion, we discuss the possible effects of initial conditions, boundary conditions, and nonlinearities on the selection of patterns in P. pallidum whorls.     

The characteristics of epidemics and invasions with thresholds

In this paper we report the development of a highly efficient numerical method for determining the principal characteristics (velocity, leading edge width, and peak height) of spatial invasions or epidemics described by deterministic one-dimensiohal reaction-diffusion models whose dynamics include a threshold or Allee effect. We prove that this methodology produces the correct results for single-component models which are generalizations of the Fisher model, and then demonstrate by numerical experimentation that analogous methods work for a wide class of epidemic and invasion models including the S-I and S-E-I epidemic models and the Rosenzweig-McArthur predator-prey model. As examplary application of this approach we consider the atto-fox effect in the classic reaction-diffusion model of rabies in the European fox population and show that the appropriate threshold for this model is within an order of magnitude of the peak disease incidence and thus has potentially significant effects on epidemic properties. We then make a careful re-parameterisation of the model and show that the velocities calculated with realistic thresholds differ surprisingly little from those calculated from threshold-free models. We conclude that an appropriately thresholded reaction-diffusion model provides a robust representation of the initial epidemic wave and thus provides a sound basis on which to begin a properly mechanistic modelling enterprise aimed at understanding the long-term persistence of the disease.     

Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels

Reproduction-Dispersal equations, called reaction-diffusion equations in the physics literature, model the growth and spreading of biological species. Integro-Difference equations were introduced to address the shortcomings of this model, since the dispersal of invasive species is often more widespread than what the classical RD model predicts. In this paper, we extend the RD model, replacing the classical second derivative dispersal term by a fractional derivative of order 1相似文献   

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.     

Bistable waves for a class of cooperative reaction-diffusion systems

In this paper, we consider a class of coupled cooperative reaction-diffusion systems, in which one population (or subpopulation) diffuses while the other is sedentary. We use the shooting method to prove the existence of the bistable travelling wave, and then obtain its global attractivity with phase shift and uniqueness (up to translation) via the dynamical system approach. The results are applied to some specific examples of reaction-diffusion population models.     

Diffusion versus network models as descriptions for the spread of prion diseases in the brain

In this paper we will discuss different modeling approaches for the spread of prion diseases in the brain. Firstly, we will compare reaction-diffusion models with models of epidemic diseases on networks. The solutions of the resulting reaction-diffusion equations exhibit traveling wave behavior on a one-dimensional domain, and the wave speed can be estimated. The models can be tested for diffusion-driven (Turing) instability, which could present a possible mechanism for the formation of plaques. We also show that the reaction-diffusion systems are capable of reproducing experimental data on prion spread in the mouse visual system. Secondly, we study classical epidemic models on networks, and use these models to study the influence of the network topology on the disease progression.     

Reaction-diffusion pattern in shoot apical meristem of plants

A fundamental question in developmental biology is how spatial patterns are self-organized from homogeneous structures. In 1952, Turing proposed the reaction-diffusion model in order to explain this issue. Experimental evidence of reaction-diffusion patterns in living organisms was first provided by the pigmentation pattern on the skin of fishes in 1995. However, whether or not this mechanism plays an essential role in developmental events of living organisms remains elusive. Here we show that a reaction-diffusion model can successfully explain the shoot apical meristem (SAM) development of plants. SAM of plants resides in the top of each shoot and consists of a central zone (CZ) and a surrounding peripheral zone (PZ). SAM contains stem cells and continuously produces new organs throughout the lifespan. Molecular genetic studies using Arabidopsis thaliana revealed that the formation and maintenance of the SAM are essentially regulated by the feedback interaction between WUSHCEL (WUS) and CLAVATA (CLV). We developed a mathematical model of the SAM based on a reaction-diffusion dynamics of the WUS-CLV interaction, incorporating cell division and the spatial restriction of the dynamics. Our model explains the various SAM patterns observed in plants, for example, homeostatic control of SAM size in the wild type, enlarged or fasciated SAM in clv mutants, and initiation of ectopic secondary meristems from an initial flattened SAM in wus mutant. In addition, the model is supported by comparing its prediction with the expression pattern of WUS in the wus mutant. Furthermore, the model can account for many experimental results including reorganization processes caused by the CZ ablation and by incision through the meristem center. We thus conclude that the reaction-diffusion dynamics is probably indispensable for the SAM development of plants.     

Computational biology: Turing’s lessons in simplicity

Biophysical modeling of development started with Alan Turing. His two-morphogen reaction-diffusion model was a radical but powerful simplification. Despite its apparent limitations, the model captured real developmental processes that only recently have been validated at the molecular level in many systems. The precision and robustness of reaction-diffusion patterning, despite boundary condition-dependence, remain active areas of investigation in developmental biology.     

Theoretical analysis of mechanisms that generate the pigmentation pattern of animals

Mechanisms of animal skin pigment pattern formation have long been of interest to developmental and mathematical biologists. Although there has been a well-studied theoretical hypothesis-the reaction-diffusion system-that is able to reproduce the variety of skin patterns, a lack of molecular evidence has kept it just a hypothesis. In this review, we summarize the results of theoretical studies to date for researchers not familiar with their mathematical underpinnings, and we discuss future approaches that will more fully integrate mathematical models and experimental analyses.     

Discussion: Turing's theory of morphogenesis—Its influence on modelling biological pattern and form

The evolution of spatial pattern is a central issue in developmental biology. Turing's (Phil. Trans. R. Soc. Lond. B237, 37–72, 1952) chemical theory of morphogenesis is a seminal contribution. In this talk I give a personal and necessarily limited view of its impact on mathematical and developmental biology. I briefly describe some of the interesting mathematical aspects of Turing's reaction-diffusion mechanism and discuss some of the different models which Turing's vision inspired. The emphasis throughout is on the practical biological applications of the various theories.     

Formation and deformation of patterns through diffusion

Simulating various patterns exhibited on biological forms with mathematical models has become an important supplement to theoretical biology. Models based on a certain mechanism are intended to provide explanations to the formation of a basic pattern. However, in real phenomena, among a basic pattern there always exist some difference between any two individuals. Such differences are consequences of environmental factors posed during the developmental processes. These factors, such as temperature, affect the diffusion rates of corresponding morphogenes which, in turn, alter a basic pattern to certain extent. We provide, in this paper, a quantitative characterization of this effect for a class of reaction-diffusion models.Mathematically, we study the emergence of stationary patterns and their dependence on diffusion rates for this class of models (RD-equations) with no-flux boundary conditions. The results are generalized to systems with homogeneous Dirichlet boundary conditions when the kinetic terms are odd functions. Through an analysis of the phase dynamics, we show that the deformation of stationary patterns, as the diffusion rates change, is governed by the variation of certain plane curves in the phase space. A constructive proof is given which shows explicitly how to obtain such curves.Applications of this study are illustrated with three model examples. We use these models to explain the biological implications of the mathematical features we investigated. Results from computer simulations are presented and compared with physical patterns.     

On some theory of monostable and bistable pure birth-jump integro-differential equations

We study an integral-differential equation that models a pure birth-jump process, where birth and dispersal cannot be decoupled. A case has been made that these processes are more suitable for phenomena such as plant dynamics, fire propagation, and cancer cell dynamics. We contrast the dynamics of this equation with those of the classical reaction-diffusion equation, where the reaction term models either logistic growth or a strong Allee effect. Recent evidence of an Allee effect has been found in plant dynamics during the germination process (due to seed predation) but not in the generation of seeds. This motivates where the Allee effect is included in our model. We prove the global existence and uniqueness of solutions with bounded initial data and analyze some properties of the solutions. Additionally, we prove results related to the persistence or extinction of a species, which are analogous to those of the classical reaction-diffusion equation. A key finding is that in some cases a population which is initially below the Allee threshold in some area, even if small, will actually survive. This is in contrast to solutions of the classical reaction-diffusion with the same initial data. Another difference of note is the lack of regularization and an infinite number of discontinuous equilibrium solutions to the birth-jump model.     

Spatial heterogeneity in three species,plant-parasite-hyperparasite,systems

This paper addresses the question of how heterogeneity may evolve due to interactions between the dynamics and movement of three-species systems involving hosts, parasites and hyperparasites in homogeneous environments. The models are motivated by the spread of soil-borne parasites within plant populations, where the hyperparasite is used as a biological control agent but where patchiness in the distribution of the parasite occurs, even when environmental conditions are apparently homogeneous. However, the models are introduced in generic form as three-species reaction-diffusion systems so that they have broad applicability to a range of ecological systems. We establish necessary criteria for the occurrence of population-driven patterning via diffusion-driven instability. Sufficient conditions are obtained for restricted cases with no host movement. The criteria are similar to those for the well-documented two-species reaction-diffusion system, although more possibilities arise for spatial patterning with three species. In particular, temporally varying patterns, that may be responsible for the apparent drifting of hot-spots of disease and periodic occurrence of disease at a given location, are possible when three species interact. We propose that the criteria can be used to screen population interactions, to distinguish those that cannot cause patterning from those that may give rise to population-driven patterning. This establishes a basic dynamical ''landscape'' against which other perturbations, including environmentally driven variations, can be analysed and distinguished from population-driven patterns. By applying the theory to a specific model example for host-parasite-hyperparasite interactions both with and without host movement, we show directly how the evolution of spatial pattern is related to biologically meaningful parameters. In particular, we demonstrate that when there is strong density dependence limiting host growth, the pattern is stable over time, whereas with less stable underlying host growth, the pattern varies with time.     

Spreading Speeds in Slowly Oscillating Environments

In this paper, we derive exact asymptotic estimates of the spreading speeds of solutions of some reaction-diffusion models in periodic environments with very large periods. Contrarily to the other limiting case of rapidly oscillating environments, there was previously no explicit formula in the case of slowly oscillating environments. The knowledge of these two extremes permits to quantify the effect of environmental fragmentation on the spreading speeds. On the one hand, our analytical estimates and numerical simulations reveal speeds which are higher than expected for Shigesada–Kawasaki–Teramoto models with Fisher-KPP reaction terms in slowly oscillating environments. On the other hand, spreading speeds in very slowly oscillating environments are proved to be 0 in the case of models with strong Allee effects; such an unfavorable effect of aggregation is merely seen in reaction-diffusion models.     

Mixed-mode pattern in Doublefoot mutant mouse limb--Turing reaction-diffusion model on a growing domain during limb development

It has been suggested that the Turing reaction-diffusion model on a growing domain is applicable during limb development, but experimental evidence for this hypothesis has been lacking. In the present study, we found that in Doublefoot mutant mice, which have supernumerary digits due to overexpansion of the limb bud, thin digits exist in the proximal part of the hand or foot, which sometimes become normal abruptly at the distal part. We found that exactly the same behaviour can be reproduced by numerical simulation of the simplest possible Turing reaction-diffusion model on a growing domain. We analytically showed that this pattern is related to the saturation of activator kinetics in the model. Furthermore, we showed that a number of experimentally observed phenomena in this system can be explained within the context of a Turing reaction-diffusion model. Finally, we make some experimentally testable predictions.     

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     

Does reaction-diffusion support the duality of fragmentation effect?

There is a gap between single-species model predictions, and empirical studies, regarding the effect of habitat fragmentation per se, i.e., a process involving the breaking apart of habitat without loss of habitat. Empirical works indicate that fragmentation can have positive as well as negative effects, whereas, traditionally, single-species models predict a negative effect of fragmentation. Within the class of reaction-diffusion models, studies almost unanimously predict such a detrimental effect. In this paper, considering a single-species reaction-diffusion model with a removal – or similarly harvesting – term, in two dimensions, we find both positive and negative effects of fragmentation of the reserves, i.e., the protected regions where no removal occurs. Fragmented reserves lead to higher population sizes for time-constant removal terms. On the other hand, when the removal term is proportional to the population density, higher population sizes are obtained on aggregated reserves, but maximum yields are attained on fragmented configurations, and for intermediate harvesting intensities.     

Transition from polar to duplicate patterns

The existence of symmetric nonuniform solutions in nonlinear reaction-diffusion systems is examined. In the first part of the paper, we establish systematically the bifurcation diagram of small amplitude solutions in the vicinity of the two first bifurcation points. It is shown that:

1. The system can adopt a stable symmetric solution (basic wave number 2) if the value of the bifurcation parameter is changed or if the initial polar structure (basic wave number 1) is sufficiently perturbed.
2. This behavior is independent of the particular reaction-diffusion model proposed and of the number of intermediate components (?2) involved.

In the second part of the paper, analogies are established between the possibilities offered by the bifurcation diagrams, involving only the two first primary branches, and the observation that in the early development of different organisms, appropriate experimental manipulations may switch the normal (polar) developmental pattern to a duplicate structure.