## No full-text available

To read the full-text of this research,

you can request a copy directly from the author.

It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue,
is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous,
may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random
disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically
convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is
found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear
on the ring. It is suggested that this might account, for instance, for the tentacle patterns onHydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account
for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested
that stationary waves in two dimensions could account for the phenomena of phyllotaxis.
The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure
of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical
laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics,
some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number
of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.

To read the full-text of this research,

you can request a copy directly from the author.

... The classical explanation of spatial pattern formation comes from Turing's pioneering work in reaction-diffusion (RD) system for chemical morphogenesis [12]. The framework of the diffusiondriven instability in Turing's work needs two basic criteria: an activator-inhibitor interaction structure of at least two interacting species, and an order of magnitude difference in their dispersal coefficient. ...

... Remark 1. From Theorem 5, we observe that E * is asymptotically stable for σ > σ S and unstable for σ < σ H independent of the diffusion parameter value d. Hence, the system (12) shows bistability between the trivial steady state E 0 and the coexisting steady state E * for σ > σ S . The system may show various non-constant stationary solutions for σ < σ S , which we discuss in section 4. ...

... and n 2 j=n 1 +1 dimE(k j ) is odd, then model (12) has at least one positive non-constant solution. ...

Allee effect in population dynamics has a major impact in suppressing the paradox of enrichment through global bifurcation, and it can generate highly complex dynamics. The influence of the reproductive Allee effect, incorporated in the prey's growth rate of a prey-predator model with Beddington-DeAngelis functional response, is investigated here. Preliminary local and global bifurcations are identified of the temporal model. Existence and non-existence of heterogeneous steady-state solutions of the spatio-temporal system are established for suitable ranges of parameter values. The spatio-temporal model satisfies Turing instability conditions, but numerical investigation reveals that the heterogeneous patterns corresponding to unstable Turing eigen modes acts as a transitory pattern. Inclusion of the reproductive Allee effect in the prey population has a destabilising effect on the coexistence equilibrium. For a range of parameter values, various branches of stationary solutions including mode-dependent Turing solutions and localized pattern solutions are identified using numerical bifurcation technique. The model is also capable to produce some complex dynamic patterns such as travelling wave, moving pulse solution, and spatio-temporal chaos for certain range of parameters and diffusivity along with appropriate choice of initial conditions Judicious choices of parametrization for the Beddington-DeAngelis functional response help us to infer about the resulting patterns for similar prey-predator models with Holling type-II functional response and ratio-dependent functional response.

... In the early 1950's Alan Turing proposed a framework for understanding the dynamics of endogenously generated spatial heterogeneity which has laid a foundation for ecologists to build upon (Turing 1952;Segel and Jackson 1972). Central to his framing was the idea of activatorinhibitor systems, where activators grow by themselves and inhibitors emerge to control the activators, both diffusing in a spatial context. ...

... When there is differential diffusion in space, typically with inhibitors diffusing faster than activators, pattern formation occurs. This process of unequal diffusion leading to pattern formation is sometimes referred to diffusive instability or Turing instability (Turing 1952). The qualitative elements of Turing's insight have provided a mechanistic framework that has been applied with great success in ecology (Alonso et al., 2002;Rietkerk and Van de Koppel 2008;Rietkerk et al., 2021) Although Turing's mechanism for the generation of spatial heterogeneity is quite broad, it has largely been successful in terms of describing intraspecific species interactions in pattern formation, with what has been called scale-dependent feedbacks (Rietkerk and Van de Koppel 2008;Pringle and Tarnita 2017). ...

... Turing and consumer-resource spatial patterns The Turing activator/inhibitor mechanism provides for a qualitative understanding of self-organization of spatial patterns (Turing 1952;Nijhout 2018), as applied to many areas of science from chemistry (Winfree & Strogatz 1984), to cosmology (Nozakura & Ikeuchi 1984), and in biology, from the cell to the ecosystem (Kondo & Miura 2010, Kefi et al. 2007). ...

Heterogeneity is a ubiquitous feature of ecosystems and perhaps an important contributing factor to the oft noted difficulties associated with making generalizations in community ecology. Our answers to questions regarding the origins and consequences of various types of heterogeneity in ecological systems have long been met with contingencies and context dependency, highlighting the need to continually revisit our organizing metaphors. The work presented in this dissertation is concerned with these metaphors and especially those associated with the ecological processes that generate spatial heterogeneity in ecosystems. In eleven case studies, we attempt to understand the generation and subsequent implications of spatial heterogeneity for the assembly and functioning of ecological communities in agroecosystems. We first address how ecological interactions create spatial pattern in Chapter 1 by presenting a novel demographic framework for understanding consumer-resource generated spatial patterns. We then explore how spatial heterogeneity influences ecological interactions in Chapter 2 and Chapter 3. Whereas in the former we ask how basic ecological interactions are influenced by dynamic patterns of heterogeneity in ecosystems, in the later we ask how changes in spatial structure influences pathogen epidemics. Chapter 4 then empirically explores how dispersal differentially alters community structure in leaf-litter metacommunities and Chapter 5 explores the use of coupled oscillators as a metaphor for ecological communities. These first five chapters represent an attempt to understand the feedbacks between ecological interactions that create spatial heterogeneity and how spatial heterogeneity structures ecological communities. The dissertation then shifts focus to a fungal pathogen of coffee, the coffee leaf rust, and uses its community of consumers as a model system to understand how spatial heterogeneity influences community structure and how community structure influences biological control of the pathogen. Chapter 6 gives a brief overview of the history and ecology of the pathogen and its community, and Chapters 7-9 explore the assembly and organization of these communities, highlighting their interactions with the pathogen as well as among themselves. Finally, Chapters 10 and 11 are concerned with the structure of interaction networks associated with the coffee leaf rust and the provisioning of top-down control of the coffee leaf rust pathogen in both Mexico and Puerto Rico. Taken together, this dissertation contributes to our understanding of how ecological communities create and are impacted by the heterogeneous environments they occupy. Furthermore, this work attempts to highlight the importance of such concepts in an agroecological context where questions of community structure and population regulation have the potential for practical significance.

... In 1952, Turing explained that the pattern on animal surfaces is due to diffusion-driven instability [1], the pattern is named Turing pattern and the instability is called Turing instability. Since then, numerous studies have focused on the theory of chemical and biological pattern formation [2][3][4][5][6][7]. ...

... Murray [2] gave a method for computing the param-B Yong Wang ywang@tjufe.edu.cn 1 eter domain for Turing instability, and showed that major differences between different diffusion mechanisms. Yadav et al. [6] obtained the conditions for the existence of Turing instability in activator-inhibitor systems and applied their conclusions to the Schnakenberg model and the Gray-Scott model. ...

... The connection is illustrated in Table 1. Therefore, we get the following results Theorem 4 Assume that p 2 > p 1 > 0, in the Turing instability region, varying the value of λ and fixing the other parameters, then (I) when μ 2 < μ < μ 3 , the system (31) only exists a stable E (π ) 3 , therefore, the system (1) only appear spot patterns, (II) when μ 3 < μ < μ 4 , the system (31) exists stable E 2 and E (π ) 3 , therefore, the system (1) will appear mixed patterns(coexistence of spot patterns and stripe patterns), (III) when μ 4 < μ, the system (31) only exists a stable E 2 , therefore, the system (1) only appears the stripe patterns. To make Theorem 4 more instinctive, we numerically show the results in Fig. 5, in which the red lines, the purple lines, and the green line together indicate μ concerning λ. ...

This paper considers a chemical reaction-diffusion model for studying pattern formation with the Sel’kov–Schnakenberg model. Firstly, the stability conditions of the positive equilibrium and the existing conditions of the Hopf bifurcation are established for the local system. Then, Turing instability (diffusion-driven), which causes the spatial pattern is investigated and the existing condition of the Turing bifurcation is obtained. In addition, the dynamic behaviors near the Turing bifurcation are also studied by employing the method of weakly nonlinear analysis. The theoretical analysis shows that spatio-temporal patterns change from the spot, mixed (spot-stripe) to stripe with the variation of parameters, which can be verified by a series of numerical simulations. These numerical simulations give a visual representation of the evolution of spatial patterns. Our results not only explain the evolution process of reactant concentration, but also reveal the mechanism of spatio-temporal patterns formation.

... The theory of pattern formation was first explored mathematically in 1952 by Alan Turing in his seminal work. 10 In 1958, spontaneous, dynamic, and oscillating circular and spiral chemical patterns were first observed experimentally, famously known as, Belousov−Zhabotinsky (BZ) reactions. 11,12 In 1972, Gierer and Meinhardt 13 developed a theoretical model of the Activator−Inhibitor system, based on Turing's theory of 1952, 10 and also devised it experimentally. ...

... 10 In 1958, spontaneous, dynamic, and oscillating circular and spiral chemical patterns were first observed experimentally, famously known as, Belousov−Zhabotinsky (BZ) reactions. 11,12 In 1972, Gierer and Meinhardt 13 developed a theoretical model of the Activator−Inhibitor system, based on Turing's theory of 1952, 10 and also devised it experimentally. 14−16 In this two-component system an activator triggers the inhibitor while the inhibitor represses the activator in a long-range way. ...

... Other theoretical models 17−19 also generate similar patterns thus, augmenting that mathematical models like Turing's drive development of theory for biological systems. 10 Mathematical models incorporating diffusion of only one of the two components have already been developed. 20,21 Moreover, experiments have been designed in various contexts of a single diffusing molecule− 22,23 spatial arrangement and manipulation of inducers, 24,25 and quorum sensing molecules, 26,27 highlighting the importance of the Turing model in exploring pattern formation in many systems. ...

Spatiotemporal pattern formation plays a key role in various biological phenomena including embryogenesis and neural network formation. Though the reaction-diffusion systems enabling pattern formation have been studied phenomenologically, the biomolecular mechanisms behind these processes have not been modeled in detail. Here, we study the emergence of spatiotemporal patterns due to simple, synthetic and commonly observed two- and three-node gene regulatory network motifs coupled with their molecular diffusion in one- and two-dimensional space. We investigate the patterns formed due to the coupling of inherent multistable and oscillatory behavior of the toggle switch, toggle switch with double self-activation, toggle triad, and repressilator with the effect of spatial diffusion of these molecules. We probe multiple parameter regimes corresponding to different regions of stability (monostable, multistable, oscillatory) and assess the impact of varying diffusion coefficients. This analysis offers valuable insights into the design principles of pattern formation facilitated by these network motifs, and it suggests the mechanistic underpinnings of biological pattern formation.

... Alan M. Turing's last published work and some posthumously published manuscripts were dedicated to the development of his theory of organic pattern formation. In his 1952 "The Chemical Basis of Morphogenesis" [1], Turing provided an elaborated mathematical formulation of the theory of the origins of biological form first proposed in 1917 and 1942 by Sir D'Arcy Wendworth Thompson in On Growth and Form [2]. In this influential work in developmental biology, Thompson highlighted the complexity of organic forms and their accessibility to mathematical descriptions while playing down the importance of mechanisms of natural selection. ...

... His approach found a considerable degree of adoption in developmental biology. Some authors claim that these accomplishments made Turing [1] "one of the most influential theoretical papers ever written in developmental biology" ( [12], p. 183), and that "Turing seems to have identified one of nature's general mechanisms for generating order from macroscopic uniformity and microscopic disorder." ( [13], p. 9). ...

... Not even the laws of genetics-if they had been fully understood at the time-or other parochially biological laws needed to be invoked. More specifically and positively, the transformations involved in morphogenetic processes can be described by recourse to changes of velocity and position as described by Newton's laws of motion, to elasticities, osmotic pressures, and diffusion reactions ( [1], p. [37][38]. ...

Alan M. Turing’s last published work and some posthumously published manuscripts were dedicated to the development of his theory of organic pattern formation. In “The Chemical Basis of Morphogenesis” (1952), he provided an elaborated mathematical formulation of the theory of the origins of biological form that had been first proposed by Sir D’Arcy Wendworth Thompson in On Growth and Form (1917/1942). While arguably his most mathematically detailed and his systematically most ambitious effort, Turing’s morphogenetical writings also form the most thematically self-contained and least philosophically explored part of his work. We dedicate our inquiry to the reasons and the implications of Turing’s choice of biological topic and viewpoint. We will probe for possible factors in Turing’s choice that go beyond availability and acquaintance with On Growth and Form. On these grounds, we will explore how and to what extent his theory of morphogenesis actually ties in with his concept of mechanistic computation. Notably, Thompson’s pioneering work in biological ‘structuralism’ was organicist in outlook and explicitly critical of the Darwinian approaches that were popular with Turing’s cyberneticist contemporaries—and partly used by Turing himself in his proto-connectionist models of learning. Resolving this apparent dichotomy, we demonstrate how Turing’s quest for mechanistic explanations of how organisation emerges in nature leaves room for a non-mechanist view of nature.

... Turing pattern induced by diffusion is ubiquitous and used to explain the biological mechanisms [13], which has become an important direction of nonlinear dynamics [14][15][16][17][18][19][20][21]. With the development of network dynamics, more attention has been paid to the network-organized system, where the network nodes represent the interacting species points is analyzed. ...

... When 1 = 0, one has 2 + 11 + ( 12 + 13 ) − + 14 = 0, (8), we obtain − 2 + cos( ) 13 + sin( ) 12 + 14 + ( 11 + cos( ) 12 − sin( ) 13 ) = 0, where = , and we know − 2 + cos( ) 13 + sin( ) 12 + 14 = 0, 11 + cos( ) 12 − sin( ) 13 = 0, namely cos( ) 13 + sin( ) 12 = 2 − 14 , cos( ) 12 − sin( ) 13 ...

In this paper, we show the impact of both network and time delays on Turing instability and demarcate the role of diffusion in the epidemic. The stability and bifurcation of equilibrium points are analyzed to reveal the epidemic state, which is the precondition of pattern formation. The network could lead to the transition from the endemic to the periodic outbreak via negative wavenumber, which provides a way to prevent significant harm or decrease the damage of the epidemic to humans by the delay, the connection rate, and the infection rate. Also, the threshold value of time delay is proportional to the minimum eigenvalue of the network matrix, which provides a way to control the periodic behavior. Finally, numerical simulations validate these analytical results and the mechanisms of frequent outbreaks.

... To avoid ambiguities in my use of Turing's concepts of discreteness and continuity it is better to explain the meaning in this context of the adjective "continuous": Turing himself understands that the nervous system is surely not a DSM (Discrete-State Machine) that is, in modern terminology, the brain rather could be better described as a kind of dynamical system that is sensitive to initial or limit conditions, more complex than any n-body physical system or turbulent stream. Turing calls a systems such as this one "continuous" both in his 1950 article (Turing 1950) and in his 1952 (Turing 1952) article on morphogenesis (Longo 2009a p. 380). 4 Moreover, to also better clarify the concept of discreteness, we have to say that since Turing computer science suggests a perspective in which physical artefactual entities are "domesticated" to process the organization of information and knowledge into little boxes, bits, and pixels, which present discrete precision (each datum is well separated and accessible and each measure exact, in contrast to what happens for all-classical-physical processes), exactly, with no fuzziness and no contingency. ...

... Finally, it is important to remember again that, in 1952, though, Turing too dealt with the problems of morphology (Turing 1952). He indeed published an article on morphogenesis which presented a very original, but non computational, non-linear system of action-reaction and dynamic diffusion, in which he proposed what he called a model of the physical phenomenon in question. ...

Locked and unlocked strategies are at the center of this article as ways of shedding new light on the cognitive aspects of deep learning machines. The character and the role of these cognitive strategies, which are occurring both in humans and in computational machines, is indeed strictly related to the generation of cognitive outputs, which range from weak to strong level of knowledge creativity. The author maintains that these differences lead to important consequences when we analyze computational AI programs, such as AlphaGo, which aim at performing various kinds of abductive hypothetical reasoning. In these cases, the programs are characterized by locked abductive strategies: they deal with weak (even if sometimes amazing) kinds of hypothetical creative reasoning because they are limited in eco-cognitive openness, which instead qualifies human thinkers who are performing higher kinds of abductive creative reasoning, where cognitive strategies are instead unlocked.

... British mathematician Alan Turing was perhaps the first modern scientist to formulate the concept of complexity in a hallmark paper he wrote in the early 1940s about the growth of biological systems. In it, he put forward the idea of "morphogenesis" (Turing, 1952). He showed that a biological system described by two simple equations with feedback loops among the variables was capable of complexly patterned and totally unpredictable behavior. ...

... While patterns were easy to generate via computers, describing the patterned behavior of such system mathematically was exceedingly difficult. Whereas Turing (1952) had used two simple equations with feedback interaction to describe the growth of biological systems, Mandelbrot (1982) used only one such equation to generate similar unpredictable, infinitely complex patterns. Mandelbrot and Hudson (2004) also made an equally significant contribution to finance in their contention that many financial series do not validate the Gaussian (normal) probability distribution underlying the Brownian motion usually assumed by financial market analysts. ...

Evidence has been mounting that the interest-based debt financing regime is under increasing distress. Evidence also suggests that financial crises—despite the various labels assigned to them: exchange rate crisis or banking crisis—have been debt crises in essence. At present, data suggest that the debt-to-GDP ratio of the richest members of the G-20 is expected to reach the 120% mark by 2014. There is also evidence that, out of securities worth US$ 200 trillion in the global economy, no less than three-fourths represent interest-based debt. It is difficult to see how this massive debt volume can be validated by the underlying productive capacity of the global economy. This picture becomes more alarming considering the anemic state of global economic growth. There is great uncertainty with regard to interest rates. Although policy-driven interest rates are near zero, there is no assurance that they will not rise as the risk and inflation premiums become significant. Hence, a more serious financial crisis may be in the offing and a general collapse of asset prices may occur. This paper argues that the survival of the interestbased debt regime is becoming less tenable, as is the process of financialization that has accompanied the growth of global finance over the last four decades. It further argues that Islamic finance, with its core characteristic of risk sharing, may well be a viable alternative to the present interest-based debt financing regime.

... If these diffusible molecules couple in a Turing-like Activator (A)-Repressor (B) form, such that A activates the production of B, and B limits the production of A, concentration profiles of these molecules can self-organize within their spatial domain (Turing 1952). Although these models can simulate conditions for spontaneous symmetry breaking (Ishihara and Tanaka 2018;Sozen, Cornwall-Scoones, and Zernicka-Goetz 2021), they have so far only been used to explain the formation of the observed centro-symmetrical patterns in in-vitro adherent PSC colonies (Tewary et al. 2017;Brassard and Lutolf 2019;Fattah et al. 2021;Kaul et al. 2022) and to mimic polarized patterns under asymmetric induction conditions in micro-fluidic systems (Manfrin et al. 2019). ...

... ;https://doi.org/10.1101https://doi.org/10. /2022 We thus developed a minimal 2-component reaction-diffusion system of the Turing-like activator-repressor type (Turing 1952;Werner et al. 2015) with reactive boundary conditions (Dillon, Maini, and Othmer 1994;Erban and Chapman 2007), where we represent the influence of the activating signal by applying signal-intensity-dependent flux at the colony boundary (see Eqs. 1-4). Under certain parametric conditions, this model can self-organize concentration patterns of the activator and repressor molecules. ...

The emergence of the anterior-posterior body axis during early gastrulation constitutes a symmetry-breaking event, which is key to the development of bilateral organisms, and its mechanism remains poorly understood. Two-dimensional gastruloids constitute a simple and robust framework to study early developmental events in vitro. Although spontaneous symmetry breaking has been observed in three dimensional (3D) gastruloids, the mechanisms behind this phenomenon are poorly understood. We thus set out to explore whether a controllable 2D system could be used to reveal the mechanisms behind the emergence of asymmetry in patterned cellular structures. We first computationally simulated the emergence of organization in micro-patterned mouse pluripotent stem cell (mPSC) colonies using a Turing-like activator-repressor model with activator-concentration-dependent flux boundary condition at the colony edge. This approach allows the self-organization of the boundary conditions, which results in a larger variety of patterns than previously observed. We found that this model recapitulated previous results of centrosymmetric patterns in large colonies, and also that in simulated small colony sizes, patterns with spontaneous asymmetries emerged. Model analysis revealed reciprocal effects between diffusion and size of the colony, with model-predicted asymmetries in small pattern sizes being dominated by diffusion, and centrosymmetric patterns being size-dominated. To test these predictions, we performed experiments on micro-patterned mPSC colonies of different sizes stimulated with Bone Morphogenetic Protein 4 (BMP4), and used Brachyury (BRA)-GFP expressing cells as pattern readout. We found that while large colonies showed centrosymmetric BRA patterns, the probability of colony polarization increased with decreasing sizes, with a maximum polarization frequency of 35% at ~200μm. These results indicate that a simple molecular activator-repressor system can provide cells with collective features capable of initiating a body-axes plan, and constitute a theoretical foundation for the engineering of asymmetry in developmental systems.

... In fact, it is possible to look for conditions guaranteeing that an equilibrium, stable in the absence of diffusion, becomes unstable when diffusion is allowed. The diffusion-driven instability is called Turing instability and has been widely studied in literature, especially to investigate for the Turing patterns formation ( [27]). This approach can be extended to other interacting models with different functional responses, and also in other fields of applied mathematics where nonlinear mathematical models having a similar structure are considered ( [28,29,30,31]). ...

A reaction-diffusion Leslie-Gower predator-prey model, incorporating the fear effect and prey refuge, with Beddington-DeAngelis functional response, is introduced. A qualitative analysis of the solutions of the model and the stability analysis of the coexistence equilibrium, are performed. Sufficient conditions guaranteeing the occurrence of Turing instability have been determined either in the case of self-diffusion or in the case of cross-diffusion. Different types of Turing patterns, representing a spatial redistribution of population in the environment, emerge for different values of the model parameters.

... Here, we propose the alternative approach of modelling PLK4 symmetry breaking as a "Turing system"; which we define here as a two-component reaction-diffusion system that breaks symmetry through activator-inhibitor dynamics. Such systems fall into one of two classes in which symmetry is broken through "long-range activation, short range inhibition" (Turing, 1952;Gierer and Meinhardt, 1972). Turing systems are famous for their ability to produce complex spatial patterns, and have been studied in relation to numerous phenomena including the formation of animal coats, predator-prey dynamics, and the spread of disease (Bard, 1981;Levin and Segel, 1985;Mimura and Murray, 1978;Sun, 2012). ...

Centrioles are barrel-shaped structures that duplicate when a mother centriole gives birth to a single daughter that grows from its side. Polo-like-kinase 4 (PLK4), the master regulator of centriole biogenesis, is initially recruited around the mother centriole but it quickly concentrates at a single focus that defines the daughter centriole assembly site. How this PLK4 asymmetry is generated is unclear. Two previous studies used different molecular and mathematical models to simulate PLK4 symmetry breaking. Here, we extract the core biological ideas from both models to formulate a new and much simpler mathematical model where phosphorylated and unphosphorylated species of PLK4 (either on their own, or in complexes with other centriole duplication proteins) form the two-components of a classic Turing reaction-diffusion system. These two components bind/unbind from the mother at different rates, and so effectively diffuse around the mother at different rates. This allows a slow-diffusing activator species to accumulate at a single site on the mother, while a fast-diffusing inhibitor species rapidly diffuses around the centriole to suppress activator accumulation. Our analysis suggests that phosphorylated and unphosphorylated species of PLK4 can form a Turing reaction-diffusion system to break symmetry and generate a single daughter centriole.

... Both theoretical and empirical studies show that the synaptic matrix of the MEC-grid cells of young mammals perform heavy self-organizing path-integration computations, similar to Turing's symmetry-breaking operation 5 , while the 5 A landmark Turing's paper [51] demonstrating that symmetry breaking can occur in the simple reaction-diffusion system, that results in spatially periodic structures can account for pattern formation in nature. ...

... These systems show that diffusion can produce the spontaneous formation of spatio-temporal patterns. For details, interested reader can refer to the work of Turing [5]. A general model for reaction -diffusion systems is discussed by Henry and Wearne [6]. ...

... For example, Burrows and Smith (2003) found greater complexity in the facial muscles of Otolemur than previously reported, while Burrows et al. (2006) advised that we are not allowed to claim greater complexity in Homo facial expression musculature compared with Pan troglodytes. (Turing 1952). RD describes a system of chemical substances where random disturbances are caused by the competition between two active components termed activators and inhibitors (Deca, 2017). ...

Set theory faces two difficulties: formal definitions of sets/subsets are incapable of assessing biophysical issues; formal axiomatic systems are complete/inconsistent or incomplete/consistent. To overtake these problems reminiscent of the old-fashioned principle of individuation, we provide formal treatment/validation/operationalization of a methodological weapon termed “outer approach” (OA). The observer’s attention shifts from the system under evaluation to its surroundings, so that objects are investigated from outside. Subsets become just “holes” devoid of information inside larger sets. Sets are no longer passive containers, rather active structures enabling their content’s examination. Consequences/applications of OA include: a) operationalization of paraconsistent logics, anticipated by unexpected forerunners, in terms of advanced truth theories of natural language, anthropic principle and quantum dynamics; b) assessment of embryonic craniocaudal migration in terms of Turing’s spots; c) evaluation of hominids’ social behaviors in terms of evolutionary modifications of facial expression’s musculature; d) treatment of cortical action potentials in terms of collective movements of extracellular currents, leaving apart what happens inside the neurons; e) a critique of Shannon’s information in terms of the Arabic thinkers’ active/potential intellects. Also, OA provides an outer view of a) humanistic issues such as the enigmatic Celestino of Verona’s letter, Dante Alighieri’s “Hell” and the puzzling Voynich manuscript; b) historical issues such as Aldo Moro’s death and the Liston/Clay boxing fight. Summarizing, the safest methodology to quantify phenomena is to remove them from our observation and tackle an outer view, since mathematical/logical issues such as selective information deletion and set complement rescue incompleteness/inconsistency of biophysical systems.

... It is a signal molecule that determines the location, differentiation and fate of many surrounding cells [9]. In [26], Turing showed that two diffusible morphogens could instigate diffusiondriven symmetry breaking and bifurcation. Diffusion can destroy the stability of spatial homogeneous steady state, that is, the stability process can evolve into an instability with diffusion effect. ...

In this paper, we consider the dynamics of delayed Gierer–Meinhardt system, which is used as a classic example to explain the mechanism of pattern formation. The conditions for the occurrence of Turing, Hopf and Turing–Hopf bifurcation are established by analyzing the characteristic equation. For Turing–Hopf bifurcation, we derive the truncated third-order normal form based on the work of Jiang et al. [11], which is topologically equivalent to the original equation, and theoretically reveal system exhibits abundant spatial, temporal and spatiotemporal patterns, such as semistable spatially inhomogeneous periodic solutions, as well as tristable patterns of a pair of spatially inhomogeneous steady states and a spatially homogeneous periodic solution coexisting. Especially, we theoretically explain the phenomenon that time delay inhibits the formation of heterogeneous steady patterns, found by S. Lee, E. Gaffney and N. Monk [The influence of gene expression time delays on Gierer–Meinhardt pattern formation systems, Bull. Math. Biol., 72(8):2139–2160, 2010.]

... The scientific community has a wide range of interest in the formation mechanism and structural characteristics behind the pattern. Consequently, theories of pattern dynamics have been deeply studied and form a more systematic theoretical research area [10][11][12][13][14][15][16][17][18][19][20][21]. Here, we present some typical works on this topic. ...

Pattern structures are usually used to describe the spatial and temporal distribution characteristics of individuals. However, the corresponding relationship between the pattern structure and system robustness is not well understood. In this work, we use geostatistical method–semivariogram to study system robustness for different pattern structures based on three dynamical models in different fields. The results show that the structural ratio of different pattern structures including the mixed state of spot and stripe, cold spot, stripe only, and hot spot are more than 75 % , which indicated those patterns all have strong spatial dependence and heterogeneity. It was revealed that the systems corresponding to the mixed state of spot and stripe or cold spot are more robust. This article proposed a method to characterize the robustness of the system corresponding to the pattern structure and also provided a feasible approach for the study of “how structures determine their functions.”

... Others are interlaced bands of high or low population densities. Some are non-stationary with respect to space and time, where the species populations move from one region to another following a periodic or aperiodic nature [27,28,29,30,31]. ...

Propagation of a disease through a spatially varying population poses complex questions about disease spread and population survival. We consider a spatio-temporal predator-prey model in which a disease only affects the predator. Diffusion-driven instability conditions are analytically derived for the spatio-temporal model. We perform numerical simulation using experimental data given in previous studies and demonstrate that travelling waves, periodic and chaotic patterns are possible. We show that the introduction of disease in the predator species makes the standard Rosenzweig-MacArthur model capable of producing Turing patterns, which is not possible without disease. However, in the absence of disease, both species can coexist in spiral non-Turing patterns. It follows that disease persistence may be predictable, while eradication may not be.

... Pattern formation by reaction-diffusion systems has been an intensively studied field for decades. In 1952, Turing proposed the idea of diffusion-driven instability [47], in which simple mechanisms evolve from a homogeneous state into spatial heterogeneous patterns. In recent years, a variety of application areas have , (d) cheetah [46]. ...

We are interested in simulating patterns on rough surfaces. First, we consider periodic rough surfaces with analytic parametric equations, which are defined by some superposition of wave functions with random frequencies and angles of propagation. The amplitude of such surfaces is also an important variable in the provided eigenvalue analysis for the Laplace-Beltrami operator and in our numerical studies. Simulations show that the patterns become irregular as the amplitude and frequency of the rough surface increase. Next, for the sake of easy generalization to closed manifolds, we propose another construction method of rough surfaces by using random nodal values and discretized heat filters. We provide numerical evidence that both surface constructions yield comparable patterns to those found in real-life animals.

... The formation of patterns and structures in a chemically reacting medium due to differential rates of diffusion of an otherwise stable equilibrium was first proposed in the last published work of Turing (1952). It seems Turing was on the verge of explaining how this instability underlies the formation of structures in plants such as the daisy; see Dawes (2016). ...

Necessary and sufficient conditions are provided for a diffusion-driven instability of
a stable equilibrium of a reaction–diffusion system with n components and diagonal
diffusion matrix. These can be either Turing or wave instabilities. Known necessary
and sufficient conditions are reproduced for there to exist diffusion rates that cause
a Turing bifurcation of a stable homogeneous state in the absence of diffusion. The
method of proof here though, which is based on study of dispersion relations in the
contrasting limits in which thewavenumber tends to zero and to∞, gives a constructive method for choosing diffusion constants. The results are illustrated on a 3-component FitzHugh–Nagumo-like model proposed to study excitable wavetrains, and for two different coupled Brusselator systems with 4-components.

... Gierer and Meinhardt used the results obtained from a series of transplantation experiments to propose a general mathematical model of morphogenesis 14 . Their model revisits the Turing model based on the reaction-diffusion model, where two substances that exhibit distinct diffusion properties and interact with each other, form a minimal regulatory loop that suffices for de novo pattern formation 15 . Gierer and Meinhardt posed that the activation component acts over short-range distance and the inhibition one over long-range distance. ...

Polyps of the cnidarian Hydra maintain their adult anatomy through two developmental organizers, the head organizer located apically and the foot organizer basally. The head organizer is made of two antagonistic cross-reacting components, an activator, driving apical differentiation and an inhibitor, preventing ectopic head formation. Here we characterize the head inhibitor by comparing planarian genes down-regulated when β-catenin is silenced to Hydra genes displaying a graded apical-to-basal expression and an up-regulation during head regeneration. We identify Sp5 as a transcription factor that fulfills the head inhibitor properties: leading to a robust multiheaded phenotype when knocked-down in Hydra , acting as a transcriptional repressor of Wnt3 and positively regulated by Wnt/β-catenin signaling. Hydra and zebrafish Sp5 repress Wnt3 promoter activity while Hydra Sp5 also activates its own expression, likely via β-catenin/TCF interaction. This work identifies Sp5 as a potent feedback loop inhibitor of Wnt/β-catenin signaling, a function conserved across eumetazoan evolution.

... Perhaps one argument is that morphogenesis is always accompanied by large and soft deformation in order to adapt to the external environment. In addition, many scientists recognize [75] that stresses play a vital role during the development of an embryo. At the macroscopic level, differential growth generates residual stress creating the circumvolutions of the intestine [76,77,78], the brain cortex [79,80,81], and the fingerprints of skin [82,83]. ...

Shapes in nature, from cauliflower to brain, come from growth of tissues. Understanding ultimately the emergence of shapes, in particular in biological tissues, requires to capture the mechanical feature that are inherent to the living material. Besides, many thin living objects are composed of several layers which have variant elastic properties and also follow different rules of growth. In this thesis, we try to employ nonlinear elasticity to model the growing epithelium of C.elegans and Drosophila. We first study on the mechanism of C.elegans embryonic elongation. We investigate the effect of the application of laser ablation technique for stress assessment on the external epidermis. Hyperelasticity with fiber reinforcement is considered in the constitutive relation. Inner stress induced by active network is examined in the calculation. In addition, a modified formula is proposed for crack open on the animal surface. Results with experimental data prove the validity of our approach. Then we try to simulate buckling of the epithelial tissue of Drosophila wing disc. A bilayer Föppl-Von Kármán model with growth is obtained for thin lamina with extracellular matrix (ECM). Bending contribution by active network is included in the model. We acquire an analytical solution of the nonlinear governing equations. Defect of ECM is taken into account for the experiment. We employ commercial software COMSOL for numerical simulation in 3D. A finite element platform is also prepared in MATLAB for research purpose.

... systems (Mitchell, 2009). Cellular automata (CAs) (von Neumann, 1963) represent a natural playground for studying collective intelligence and morphogenesis (shape-forming processes), because of their discrete-time and Markovian dynamics (Turing, 1990). CAs are computational models inspired by the biological behaviors of cellular growth. ...

Cellular automata (CAs) are computational models exhibiting rich dynamics emerging from the local interaction of cells arranged in a regular lattice. Graph CAs (GCAs) generalise standard CAs by allowing for arbitrary graphs rather than regular lattices, similar to how Graph Neural Networks (GNNs) generalise Convolutional NNs. Recently, Graph Neural CAs (GNCAs) have been proposed as models built on top of standard GNNs that can be trained to approximate the transition rule of any arbitrary GCA. Existing GNCAs are anisotropic in the sense that their transition rules are not equivariant to translation, rotation, and reflection of the nodes' spatial locations. However, it is desirable for instances related by such transformations to be treated identically by the model. By replacing standard graph convolutions with E(n)-equivariant ones, we avoid anisotropy by design and propose a class of isotropic automata that we call E(n)-GNCAs. These models are lightweight, but can nevertheless handle large graphs, capture complex dynamics and exhibit emergent self-organising behaviours. We showcase the broad and successful applicability of E(n)-GNCAs on three different tasks: (i) pattern formation, (ii) graph auto-encoding, and (iii) simulation of E(n)-equivariant dynamical systems.

... Interest in the study of collective dynamics of ring-coupled oscillators grew significantly after publication of the Turing's pioneering paper on morphogenesis [17]. Later, ring geometry was widely explored in a number of physiological and biochemical applications [18,19] because ring-shaped network motifs often occur in biological systems, for example, in peripheral nervous systems and locomotion [20][21][22]. ...

We study dynamics of a unidirectional ring of three Rulkov neurons coupled by chemical synapses. We consider both deterministic and stochastic models. In the deterministic case, the neural dynamics transforms from a stable equilibrium into complex oscillatory regimes (periodic or chaotic) when the coupling strength is increased. The coexistence of complete synchronization, phase synchronization, and partial synchronization is observed. In the partial synchronization state either two neurons are synchronized and the third is in antiphase, or more complex combinations of synchronous and asynchronous interaction occur. In the stochastic model, we observe noise-induced destruction of complete synchronization leading to multistate intermittency between synchronous and asynchronous modes. We show that even small noise can transform the system from the regime of regular complete synchronization into the regime of asynchronous chaotic oscillations.

... In nonlinear science related to the process of periodic energy exchange between the modulation and the continuous wave (CW) background, MI is a natural phenomenon that can be seen everywhere. In ocean dynamics it is called the Benjamin-Feir instability 19 ; in optics it is called the Bespalov-Talanov instability 20 ; and in biology it is called the Turing instability 21 . ...

In the context of the parallel flow hypothesis, we derive the higher-order generalized cubic-quintic complex Ginzburg--Landau (GCQ-CGL) equation to describe the amplitude evolution of shallow wake flow from the dimensionless shallow water equations by using multi-scale analysis, perturbation expansion, and weak nonlinear theory. The evolution model includes not only the slowly changing envelope approximation but also the influence of higher-order dissipation, dispersion, and cubic and quintic nonlinear effects. We give the analytical solution of the higher-order GCQ-CGL equation based on the ansatz and coordinate transformation methods, and we discuss the influence of the higher-order dissipation coefficient on the amplitude and frequency of the wake flow by means of three-dimensional diagrams, contour maps, and plane graphs. The subsequent linear stability analysis gives a theoretical basis for the modulation instability (MI) of plane waves, and the linear theory predicts the instability of any amplitude of the main waves. Finally, we focus on the MI of shallow wake flows. Results show that the MI gain function is internally related to the background wave number, disturbance wave number, background amplitude, disturbance expansion parameter, and dissipation coefficient. The area of the MI decreases as the higher-order dissipation coefficient decreases.

... Solutions of reaction-diffusion systems exhibit a wide variety of patterns, which makes them ubiquitous in modeling chemical, biological and ecological models [18]. For example, Turing patterns are potential mechanism for the emergence of stripes and spots on animal coats [24]. In chemistry, spontaneous pattern generation occurs in experiments of the Belousov-Zhabotinsky reaction [25] and in numerical simulations of model systems, in which both rigidly-rotating spiral waves and spiral waves exhibiting one or more line defects have been observed (see Figure 1). ...

Contact defects are time-periodic patterns in one space dimension that resemble spatially homogeneous oscillations with an embedded defect in their core region. For theoretical and numerical purposes, it is important to understand whether these defects persist when the domain is truncated to large spatial intervals, supplemented by appropriate boundary conditions. The present work shows that truncated contact defects exist and are unique on sufficiently large spatial intervals.

... Generalized concepts of "computing" have been sought since long in several disciplines, especially in theoretical biology and neuroscience, theoretical physics and various unconventional niches in computer science. It is maybe more than a historical curiosity that Turing himself in his last works laid the foundations for self-organized pattern formation in biological systems (Turing, 1952), addressing a family of phenomena that has later been recruited as a basis for unconventional or brain-like kinds of "computing" (Adamatzky, 2011;Lins & Schöner, 2014). The quest for a generalized theory of "computing" has recently gained fresh momentum (Adamatzky, 2017;Stepney, Rasmussen, & Amos, 2018;Jaeger, 2021). ...

Tasks that one wishes to have done by a computer often come with conditions that relate to timescales. For instance, the processing must terminate within a given time limit; or a signal processing computer must integrate input information across several timescales; or a robot motor controller must react to sensor signals with a short latency. For classical digital computing machines such requirements pose no fundamental problems as long as the physical clock rate is fast enough for the fastest relevant timescales. However, when digital microchips become scaled down into the few-nanometer range where quantum noise and device mismatch become unavoidable, or when the digital computing paradigm is altogether abandoned in analog neuromorphic systems or other unconventional hardware bases, it can become difficult to relate timescale conditions to the physical hardware dynamics. Here we explore the relations between task-defined timescale conditions and physical hardware timescales in some depth. The article has two main parts. In the first part we develop an abstract model of a generic computational system that admits a unified discussion of computational timescales. This model is general enough to cover digital computing systems as well as analog neuromorphic or other unconventional ones. We identify four major types of timescales which require separate considerations: causal physical timescales; timescales of phenomenal change which characterize the ``speed'' of how something changes in time; timescales of reactivity which describe how fast a computing system can react to incoming trigger information; and memory timescales. In the second part we survey twenty known computational mechanisms that can be used to obtain desired task-related timescale characteristics from the physical givens of the available hardware.

... The first characteristic lies in modulation instability as the solution encounters abnormal initial input that will increase exponentially [6]. On the other hand, Bespalov-Talanov instability does not last long and will return the solution to its initial state [7]. As a result, in a conservative nonlinear system, each wave that "appears from nowhere" must "disappear without a trace" [8]. ...

This paper investigates rational solutions of an extended Camassa-Holm-Kadomtsev-Petviashvili equation, which simulates dispersion's role in the development of patterns in a liquid drop, and describes left and right traveling waves like the Boussinesq equation. Through its bilinear form and symbolic computation, we derive some multiple order rational and generalized rational solutions and analyze their dynamic features, such as the connection between rational solution and bilinear equation, scatter behavior, moving path, and exact location of the soliton. The obtained solutions demonstrate two wave forms: multi-lump and multi-wave that consist of three, six and eight lump waves or two, three and four line waves. Moreover, different from the multi-wave solitons, stationary multiple dark waves are presented.

... Numerous examples of instabilities exist in geomorphology, such as in the formation of sand dunes (Elbelrhiti et al., 2005), rills on hillslopes (Smith & Bretherton, 1972), and cusps on beaches (Werner & Fink, 1993). Elsewhere in nature, instabilities are responsible for numerous patterns including in clouds (Baumgarten & Fritts, 2014), animal stripes (Turing, 1952), snowflake formation (Libbrecht, 2003), and filaments within the structure of nebulae (Hester, 2008). A common denominator between these phenomena is the large-scale regularity of the patterns they form, with a preferred spacing between component elements; a characteristic that also exists for subglacial bedforms (Clark, Ely, Spagnolo, et al., 2018b). ...

Initially a matter of intellectual curiosity, but now important for understanding ice‐sheet dynamics, the formation of subglacial bedforms has been a subject of scientific enquiry for over a century. Here, we use a numerical model of the coupled flow of ice, water, and subglacial sediment to explore the formation of subglacial ribs (i.e. ribbed moraine), drumlins and mega‐scale glacial lineations (MSGL). The model produces instabilities at the ice‐bed interface, which result in landforms resembling subglacial ribs and drumlins. We find that a behavioural trajectory is present. Initially subglacial ribs form, which can either develop into fields of organised drumlins, or herringbone‐type structures misaligned with ice flow. We present potential examples of these misaligned bedforms in deglaciated landscapes, the presence of which means caution should be taken when interpreting cross‐cutting bedforms to reconstruct ice flow directions. Under unvarying ice flow parameters, MSGL failed to appear in our experiments. However, drumlin fields can elongate into MSGL in our model if low ice‐bed coupling conditions are imposed. The conditions under which drumlins elongate into MSGL are analogous to those found beneath contemporary ice streams, providing the first mechanism, rather than just an association, for linking MSGL with ice stream flow. We conclude that the instability theory, as realised in this numerical model, is sufficient to explain the fundamental mechanics and process‐interactions that lead to the initiation of subglacial bedforms, the development of the distinctive types of bedform patterns, and their evolutionary trajectories. We therefore suggest that the first part of the longstanding ‘drumlin problem’ – how and why they come into existence ‐ is now solved. However, much remains to be discovered regarding the exact sedimentary and hydrological processes involved.

... Here, a cell being "dead" is understood to be at rest while being "alive" means its enzyme concentrations vary periodically. As Smale points out, the underlying model goes back even further to Turing's seminal paper [Tur53] on reaction-diffusion systems in biology, originally published in 1953. ...

Die vorliegende Arbeit behandelt die kollektive Dynamik identischer Klasse-I-anregbarer Elemente. Diese können im Rahmen der nichtlinearen Dynamik als Systeme nahe einer Sattel-Knoten-Bifurkation auf einem invarianten Kreis beschrieben werden. Der Fokus der Arbeit liegt auf dem Studium aktiver Rotatoren als Prototypen solcher Elemente. In Teil eins der Arbeit besprechen wir das klassische Modell abstoßend gekoppelter aktiver Rotatoren von Shinomoto und Kuramoto und generalisieren es indem wir höhere Fourier-Moden in der internen Dynamik der Rotatoren berücksichtigen. Wir besprechen außerdem die mathematischen Methoden die wir zur Untersuchung des Aktive-Rotatoren-Modells verwenden. In Teil zwei untersuchen wir Existenz und Stabilität periodischer Zwei-Cluster-Lösungen für generalisierte aktive Rotatoren und beweisen anschließend die Existenz eines Kontinuums periodischer Lösungen für eine Klasse Watanabe-Strogatz-integrabler Systeme zu denen insbesondere das klassische Aktive-Rotatoren-Modell gehört und zeigen dass (i) das Kontinuum eine normal-anziehende invariante Mannigfaltigkeit bildet und (ii) eine der auftretenden periodischen Lösungen Splay-State-Dynamik besitzt. Danach entwickeln wir mit Hilfe der Averaging-Methode eine Störungstheorie für solche Systeme. Mit dieser können wir Rückschlüsse auf die asymptotische Dynamik des generalisierten Aktive-Rotatoren-Modells ziehen. Als Hauptergebnis stellen wir fest dass sowohl periodische Zwei-Cluster-Lösungen als auch Splay States robuste Lösungen für das Aktive-Rotatoren-Modell darstellen. Wir untersuchen außerdem einen "Stabilitätstransfer" zwischen diesen Lösungen durch sogenannte Broken-Symmetry States. In Teil drei untersuchen wir Ensembles gekoppelter Morris-Lecar-Neuronen und stellen fest, dass deren asymptotische Dynamik der der aktiven Rotatoren vergleichbar ist was nahelegt dass die Ergebnisse aus Teil zwei ein qualitatives Bild für solch kompliziertere und realistischere Neuronenmodelle liefern.

... 195 A prime example in this context are reaction-diffusion systems with an activator and an inhibitor, whose antagonistic and nonlinear reaction terms can lead to the emergence of stationary spatial patterns when diffusional mixing is sufficiently poor (see Fig. 6 for an example). 196 Such spatially non-uniform steady states are named ''Turing patterns'', in honour of A. M. Turing. By now, many similar pattern formation systems have been observed experimentally, and a multitude of pattern forming systems formulated mathematically. ...

A panoply of new tools for tracking single particles and molecules has led to an explosion of experimental data, leading to novel insights into physical properties of living matter governing cellular development and function, health and disease. In this Perspective, we present tools to investigate the dynamics and mechanics of living systems from the molecular to cellular scale via single-particle techniques. In particular, we focus on methods to measure, interpret, and analyse complex data sets that are associated with forces, materials properties, transport, and emergent organisation phenomena within biological and soft-matter systems. Current approaches, challenges, and existing solutions in the associated fields are outlined in order to support the growing community of researchers at the interface of physics and the life sciences. Each section focuses not only on the general physical principles and the potential for understanding living matter, but also on details of practical data extraction and analysis, discussing limitations, interpretation, and comparison across different experimental realisations and theoretical frameworks. Particularly relevant results are introduced as examples. While this Perspective describes living matter from a physical perspective, highlighting experimental and theoretical physics techniques relevant for such systems, it is also meant to serve as a solid starting point for researchers in the life sciences interested in the implementation of biophysical methods.

... In contrast to the D. guttifera wing spot pattern, the abdominal spots develop in the absence of any visible landmark structures. We speculate that the spots on the abdomen are established in a self-regulating manner, i.e., a Turing pattern [29], with the possible interplay of the morphogens Wg, Dpp, Hh, and perhaps other upstream regulators that co-regulate these developmental genes to assemble the complete pattern. Thus, the abdominal color pattern of D. guttifera may be regulated by multiple developmental pathways consisting of activators and repressors acting in parallel, possibly targeting pigmentation genes other than y as well [14,15,27,30]. ...

Changes in the control of developmental gene expression patterns have been implicated in the evolution of animal morphology. However, the genetic mechanisms underlying complex morphological traits remain largely unknown. Here we investigated the molecular mechanisms that induce the pigmentation gene yellow in a complex color pattern on the abdomen of Drosophila guttifera . We show that at least five developmental genes may collectively activate one cis -regulatory module of yellow in distinct spot rows and a dark shade to assemble the complete abdominal pigment pattern of Drosophila guttifera . One of these genes, wingless , may play a conserved role in the early phase of spot pattern development in several species of the quinaria group. Our findings shed light on the evolution of complex animal color patterns through modular changes of gene expression patterns.

... In 1836, Pierre Francois Verhulst formulated the logistic growth model that became a building block of modern system dynamics and machine learning algorithms. In the 1950s, the pioneers of modern Computer Science Alan Turing and John von Neumann not only developed their one-dimensional computing machine models but also discovered two-dimensional cellular automata, which explain distributed textural patterns and self-reproduction in nature [1,2,6]. In collaborating with neuroscientists and engineers in the 1960s, Norbert Wiener developed mathematical models for feedback control, nervous oscillation movements, self-reproduction, learning, and even Gestalt phenomena which he called "Cybernetics" [3]. ...

There are at least 45 definitions of complexity according to Seth Lloyd as reported in The End of Science (Horgan, 1997, pp. 303–305). Rosser Jr. (1999) argued for the usefulness in studying economics of a definition he called dynamiccomplexity that was originated by Day (1994). This is that a dynamical economic system fails to generate convergence to a point, a limit cycle or an explosion (or implosion) endogenously from its deterministic parts. It has been argued that nonlinearity was a necessary but not sufficient condition for this form of complexity, and that this definition constituted a suitably broad “big tent” to encompass the “four C’s” of cybernetics, catastrophe, chaos, and “small tent” (now better known as heterogeneous agents) complexity.

Herbert A. Simon developed the idea of boundedrationality from his earliest works (Simon 1947, 1955a, 1957), which is viewed as the foundation of modern behavioral economics. Behavioral economics contrasts with more conventional economics in not assuming full information rationality on the part of economic agents in their behavior. In this regard, it draws on insights regarding human behavior from other social science disciplines such as psychology and sociology, among others. Without question, one can find earlier economists who argued that people are motivated by more than mere selfish maximization. Indeed, from the very beginnings of economics with Aristotle, who put economic considerations into a context of moral philosophy and proper conduct, through the father of political economy, Adam Smith in his Theory of Moral Sentiments (1759), to later institutional economists such as Thorstein Veblen (1899) and Karl Polanyi (1944) who saw peoples’ economic conduct as embedded within broader social and political contexts. Nevertheless, it was Simon who coined both of these terms and established modern behavioral economics.

Urbanization is one of the defining trends of our time and appropriate models are needed to anticipate the changes in cities, which are largely determined by human behavior. In the social sciences, where the task of describing human behavior falls, a distinction is made between quantitative and qualitative approaches, each of which has its own advantages and disadvantages. While the latter often provide descriptions of exemplary processes in order to describe phenomena as holistically as possible, the goal of mathematically motivated modeling is primarily to make a problem tangible. Both approaches are discussed in terms of the temporal evolution of one of the dominant settlement types in the world today: informal settlements. These areas have been modeled in conceptual works as self-organizing entities and in mathematical works as Turing systems. It is shown that the social issues surrounding these areas need to be understood both qualitatively and quantitatively. Inspired by the philosopher C. S. Peirce, a framework is proposed in which the various modeling approaches describing these settlements can be combined to arrive at a more holistic understanding of this phenomenon by using the language of mathematical modeling.

In the previous chapter, we discussed the mathematical model of ordinary differential equations, which considers only the time derivative without considering the spatial extension. However, when considering life phenomena in multicellular organisms, it is necessary to consider the interaction between cells laid out in space. In addition, the substances produced by each cell do not necessarily stay in that cell. Many secreted ligands are released and diffuse out of the cell and then bind to receptors on surrounding cells to transmit information between cells (Fig. 4.1). In this section, we will consider the phenomenon of secretory ligands diffusing in the extracellular space.

Self-organization of cells into higher-order structures is key for multicellular organisms, e.g. via repetitive replication of template-like founder cells or syncytial energids. Yet, very similar spatial arrangements of cell-like compartments ('protocells') are also seen in a minimal model system of Xenopus egg extracts in the absence of template structures and chromatin, with dynamic microtubule assemblies driving the self-organization process. Quantifying geometrical features over time, we show here that protocell patterns are highly organized with a spatial arrangement and coarsening dynamics like two-dimensional foams but without the long-range ordering expected for hexagonal patterns. These features remain invariant when enforcing smaller protocells by adding taxol, i.e. patterns are dominated by a single, microtubule-derived length scale. Comparing our data to generic models, we conclude that protocell patterns emerge by simultaneous formation of randomly assembling protocells that grow at a uniform rate towards a frustrated arrangement before fusion of adjacent protocells eventually drives coarsening. The similarity of protocell patterns to arrays of energids and cells in developing organisms, but also to epithelial monolayers, suggests generic mechanical cues to drive self-organized space compartmentalization.

Pattern formation has been extensively studied in the context of evolving (time-dependent) domains in recent years, with domain growth implicated in ameliorating problems of pattern robustness and selection, in addition to more realistic modelling in developmental biology. Most work to date has considered prescribed domains evolving as given functions of time, but not the scenario of concentration-dependent dynamics, which is also highly relevant in a developmental setting. Here, we study such concentration-dependent domain evolution for reaction–diffusion systems to elucidate fundamental aspects of these more complex models. We pose a general form of one-dimensional domain evolution and extend this to N -dimensional manifolds under mild constitutive assumptions in lieu of developing a full tissue-mechanical model. In the 1D case, we are able to extend linear stability analysis around homogeneous equilibria, though this is of limited utility in understanding complex pattern dynamics in fast growth regimes. We numerically demonstrate a variety of dynamical behaviours in 1D and 2D planar geometries, giving rise to several new phenomena, especially near regimes of critical bifurcation boundaries such as peak-splitting instabilities. For sufficiently fast growth and contraction, concentration-dependence can have an enormous impact on the nonlinear dynamics of the system both qualitatively and quantitatively. We highlight crucial differences between 1D evolution and higher-dimensional models, explaining obstructions for linear analysis and underscoring the importance of careful constitutive choices in defining domain evolution in higher dimensions. We raise important questions in the modelling and analysis of biological systems, in addition to numerous mathematical questions that appear tractable in the one-dimensional setting, but are vastly more difficult for higher-dimensional models.

A central goal of ecology is identifying the mechanisms that allow large, complex food webs to persist. Spatial mechanisms resulting from dispersal connections among local food webs are one factor shown to play a significant role in enabling species persistence, particularly by driving asynchrony in the dynamics among local food webs. However, it is still unknown how these spatial persistence mechanisms operate across food webs. Using simulations of full non-linear food web models, we investigate how spatial persistence mechanisms emerge in multi-species food webs that possess different structural metrics. Specifically, we ask whether 1) spatial persistence mechanisms work similarly across food webs, and 2) if differences can be explained by food web features influencing stability in the absence of dispersal, particularly trophic structure. Food web structures are generated using the allometric niche model that is capable of reproducing realistic feeding patterns and interaction strengths. Our analyses quantify the tendency of modeled food webs to achieve asynchrony in the presence of dispersal and show that this positively affects the ability of species in the food web to persist. We observe an inverse relationship between the ability of food webs to persist when isolated and their tendency to be asynchronous when spatial, indicating a limited ability of food webs that persist when isolated to benefit from spatial persistence mechanisms. Our results demonstrate a relatively unexplored layer of food web properties which determine the ability of a food web to capitalize on the stabilizing opportunities created by dispersal, specifically those that influence the tendency for dispersal-linked food webs to be asynchronous. Future studies should expand on our results by examining how properties of spatial connections and food webs influence the ability of food webs to achieve asynchrony.

The ageing process is highly complex involving multiple processes operating at different biological levels. Systems Biology presents an approach using integrative computational and laboratory study that allows us to address such complexity. The approach relies on the computational analysis of knowledge and data to generate predictive models that may be validated with further laboratory experimentation. Our understanding of ageing is such that translational opportunities are within reach and systems biology offers a means to ensure that optimal decisions are made. We present an overview of the methods employed from bioinformatics and computational modelling and describe some of the insights into ageing that have been gained.

Turing instability is a general and straightforward mechanism of pattern formation in reaction–diffusion systems, and its relevance has been demonstrated in different biological phenomena. Still, there are many open questions, especially on the robustness of the Turing mechanism. Robust patterns must survive some variation in the environmental conditions. Experiments on pattern formation using chemical systems have shown many reaction–diffusion patterns and serve as relatively simple test tools to study general aspects of these phenomena. Here, we present a study of sinusoidal variation of the input feed concentrations on chemical Turing patterns. Our experimental, numerical and theoretical analysis demonstrates that patterns may appear even at significant amplitude variation of the input feed concentrations. Furthermore, using time-dependent feeding opens a way to control pattern formation. The patterns settled at constant feed may disappear, or new patterns may appear from a homogeneous steady state due to the periodic forcing. The generation of stationary patterns is often studied under constant experimental conditions, but in biological systems parameters such as chemical flow are not stationary. Here, the authors use experiments and numerical analyses to elucidate the mechanisms controlling Turing patterns under periodic variations in chemical feed concentration.

L’objet de ce mémoire est de clarifier la signification du concept de contraintes de développement dans deux approches différentes de la théorie de l’évolution que sont l’approche structuraliste et l’approche adaptationniste. Un débat a en effet eu lieu au cours des 30 dernières années entre les tenants de ces approches, au sujet de l’importance et du rôle des contraintes de développement dans la détermination de la direction prise par l’évolution. Etant donné que ces deux approches de l’évolution se situent à des niveaux d’organisation différents et s’intéressent à des caractères différents, nous soutenons la thèse que les désaccords autour de la notion de contraintes de développement proviennent de divergences au niveau des intérêts explicatifs et des méthodologies de ces deux approches. Après avoir illustré l’importance du concept de contraintes de développement dans les problématiques évolutionnistes actuelles, en présentant l’émergence et les concepts d’une nouvelle discipline nommée EvoDevo, nous présentons le chemin par lequel le concept de contraintes de développement est entré dans les problématiques évolutionnistes, chemin dont le début se situe avec la critique de l’adaptationnisme émise par Gould et Lewontin. En plus de proposer une approche pluraliste des études adaptationnistes, Gould et Lewontin proposent d’aborder les phénomènes évolutifs également sous une approche structuraliste. Cette dernière a pour objectif de définir, au sein d’un groupe taxonomique, l’ensemble des formes possibles pour un caractère donné. Cet ensemble de formes est représenté par une construction théorique abstraite appellée Bauplan et il est définit sur la base de nos connaissances des mécanismes du développement responsables de la formation du caractère considéré. Dans ce cadre, nous montrons qu’une connaissance de la façon dont ces mécanismes fonctionnent et définissent le Bauplan passe par une compréhension de la façon dont ils sont contraints par l’histoire du groupe taxonomique considéré, ainsi que par des lois formelles internes (allométrie) et externes (lois universelles de la physique) à ces organismes. Nous voyons ensuite la façon dont les adaptationnistes ont détourné le concept de contraintes de développement de sa définition initiale, afin de l’intégrer dans leur objectif explicatif à savoir l’explication de l’adaptation. Alors que le concept de Bauplan acquière une forte valeur explicative dans le cadre de l’objectif des structuralistes qui est l’explication de la façon dont les mécanismes du développement construisent la diversité des formes, il n’est que faiblement explicatif dans le cadre de l’objectif des adaptationnistes, même dans sa version dénaturée. La valeur explicative du concept dépend donc bien de l’objectif fixé. Nous concluons en proposant de coupler les approches adaptationnistes et structuralistes afin d’améliorer notre capacité explicative et prédictive des phénomènes évolutif

Introduction: the three divisions of electromagnetism; Part I.
Electrostatics and Current Electricity: 1. Physical principles; 2. The
electrostatic field of force; 3. Conductors and condensers; 4. Systems
of conductors; 5. Dielectrics and inductive capacity; 6. The state of
the medium in the electrostatic field; 7. General analytical theorems;
8. Methods for the solution of special problems; 9. Steady currents in
linear conductors; 10. Steady currents in continuous media; Part II.
Magnetism: 11. Permanent magnetism; 12. Induced magnetism; Part III.
Electromagnetism: 13. The magnetic field produced by electric currents;
14. Induction of currents in linear circuits; 15. Induction of currents
in continuous media; 16 Dynamical theory of currents; 17. Displacement
currents and electromagnetic waves; 18. The electromagnetic theory of
light; 19. The motion of electrons; 20. The theory of relativity; 21.
The electrical structure of matter; Index.

Arcy 1942 On growth andform

- Thompson
- D Sir

Thompson, Sir D'Arcy 1942 On growth andform, 2nd ed. Cambridge University Press.