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Journal articles on the topic 'Reaction systems'

Author: Grafiati

Published: 4 June 2021

Last updated: 10 March 2023

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1

Manzoni, Luca, Antonio E. Porreca, and Grzegorz Rozenberg. "Facilitation in reaction systems." Journal of Membrane Computing 2, no. 3 (August 31, 2020): 149–61. http://dx.doi.org/10.1007/s41965-020-00044-0.

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Abstract Reaction systems is a formal model of computation which originated as a model of interactions between biochemical reactions in the living cell. These interactions are based on two mechanisms, facilitation and inhibition, and this is well reflected in the formulation of reaction systems. In this paper, we investigate the facilitation aspect of reaction systems, where the products of a reaction may facilitate other reactions by providing some of their reactants. This aspect is formalized through positive dependency graphs which depict explicitly such facilitating interactions. The focus of the paper is on demonstrating how structural properties of reaction systems defined through the properties of their positive dependency graphs influence the behavioural properties of (suitable subclasses of) reaction systems, which, as usual, are defined through their transition graphs.

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2

EHRENFEUCHT, ANDRZEJ, MICHAEL MAIN, and GRZEGORZ ROZENBERG. "FUNCTIONS DEFINED BY REACTION SYSTEMS." International Journal of Foundations of Computer Science 22, no. 01 (January 2011): 167–78. http://dx.doi.org/10.1142/s0129054111007927.

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Reaction systems are a formal model of interactions between biochemical reactions. They consist of sets of reactions, where each reaction is classified by its set of reactants (needed for the reaction to take place), its set of inhibitors (each of which prevents the reaction from taking place), and its set of products (produced when the reaction takes place) – the set of reactants and inhibitors form the resources of the reaction. Each reaction system defines a (transition) function on its set of states. (States here are subsets of an a priori given set of biochemical entities.) In this paper we investigate properties of functions defined by reaction systems. In particular, we investigate how the power of defining functions depends on available resources, and we demonstrate that with small resources one can define functions exhibiting complex behavior.

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3

BRIJDER, ROBERT, ANDRZEJ EHRENFEUCHT, MICHAEL MAIN, and GRZEGORZ ROZENBERG. "A TOUR OF REACTION SYSTEMS." International Journal of Foundations of Computer Science 22, no. 07 (November 2011): 1499–517. http://dx.doi.org/10.1142/s0129054111008842.

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Reaction systems are a formal framework for investigating processes carried out by biochemical reactions. This paper is an introduction to reaction systems. It provides basic notions together with the underlying intuition and motivation as well as two examples (a binary counter and transition systems) of "programming" with reaction systems. It also provides a tour of some research themes.

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4

Volpert, V. A., Y. Nec, and A. A. Nepomnyashchy. "Fronts in anomalous diffusion–reaction systems." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1982 (January 13, 2013): 20120179. http://dx.doi.org/10.1098/rsta.2012.0179.

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A review of recent developments in the field of front dynamics in anomalous diffusion–reaction systems is presented. Both fronts between stable phases and those propagating into an unstable phase are considered. A number of models of anomalous diffusion with reaction are discussed, including models with Lévy flights, truncated Lévy flights, subdiffusion-limited reactions and models with intertwined subdiffusion and reaction operators.

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5

MANZONI, LUCA, DIOGO POÇAS, and ANTONIO E. PORRECA. "SIMPLE REACTION SYSTEMS AND THEIR CLASSIFICATION." International Journal of Foundations of Computer Science 25, no. 04 (June 2014): 441–57. http://dx.doi.org/10.1142/s012905411440005x.

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Reaction systems are a model of computation inspired by biochemical reactions involving reactants, inhibitors and products from a finite background set. We define a notion of multi-step simulation among reaction systems and derive a classification with respect to the amount of resources (reactants and inhibitors) involved in each reaction. We prove that “simple” reaction systems, having at most one reactant and one inhibitor per reaction, suffice in order to simulate arbitrary systems. Finally, we show that the equivalence relation of mutual simulation induces exactly five linearly ordered classes of reaction systems characterizing well-known subclasses of the functions over Boolean lattices, such as the constant, additive (join-semilattice endomorphisms), monotone, and antitone functions.

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6

Ehrenfeucht, Andrzej, Jetty Kleijn, Maciej Koutny, and Grzegorz Rozenberg. "Evolving reaction systems." Theoretical Computer Science 682 (June 2017): 79–99. http://dx.doi.org/10.1016/j.tcs.2016.12.031.

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7

Nicolis, Gregoire, and Anne Wit. "Reaction-diffusion systems." Scholarpedia 2, no. 9 (2007): 1475. http://dx.doi.org/10.4249/scholarpedia.1475.

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8

Bagossy, Attila, and György Vaszil. "Simulating reversible computation with reaction systems." Journal of Membrane Computing 2, no. 3 (September 8, 2020): 179–93. http://dx.doi.org/10.1007/s41965-020-00049-9.

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Abstract Reaction systems are a formal model of computation providing a framework for investigating biochemical reactions inside living cells. We look at the functioning of these systems as a process producing a series of different possible sets of entities representing states which can be changed by the application of reactions, and we study reversibility and its simulation in this framework. Our goal is to establish an Undo-Redo-Do-like semantics of reversibility with environmental control over the direction of the computation following a so-called no-memory approach, that is, without introducing modifications to the model of reaction systems itself. We first establish requirements the systems must satisfy in order to produce processes consisting of states with unique predecessors, then define reversible reaction systems in terms of reversible interactive processes. For such reversible systems, we also construct simulator systems that can traverse between the states of reversible interactive processes back and forth based on the input of a special “rollback” symbol from the environment.

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9

Park, Joon Sik, and Jeong Min Kim. "Interface Reactions and Synthetic Reaction of Composite Systems." Materials 3, no. 1 (January 8, 2010): 264–95. http://dx.doi.org/10.3390/ma3010264.

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10

Westerlund, Tapio, and Tapio Salmi. "Factorization of reaction systems applied to catalytic reactions." Chemical Engineering Science 45, no. 1 (1990): 237–41. http://dx.doi.org/10.1016/0009-2509(90)87095-a.

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11

Gurevich, Pavel, and Sergey Tikhomirov. "Systems of reaction-diffusion equations with spatially distributed hysteresis." Mathematica Bohemica 139, no. 2 (2014): 239–57. http://dx.doi.org/10.21136/mb.2014.143852.

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12

Drábek, Pavel, Milan Kučera, and Marta Míková. "Bifurcation points of reaction-diffusion systems with unilateral conditions." Czechoslovak Mathematical Journal 35, no. 4 (1985): 639–60. http://dx.doi.org/10.21136/cmj.1985.102055.

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13

EHRENFEUCHT, ANDRZEJ, MICHAEL MAIN, GRZEGORZ ROZENBERG, and ALLISON THOMPSON BROWN. "STABILITY AND CHAOS IN REACTION SYSTEMS." International Journal of Foundations of Computer Science 23, no. 05 (August 2012): 1173–84. http://dx.doi.org/10.1142/s0129054112500177.

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Reaction systems are an abstract model of biochemical reactions in the living cell within a framework of finite (though often large) discrete dynamical systems. In this setting, this paper provides an analytical and experimental study of stability. The notion of stability is defined in terms of the way in which small perturbations to the initial state of a system are likely to change the system's eventual behavior. At the stable end of the spectrum, there is likely to be no change; but at the unstable end, small perturbations take the system into a state that is probabilistically the same as a randomly selected state, similar to chaotic behavior in continuous dynamical systems.

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14

Shang, Zeyi, Sergey Verlan, Ion Petre, and Gexiang Zhang. "Reaction Systems and Synchronous Digital Circuits." Molecules 24, no. 10 (May 21, 2019): 1961. http://dx.doi.org/10.3390/molecules24101961.

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A reaction system is a modeling framework for investigating the functioning of the living cell, focused on capturing cause–effect relationships in biochemical environments. Biochemical processes in this framework are seen to interact with each other by producing the ingredients enabling and/or inhibiting other reactions. They can also be influenced by the environment seen as a systematic driver of the processes through the ingredients brought into the cellular environment. In this paper, the first attempt is made to implement reaction systems in the hardware. We first show a tight relation between reaction systems and synchronous digital circuits, generally used for digital electronics design. We describe the algorithms allowing us to translate one model to the other one, while keeping the same behavior and similar size. We also develop a compiler translating a reaction systems description into hardware circuit description using field-programming gate arrays (FPGA) technology, leading to high performance, hardware-based simulations of reaction systems. This work also opens a novel interesting perspective of analyzing the behavior of biological systems using established industrial tools from electronic circuits design.

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15

Teh, Wen Chean, and Adrian Atanasiu. "Irreducible reaction systems and reaction system rank." Theoretical Computer Science 666 (March 2017): 12–20. http://dx.doi.org/10.1016/j.tcs.2016.08.021.

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16

Field, Richard J. "Chaos in the Belousov–Zhabotinsky reaction." Modern Physics Letters B 29, no. 34 (December 20, 2015): 1530015. http://dx.doi.org/10.1142/s021798491530015x.

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The dynamics of reacting chemical systems is governed by typically polynomial differential equations that may contain nonlinear terms and/or embedded feedback loops. Thus the dynamics of such systems may exhibit features associated with nonlinear dynamical systems, including (among others): temporal oscillations, excitability, multistability, reaction-diffusion-driven formation of spatial patterns, and deterministic chaos. These behaviors are exhibited in the concentrations of intermediate chemical species. Bifurcations occur between particular dynamic behaviors as system parameters are varied. The governing differential equations of reacting chemical systems have as variables the concentrations of all chemical species involved, as well as controllable parameters, including temperature, the initial concentrations of all chemical species, and fixed reaction-rate constants. A discussion is presented of the kinetics of chemical reactions as well as some thermodynamic considerations important to the appearance of temporal oscillations and other nonlinear dynamic behaviors, e.g., deterministic chaos. The behavior, chemical details, and mechanism of the oscillatory Belousov–Zhabotinsky Reaction (BZR) are described. Furthermore, experimental and mathematical evidence is presented that the BZR does indeed exhibit deterministic chaos when run in a flow reactor. The origin of this chaos seems to be in toroidal dynamics in which flow-driven oscillations in the control species bromomalonic acid couple with the BZR limit cycle.

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17

Aghamohammadi, A., A. H. Fatollahi, M. Khorrami, and A. Shariati. "Multispecies reaction-diffusion systems." Physical Review E 62, no. 4 (October 1, 2000): 4642–49. http://dx.doi.org/10.1103/physreve.62.4642.

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18

Bottoni, Paolo, Anna Labella, and Grzegorz Rozenberg. "Networks of Reaction Systems." International Journal of Foundations of Computer Science 31, no. 01 (January 2020): 53–71. http://dx.doi.org/10.1142/s0129054120400043.

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In this paper, we study the behavior (processes) of reaction systems where the context is not arbitrary, but it has its own structure. In particular, we consider a model where the context for a reaction system originates from a network of reaction systems. Such a network is formalized as a graph with reaction systems residing at its nodes, where each reaction system contributes to defining the context of all its neighbors. This paper provides a framework for investigating the behavior of reaction systems receiving contexts from networks of reaction systems, provides a characterisation of their state sequences, and considers different topologies of context networks.

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19

Ivanov, Sergiu, and Ion Petre. "Controllability of reaction systems." Journal of Membrane Computing 2, no. 4 (November 19, 2020): 290–302. http://dx.doi.org/10.1007/s41965-020-00055-x.

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20

Eisner, Jan. "Reaction-diffusion systems: Destabilizing effect of conditions given by inclusions." Mathematica Bohemica 125, no. 4 (2000): 385–420. http://dx.doi.org/10.21136/mb.2000.126272.

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21

EHRENFEUCHT, ANDRZEJ, MICHAEL MAIN, and GRZEGORZ ROZENBERG. "COMBINATORICS OF LIFE AND DEATH FOR REACTION SYSTEMS." International Journal of Foundations of Computer Science 21, no. 03 (June 2010): 345–56. http://dx.doi.org/10.1142/s0129054110007295.

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Reaction systems are a functional model of interactions between biochemical reactions. They define functions on finite sets (over a common finite domain). In this paper, we investigate combinatorial properties of functions defined by reaction systems. In particular, we provide analytical approximations of combinatorial properties of random reaction systems, with a focus on the probability of whether a system lives or dies. Based on these results, we can create parameterized random reaction systems that rarely die. We also empirically analyze the length of time before such a system enters cyclic behavior, and find that the time is related to the behavior of completely random functions on a smaller domain.

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22

Horno, José, and Carlos F. González-Fernández. "Analysis of chemical reaction systems by means of network thermodynamics." Collection of Czechoslovak Chemical Communications 54, no. 9 (1989): 2335–44. http://dx.doi.org/10.1135/cccc19892335.

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The simple network thermodynamics approach is applied to chemical reaction systems, whereby chemical reactions can be studied avoiding complex mathematical treatment. Steady state reaction rates are obtained for two chemical reaction systems, viz. the decomposition of ozone and the reaction of hydrogen with bromine. The rate equations so obtained agree with those derived from the chemical kinetics concept.

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23

Teh, Wen Chean, and Adrian Atanasiu. "Minimal Reaction Systems Revisited and Reaction System Rank." International Journal of Foundations of Computer Science 28, no. 03 (April 2017): 247–61. http://dx.doi.org/10.1142/s0129054117500162.

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Some mathematical aspects of reaction systems introduced by Ehrenfeucht and Rozenberg are considered. Ehrenfeucht et al. have previously obtained a complete classification of functions specified by minimal reaction systems in terms of certain closure properties of the specified functions. In this work, a refined proof of this classification with slight extension is obtained. Furthemore, the recently introduced notion of reaction system rank is studied for functions belonging to this class, as well as for focus functions, the latter which play a significant role in the proof of the classification theorem.

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24

Sieniutycz, Stanisław. "A Fermat-like Principle for Chemical Reactions in Heterogeneous Systems." Open Systems & Information Dynamics 09, no. 03 (September 2002): 257–72. http://dx.doi.org/10.1023/a:1019708629128.

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We formulate a variational principle of Fermat type for chemical kinetics in heterogeneous reacting systems. The principle is consistent with the notion of ‘intrinsic reaction coordinate’ (IRC), the idea of ‘chemical resistance’ (CR) and the second law of thermodynamics. The Lagrangian formalism applies a nonlinear functional of entropy production that follows from classical (single-phase) nonequilibrium thermodynamics of chemically reacting systems or its extension for multiphase systems involving interface reactions and transports. For a chemical flux, a “law of bending” is found which implies that — by minimizing the total resistance — the chemical ray spanned between two given points takes the shape assuring its relatively large part in a region of lower chemical resistivity (a ‘rarer’ region of the medium). In effect, the chemical flux bends into the direction that ensures its shape consistent with the longest residence of the chemical complex in regions of lower resistivity. The dynamic programming method quantifies the “chemical rays” and related wavefronts along the reaction coordinate.

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25

Dr. P.vikkraman, Dr P. vikkraman, and N. Sumathi N. Sumathi. "Shopper's Reaction To Modern Retailing Systems In An Evolving Apparel Market." Indian Journal of Applied Research 1, no. 3 (October 1, 2011): 150–51. http://dx.doi.org/10.15373/2249555x/dec2011/51.

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26

Berkemeier, T., A. J. Huisman, M. Ammann, M. Shiraiwa, T. Koop, and U. Pöschl. "Kinetic regimes and limiting cases of gas uptake and heterogeneous reactions in atmospheric aerosols and clouds: a general classification scheme." Atmospheric Chemistry and Physics Discussions 13, no. 1 (January 9, 2013): 983–1044. http://dx.doi.org/10.5194/acpd-13-983-2013.

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Abstract. Heterogeneous reactions are important to atmospheric chemistry and are therefore an area of intense research. In multiphase systems such as aerosols and clouds, chemical reactions are usually strongly coupled to a complex sequence of mass transport processes and results are often not easy to interpret. Here we present a systematic classification scheme for gas uptake by aerosol or cloud particles which distinguishes two major regimes: a reaction-diffusion regime and a mass-transfer regime. Each of these regimes includes four distinct limiting cases, characterized by a dominant reaction location (surface or bulk) and a single rate-limiting process: chemical reaction, bulk diffusion, gas-phase diffusion or mass accommodation. The conceptual framework enables efficient comparison of different studies and reaction systems, going beyond the scope of previous classification schemes by explicitly resolving interfacial transport processes and surface reactions limited by mass transfer from the gas phase. The use of kinetic multi-layer models instead of resistor model approaches increases the flexibility and enables a broader treatment of the subject, including cases which do not fit into the strict limiting cases typical of most resistor model formulations. The relative importance of different kinetic parameters such as diffusion, reaction rate and accommodation coefficients in this system is evaluated by a quantitative global sensitivity analysis. We outline the characteristic features of each limiting case and discuss the potential relevance of different regimes and limiting cases for various reaction systems. In particular, the classification scheme is applied to three different data sets for the benchmark system of oleic acid reacting with ozone. In light of these results, future directions of research needed to elucidate the multiphase chemical kinetics in this and other reaction systems are discussed.

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27

Berkemeier, T., A. J. Huisman, M. Ammann, M. Shiraiwa, T. Koop, and U. Pöschl. "Kinetic regimes and limiting cases of gas uptake and heterogeneous reactions in atmospheric aerosols and clouds: a general classification scheme." Atmospheric Chemistry and Physics 13, no. 14 (July 15, 2013): 6663–86. http://dx.doi.org/10.5194/acp-13-6663-2013.

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Abstract. Heterogeneous reactions are important to atmospheric chemistry and are therefore an area of intense research. In multiphase systems such as aerosols and clouds, chemical reactions are usually strongly coupled to a complex sequence of mass transport processes and results are often not easy to interpret. Here we present a systematic classification scheme for gas uptake by aerosol or cloud particles which distinguishes two major regimes: a reaction-diffusion regime and a mass transfer regime. Each of these regimes includes four distinct limiting cases, characterised by a dominant reaction location (surface or bulk) and a single rate-limiting process: chemical reaction, bulk diffusion, gas-phase diffusion or mass accommodation. The conceptual framework enables efficient comparison of different studies and reaction systems, going beyond the scope of previous classification schemes by explicitly resolving interfacial transport processes and surface reactions limited by mass transfer from the gas phase. The use of kinetic multi-layer models instead of resistor model approaches increases the flexibility and enables a broader treatment of the subject, including cases which do not fit into the strict limiting cases typical of most resistor model formulations. The relative importance of different kinetic parameters such as diffusion, reaction rate and accommodation coefficients in this system is evaluated by a quantitative global sensitivity analysis. We outline the characteristic features of each limiting case and discuss the potential relevance of different regimes and limiting cases for various reaction systems. In particular, the classification scheme is applied to three different datasets for the benchmark system of oleic acid reacting with ozone in order to demonstrate utility and highlight potential issues. In light of these results, future directions of research needed to elucidate the multiphase chemical kinetics in this and other reaction systems are discussed.

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28

Pekař, Miloslav. "Reaction-Diffusion Systems: Self-Balancing Diffusion and the Use of the Extent of Reaction as a Descriptor of Reaction Kinetics." International Journal of Molecular Sciences 23, no. 18 (September 10, 2022): 10511. http://dx.doi.org/10.3390/ijms231810511.

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Self-balancing diffusion is a theoretical concept that restricts the introduction of extents of reactions. This concept is analyzed in detail for general mass- and molar-based balances of reaction-diffusion mixtures, in relation to non-self-balancing cases, and with respect to its practical consequences. Self-balancing is a mathematical restriction on the divergences of diffusion fluxes. Fulfilling this condition enables the proper introduction of the extents of (independent) reactions that reduce the number of independent variables in thermodynamic descriptions. A note on a recent generalization of the concept of reaction and diffusion extents is also included. Even in the case of self-balancing diffusion, such extents do not directly replace reaction rates. Concentration changes caused by reactions (not by diffusion) are properly described by rates of independent reactions, which are instantaneous descriptors. If an overall descriptor is needed, the traditional extents of reactions can be used, bearing in mind that they include diffusion-caused changes. On the other hand, rates of independent reactions integrated with respect to time provide another overall, but reaction-only-related descriptor.

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29

EHRENFEUCHT, ANDRZEJ, and GRZEGORZ ROZENBERG. "ZOOM STRUCTURES AND REACTION SYSTEMS YIELD EXPLORATION SYSTEMS." International Journal of Foundations of Computer Science 25, no. 03 (April 2014): 275–305. http://dx.doi.org/10.1142/s0129054114500142.

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In this paper we extend the framework of reaction systems by introducing (extended) zoom structures which formalize a depository of knowledge of a discipline of science. The integrating structure of such a depository (which is a well-founded partial order) allows one to deal with the hierarchical nature of biology. This leads to the notion of an exploration system [Formula: see text] which consists of (1) a static part which is a depository of knowledge given by an extended zoom structure [Formula: see text], and (2) a dynamic part given by a family of reaction systems [Formula: see text]. In this setup the depository of knowledge [Formula: see text] is explored by computations/processes provided by reaction systems from [Formula: see text], where this exploration can use/integrate knowledge present on different levels (e.g., atomic, cellular, organism, species, … levels).

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30

Fuchs, Sonja, Arumugam Jayaraman, Ivo Krummenacher, Laura Haley, Marta Baštovanović, Maximilian Fest, Krzysztof Radacki, Holger Helten, and Holger Braunschweig. "Diboramacrocycles: reversible borole dimerisation–dissociation systems." Chemical Science 13, no. 10 (2022): 2932–38. http://dx.doi.org/10.1039/d1sc06908j.

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Diboramacrocycles are a new form of borole dimers, participating in various addition reactions as “masked” boroles. The reaction of a less crowded diboramacrocycle with organic azides affords unprecedented complex heteropropellanes.

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31

Puszynski, J., S. Kumar, P. Dimitriou, and V. Hlavacek. "A Numerical and Experimental Study of Reaction Front Propagation in Condensed Phase Systems." Zeitschrift für Naturforschung A 43, no. 12 (December 1, 1988): 1017–25. http://dx.doi.org/10.1515/zna-1988-1202.

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Certain noncatalytic exothermic chemical reactions of the type solid-solid characterized by high values of activation energy and heat of reaction represent an example of strongly nonlinear chemically reacting systems. In these systems different types of propagating waves can be observed such as constant pattern, planar pulsating, and rotating waves. Numerical simulations in two and three spatial dimensions predict, qualitatively, the same behavior as experimentally observed. For geometrically large systems multihead spinning or erratic waves occur, which bifurcate from a planar pulsating front. In nonadiabatic systems the spinning wave is more resistant to extinction than the one-dimensional planar pulsating front.

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32

Friedly, John C. "Extent of reaction in open systems with multiple heterogeneous reactions." AIChE Journal 37, no. 5 (May 1991): 687–93. http://dx.doi.org/10.1002/aic.690370507.

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33

Glitzky, Annegret, and Rolf Hünlich. "Electro-Reaction-Diffusion Systems Including Cluster Reactions of Higher Order." Mathematische Nachrichten 216, no. 1 (August 2000): 95–118. http://dx.doi.org/10.1002/1522-2616(200008)216:1<95::aid-mana95>3.0.co;2-h.

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34

Kleijn, Jetty, Maciej Koutny, and Łukasz Mikulski. "Reaction Systems and Enabling Equivalence." Fundamenta Informaticae 171, no. 1-4 (October 23, 2019): 261–77. http://dx.doi.org/10.3233/fi-2020-1882.

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35

Shamseldeen, Samir A. "Instabilities in reaction-diffusion systems." Applied Mathematical Sciences 8 (2014): 7703–13. http://dx.doi.org/10.12988/ams.2014.49762.

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36

BIANCO, LUCA, FEDERICO FONTANA, and VINCENZO MANCA. "P SYSTEMS WITH REACTION MAPS." International Journal of Foundations of Computer Science 17, no. 01 (February 2006): 27–48. http://dx.doi.org/10.1142/s0129054106003681.

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Some recent types of membrane systems have shown their potential in the modelling of specific processes governing biological cell behavior. These models represent the cell as a huge and complex dynamical system in which quantitative aspects, such as biochemical concentrations, must be related to the discrete informational nature of the DNA and to the function of the organelles living in the cytosol. In an effort to compute dynamical (especially oscillatory) phenomena—so far mostly treated using differential mathematical tools—by means of rewriting rules, we have enriched a known family of membrane systems (P systems), with rules that are applied proportionally to the values expressed by real functions called reaction maps. Such maps are designed to model the dynamic behavior of a biochemical phenomenon and their formalization is best worked out inside a family of P systems called PB systems. The overall rule activity is controlled by an algorithm that guarantees the system to evolve consistently with the available resources (i.e., objects). Though radically different, PB systems with reaction maps exhibit very interesting, often similar dynamic behavior as compared to systems of differential equations. Successful simulations of the Lotka-Volterra population dynamics, the Brusselator, and the Protein Kinase C activation foster potential applications of these systems in computational systems biology.

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37

Hutson, V., and W. Moran. "Repellers in reaction–diffusion systems." Rocky Mountain Journal of Mathematics 17, no. 2 (June 1987): 301–14. http://dx.doi.org/10.1216/rmj-1987-17-2-301.

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38

Bobisud, L. E. "Steady-state reaction-diffusion systems." Applicable Analysis 20, no. 1-2 (July 1985): 151–64. http://dx.doi.org/10.1080/00036818508839566.

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39

Górak, Andrzej, and Andrzej Stankiewicz. "Intensified Reaction and Separation Systems." Annual Review of Chemical and Biomolecular Engineering 2, no. 1 (July 15, 2011): 431–51. http://dx.doi.org/10.1146/annurev-chembioeng-061010-114159.

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40

Elderfield, D., and M. Wilby. "Scaling in reaction-diffusion systems." Journal of Physics A: Mathematical and General 20, no. 2 (February 1, 1987): L77—L83. http://dx.doi.org/10.1088/0305-4470/20/2/007.

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41

Banaji, Murad. "Monotonicity in chemical reaction systems." Dynamical Systems 24, no. 1 (March 2009): 1–30. http://dx.doi.org/10.1080/14689360802243813.

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42

Krishnan, Madhavi, Vijay Namasivayam, Rongsheng Lin, Rohit Pal, and Mark A. Burns. "Microfabricated reaction and separation systems." Current Opinion in Biotechnology 12, no. 1 (February 2001): 92–98. http://dx.doi.org/10.1016/s0958-1669(00)00166-x.

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43

Kirkilionis, M. "Exploration of cellular reaction systems." Briefings in Bioinformatics 11, no. 1 (January 1, 2010): 153–78. http://dx.doi.org/10.1093/bib/bbp062.

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44

Craciun, Gheorghe, Maya Mincheva, Casian Pantea, and Polly Y. Yu. "Delay stability of reaction systems." Mathematical Biosciences 326 (August 2020): 108387. http://dx.doi.org/10.1016/j.mbs.2020.108387.

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Oshanin, G. S., and S. F. Burlatsky. "Reaction kinetics in polymer systems." Journal of Statistical Physics 65, no. 5-6 (December 1991): 1109–22. http://dx.doi.org/10.1007/bf01049601.

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Ehrenfeucht, A., and G. Rozenberg. "Introducing time in reaction systems." Theoretical Computer Science 410, no. 4-5 (February 2009): 310–22. http://dx.doi.org/10.1016/j.tcs.2008.09.043.

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Markholiya, T. P., I. I. Kozelkova, T. M. Bragina, and L. M. Aksel'rod. "Reaction-sintered carbide-nitride systems." Refractories 31, no. 9-10 (September 1990): 550–53. http://dx.doi.org/10.1007/bf01282790.

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Cho, Dong-Gyu, and Jonathan L. Sessler. "Modern reaction-based indicator systems." Chemical Society Reviews 38, no. 6 (2009): 1647. http://dx.doi.org/10.1039/b804436h.

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van Vuuren, J. "Resilience in reaction-diffusion systems." IMA Journal of Applied Mathematics 63, no. 2 (October 1, 1999): 179–97. http://dx.doi.org/10.1093/imamat/63.2.179.

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Barzykin, A. V., and M. Tachiya. "Reaction Kinetics in Microdisperse Systems." Heterogeneous Chemistry Reviews 3, no. 2 (June 1996): 105–67. http://dx.doi.org/10.1002/(sici)1234-985x(199606)3:2<105::aid-hcr56>3.0.co;2-3.

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