scholarly journals Chaos and Pattern Formation in Spatial Phytoplankton Dynamics

2020 ◽  
Author(s):  
Randhir Singh Baghel
Keyword(s):  
Wide Spectrum ◽  
Reaction Diffusion ◽  
Spatiotemporal Chaos ◽  
Two Dimensions ◽  
System Stability ◽  
Growth Fraction ◽  
Phytoplankton Dynamics ◽  
Temporal Models ◽  
Spatially Extended ◽  
Temporal Model

In this paper the dynamics of spatially extended infected phytoplankton with the Holling type II functional response and logistically growing susceptible phytoplankton system is studied. The proposed model is an extension of temporal model available [6], in spatiotemporal domain. The reaction diffusion system exhibits spatiotemporal chaos in phytoplankton dynamics. This is particularly important for the spatially extended systems that are studied in this paper as they display a wide spectrum of ecologically relevant behavior, including chaos. In this system stability of the system is studied with respect to disease contact rate and the growth fraction of infected phytoplankton indirectly rejoin the susceptible phytoplankton population. The results of numerical experiments in one dimension and two dimensions in space as well as time series in temporal models are presented using MATLAB simulation. Moreover, the stability of the corresponding temporal model is studied analytically. Finally, the comparison of the three types of numerical experimentations are discussed in conclusion.

Karhunen-Loève local characterization of spatiotemporal chaos in a reaction-diffusion system

2000 ◽  
Vol 61 (2) ◽  
pp. 1382-1385 ◽  
Author(s):  
Matthias Meixner ◽  
Scott M. Zoldi ◽  
Sumit Bose ◽  
Eckehard Schöll
Keyword(s):  
Reaction Diffusion ◽  
Spatiotemporal Chaos ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Local Characterization

scholarly journals Comparison between VAR, GSTAR, FFNN-VAR and FFNN-GSTAR Models for Forecasting Oil Production

MATEMATIKA ◽  
2018 ◽  
Vol 34 (1) ◽  
pp. 103-111 ◽  
Author(s):  
Suhartono Suhartono ◽  
Dedy Dwi Prastyo ◽  
Heri Kuswanto ◽  
Muhammad Hisyam Lee
Keyword(s):  
Forecast Accuracy ◽  
Oil Production ◽  
Training Data ◽  
Temporal Data ◽  
Monthly Data ◽  
Vector Autoregressive ◽  
Temporal Models ◽  
Accurate Forecast ◽  
Temporal Model ◽  
Spatio Temporal

Monthly data about oil production at several drilling wells is an example of spatio-temporal data. The aim of this research is to propose nonlinear spatio-temporal model, i.e. Feedforward Neural Network - Vector Autoregressive (FFNN-VAR) and FFNN - Generalized Space-Time Autoregressive (FFNN-GSTAR), and compare their forecast accuracy to linear spatio-temporal model, i.e. VAR and GSTAR. These spatio-temporal models are proposed and applied for forecasting monthly oil production data at three drilling wells in East Java, Indonesia. There are 60 observations that be divided to two parts, i.e. the first 50 observations for training data and the last 10 observations for testing data. The results show that FFNN-GSTAR(11) and FFNN-VAR(1) as nonlinear spatio-temporal models tend to give more accurate forecast than VAR(1) and GSTAR(11) as linear spatio-temporal models. Moreover, further research about nonlinear spatio-temporal models based on neural networks and GSTAR is needed for developing new hybrid models that could improve the forecast accuracy.

scholarly journals EVOLVING LOCALIZATIONS IN REACTION-DIFFUSION CELLULAR AUTOMATA

2008 ◽  
Vol 19 (04) ◽  
pp. 557-567 ◽  
Author(s):  
ANDREW ADAMATZKY ◽  
LARRY BULL ◽  
PIERRE COLLET ◽  
EMMANUEL SAPIN
Keyword(s):  
Cellular Automata ◽  
Chemical Species ◽  
Reaction Diffusion ◽  
Discrete Systems ◽  
Chemical Systems ◽  
Transition Functions ◽  
Phenomenological Studies ◽  
Spatially Extended ◽  
Limited Diffusion

We consider hexagonal cellular automata with immediate cell neighbourhood and three cell-states. Every cell calculates its next state depending on the integral representation of states in its neighbourhood, i.e., how many neighbours are in each one state. We employ evolutionary algorithms to breed local transition functions that support mobile localizations (gliders), and characterize sets of the functions selected in terms of quasi-chemical systems. Analysis of the set of functions evolved allows to speculate that mobile localizations are likely to emerge in the quasi-chemical systems with limited diffusion of one reagent, a small number of molecules are required for amplification of travelling localizations, and reactions leading to stationary localizations involve relatively equal amount of quasi-chemical species. Techniques developed can be applied in cascading signals in nature-inspired spatially extended computing devices, and phenomenological studies and classification of non-linear discrete systems.

The Chemical Basis of Morphogenesis (1952)

2004 ◽  
Author(s):  
Alan Turing
Keyword(s):  
Reaction System ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Stationary Waves ◽  
Chemical Substances ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Random Disturbances ◽  
Physical Laws ◽  
And Diffusion

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 homogenous, 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 on Hydra 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. In this section a mathematical model of the growing embryo will be described. This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge. The model takes two slightly different forms. In one of them the cell theory is recognized but the cells are idealized into geometrical points.

Radiolaria: validating the Turing theory

2017 ◽  
Author(s):  
Bernard Richards
Keyword(s):  
Diffusion Mechanism ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Black And White ◽  
Alan Turing ◽  
School Sports ◽  
The University ◽  
University Of Manchester

In his 1952 paper ‘The chemical basis of morphogenesis’ Turing postulated his now famous Morphogenesis Equation. He claimed that his theory would explain why plants and animals took the shapes they did. When I joined him, Turing suggested that I might solve his equation in three dimensions, a new problem. After many manipulations using rather sophisticated mathematics and one of the first factory-produced computers in the UK, I derived a series of solutions to Turing’s equation. I showed that these solutions explained the shapes of specimens of the marine creatures known as Radiolaria, and that they corresponded very closely to the actual spiny shapes of real radiolarians. My work provided further evidence for Turing’s theory of morphogenesis, and in particular for his belief that the external shapes exhibited by Radiolaria can be explained by his reaction–diffusion mechanism. While working in the Computing Machine Laboratory at the University of Manchester in the early 1950s, Alan Turing reignited the interests he had had in both botany and biology from his early youth. During his school-days he was more interested in the structure of the flowers on the school sports field than in the games played there (see Fig. 1.3). It is known that during the Second World War he discussed the problem of phyllotaxis (the arrangement of leaves and florets in plants), and then at Manchester he had some conversations with Claude Wardlaw, the Professor of Botany in the University. Turing was keen to take forward the work that D’Arcy Thompson had published in On Growth and Form in 1917. In his now-famous paper of 1952 Turing solved his own ‘Equation of Morphogenesis’ in two dimensions, and demonstrated a solution that could explain the ‘dappling’—the black-and-white patterns—on cows. The next step was for me to solve Turing’s equation in three dimensions. The two-dimensional case concerns only surface features of organisms, such as dappling, spots, and stripes, whereas the three-dimensional version concerns the overall shape of an organism. In 1953 I joined Turing as a research student in the University of Manchester, and he set me the task of solving his equation in three dimensions. A remarkable journey of collaboration began. Turing chatted to me in a very friendly fashion.

scholarly journals From chemical soup to computing circuit: transforming a contiguous chemical medium into a logic gate network by modulating its external conditions

2019 ◽  
Vol 16 (158) ◽  
pp. 20190190
Author(s):  
Matthew Egbert ◽  
Jean-Sébastien Gagnon ◽  
Juan Pérez-Mercader
Keyword(s):  
Logic Gate ◽  
Reaction Diffusion ◽  
Logic Gates ◽  
Full Adder ◽  
Integrated Network ◽  
Lighting Conditions ◽  
Reaction Diffusion Model ◽  
Spatially Extended ◽  
External Conditions ◽  
Spatially Extended Systems

It has been shown that it is possible to transform a well-stirred chemical medium into a logic gate simply by varying the chemistry’s external conditions (feed rates, lighting conditions, etc.). We extend this work, showing that the same method can be generalized to spatially extended systems. We vary the external conditions of a well-known chemical medium (a cubic autocatalytic reaction–diffusion model), so that different regions of the simulated chemistry are operating under particular conditions at particular times. In so doing, we are able to transform the initially uniform chemistry, not just into a single logic gate, but into a functionally integrated network of diverse logic gates that operate as a basic computational circuit known as a full-adder.

scholarly journals Spatiotemporal chaos and quasipatterns in coupled reaction–diffusion systems

2020 ◽  
Vol 409 ◽  
pp. 132475 ◽  
Author(s):  
Jennifer K. Castelino ◽  
Daniel J. Ratliff ◽  
Alastair M. Rucklidge ◽  
Priya Subramanian ◽  
Chad M. Topaz
Keyword(s):  
Reaction Diffusion ◽  
Spatiotemporal Chaos ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Coupled Reaction

scholarly journals Ecological feedback on diffusion dynamics

2019 ◽  
Vol 6 (2) ◽  
pp. 181273 ◽  
Author(s):  
Hye Jin Park ◽  
Chaitanya S. Gokhale
Keyword(s):  
Public Goods ◽  
Spatial Patterns ◽  
Spatial Organization ◽  
Evolutionary Dynamics ◽  
Evolutionary Process ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Public Goods Games ◽  
Diffusion Dynamics ◽  
Spatially Extended

Spatial patterns are ubiquitous across different scales of organization in ecological systems. Animal coat pattern, spatial organization of insect colonies and vegetation in arid areas are prominent examples from such diverse ecologies. Typically, pattern formation has been described by reaction–diffusion equations, which consider individuals dispersing between subpopulations of a global pool. This framework applied to public goods game nicely showed the endurance of populations via diffusion and generation of spatial patterns. However, how the spatial characteristics, such as diffusion, are related to the eco-evolutionary process as well as the nature of the feedback from evolution to ecology and vice versa, has been so far neglected. We present a thorough analysis of the ecologically driven evolutionary dynamics in a spatially extended version of ecological public goods games. Furthermore, we show how these evolutionary dynamics feed back into shaping the ecology, thus together determining the fate of the system.

COEXISTENCE OF LOW- AND HIGH-DIMENSIONAL SPATIOTEMPORAL CHAOS IN A CHAIN OF DISSIPATIVELY COUPLED CHUA’S CIRCUITS

1994 ◽  
Vol 04 (03) ◽  
pp. 639-674 ◽  
Author(s):  
A.L. ZHELEZNYAK ◽  
L.O. CHUA
Keyword(s):  
Phase Space ◽  
Initial Conditions ◽  
Coupling Parameter ◽  
Reaction Diffusion ◽  
Cellular Neural Network ◽  
Spatiotemporal Chaos ◽  
Matrix Phase ◽  
High Dimensional ◽  
The Matrix ◽  
A Chain

Spatiotemporal dynamics of a one-dimensional cellular neural network (CNN) made of Chua’s circuits which mimics a reaction-diffusion medium is considered. An approach is presented to analyse the properties of this reaction-diffusion CNN through the characteristics of the attractors of an associated infinite-dimensional dynamical system with a matrix phase space. Using this approach, the spatiotemporal correlation dimension of the CNN’s spatiotemporal patterns is computed over various ranges of the diffusion coupling parameter, length of the chain, and initial conditions. It is shown that in a finite-dimensional projection of the matrix phase space of the CNN, both low- and high-dimensional attractors corresponding to different initial conditions coexist.

Balanced-norm error estimates for sparse grid finite element methods applied to singularly perturbed reaction–diffusion problems

2019 ◽  
Vol 27 (1) ◽  
pp. 37-55 ◽  
Author(s):  
Stephen Russell ◽  
Martin Stynes
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Two Dimensions ◽  
Sparse Grid ◽  
Singularly Perturbed ◽  
Balanced Norm ◽  
Norm Error ◽  
Diffusion Problems

Abstract We consider a singularly perturbed linear reaction–diffusion problem posed on the unit square in two dimensions. Standard finite element analyses use an energy norm, but for problems of this type, this norm is too weak to capture adequately the behaviour of the boundary layers that appear in the solution. To address this deficiency, a stronger so-called ‘balanced’ norm has been considered recently by several researchers. In this paper we shall use two-scale and multiscale sparse grid finite element methods on a Shishkin mesh to solve the reaction–diffusion problem, and prove convergence of their computed solutions in the balanced norm.

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