globals  [ g k1 k2 name ext number full_name nn nnvar x y counter patch-state ]

;;see Info tab for full description of variables
;; g         - speed of change from infected state to sick state
;; k1        - negative tendency to be infected by infected neighbors
;; k2        - negative tendency to be infected by sick neighbors
;; name      - part of the name of the file
;; nn        - name of the file with the number
;; ext       - extension of the image files
;; number    - number of the file
;; full_name - full name of the file
;; nnvar     - name of the file where variables are stored


patches-own [state new-state g-multiplicity]

to setup
  clear-all
  ask patches
    [ set state 
      precision ( random-exponential ( (max-state + 1) * mean-position) ) rounding 
      ;; picks a state from 0 to max-state with exponential distribution with mean defined 
      ;;by the mean-position and precision to rounding number of decimal 
      set pcolor scale-color blue state 0 max-state ]
  set g g-0 
      ;;sets g to initial g level
      ;;g affect the rate by which cells get infected
  set k1 k1-0
  set k2 k2-0
      ;; set k1 and k2 to intial state
  
  set ext ".png"
      ;; sets extension for the file
  reset-ticks
  if print-images 
     [ save-image-1
       export-variables
       write-matrix ]
end

to go
  ;; first all the patches compute their new state
  ask patches [ find-new-state ]
  ;; only once all the patches have computed their new state
  ;; do they actually change state
  ask patches
  [ set state new-state ;; state of the patch is replaced by calculated new state
    set pcolor scale-color blue state 0 max-state ]
  ;; sets patches to a new calculated state
  ;set g ( g + g-decay )
  ;set k1 ( k1 + k1-decay )
  ;set k2 ( k2 + k2-decay )
  ;; changes the g, k1 and k2 value by the step g-decay 
  ;; have been commented out to speed up the calculation but this procedure is functional upon uncommenting
  if print-images 
     [ save-image 
       ;export-variables
       ;; export variables for each step was commented out to save time and disk space
       ;; this procedure is functional upon uncommenting
       write-matrix 
       ]
  tick
end

to find-new-state  ;; patch procedure
  ifelse state = max-state  ;; ill?
    [ set new-state 0 ] ;; the ill cell gets miraculously well - the PHASE TRANSITION
    [ let a count neighbors with [state > 0 and state < max-state]  ;; count infected
      let b count neighbors with [state = max-state] ;; count ill
      ifelse state = 0  ;; healthy?
      [ set new-state precision (
          ( (a / k1) + (b / k2) )
          * (1 + random-float ground-state-noise-level)) rounding                 
        ] ;; healthy cell gets immediately infected by to the average of its sick and healthy neigbours - the PHASE TRANSITION
         [ let s state + sum [state] of neighbors  ;; s is sum of infections around
           set new-state  ( precision ((s / (a + b + 1)) * (1 + random-float averaging-noise-level) + g * (1 + random-float g-noise-level)) rounding )  ] 
         ;; new level of infection is average of infections around plus g multiplied - the REACTION - DIFFUSION PROCESS
      if new-state > max-state   ;; don't exceed the maximum state
        [ set new-state max-state ] ]
end

to save-image ;;export images
  set number ( ticks + 1) 
  if  (number < 10)
    [set name "img4\\BZ000"] 
  if (9 < number) and (number < 100)
    [set name "img4\\BZ00"] 
  if (99 < number) and (number < 1000)
    [set name "img4\\BZ0"] 
  if (999 < number)
    [set name "img4\\BZ"]   
    
  set nn word name number
  set full_name  word nn ext
  export-view full_name 
end

to save-image-1 ;;export images
  set number ticks 
  if  (number < 10)
    [set name "img4\\BZ000"] 
  if (9 < number) and (number < 100)
    [set name "img4\\BZ00"] 
  if (99 < number) and (number < 1000)
    [set name "img4\\BZ0"] 
  if (999 < number)
    [set name "img4\\BZ"]   
    
  set nn word name number
  set full_name  word nn ext
  export-view full_name 
end

to export-variables ;;export variables
  
  set nnvar word nn "-variable.txt"
  
  file-open nnvar
  
  file-write "Exponential distribution of initial states" 

  file-write "Max State "
  file-print max-state
  
  file-write "k1-0 "
  file-print k1-0
  
  file-write "k1-decay "
  file-print k1-decay

  file-write "k1 "
  file-print k1
  
  file-write "k2-0 "
  file-print k2-0
  
  file-write "k2-decay "
  file-print k2-decay

  file-write "k2 "
  file-print k2

  file-write "g-0 "
  file-print g-0  
  
  file-write "g-decay "
  file-print g-decay
    
  file-write "g "
  file-print g 
  
  file-write "rounding "
  file-print rounding
  
  file-write "mean "
  file-print mean-position
  
  file-write "initial state noise "
  file-print ground-state-noise-level
  
  file-write "average state noise"
  file-print averaging-noise-level
  
  file-write "g noise "
  file-print g-noise-level
  
  file-write "model size"
  file-print max-pycor

    
  file-flush
  file-close-all
end

to write-matrix ;; export state values in a x-y matrix
  
  set nnvar word nn "-matrix.txt"
  
  file-open nnvar
    
  set counter 0
  
  let x-extent n-values world-width  [? + min-pxcor] 
  let y-extent n-values world-height [? + min-pycor] 
  foreach x-extent 
  [ let x1 ? 
    foreach y-extent 
    [ file-type (word [state] of patch x1 ? ",") ] 
    file-print "" ] 
  file-close 
end 
  
  
 

; Copyright 2003 Uri Wilensky.
; Copyright 2014 Dalibor Stys 
; See Info tab for full copyright and license.
@#$#@#$#@
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ticks
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SLIDER
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setup
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OBSERVER
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k2-0
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SLIDER
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g-decay
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SLIDER
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g-0
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MONITOR
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g evolution
g
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SLIDER
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rounding
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SLIDER
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mean-position
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SLIDER
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k1-decay
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SLIDER
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MONITOR
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k1 evolution
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MONITOR
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k2 evolution
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SLIDER
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ground-state-noise-level
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SWITCH
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print-images
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PLOT
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state of patch 20 20
NIL
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true
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"" ""
PENS
"default" 1.0 0 -16777216 true "" "plot [state] of patch 20 20"

MONITOR
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[state] of patch 20 20
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SLIDER
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g-noise-level
g-noise-level
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SLIDER
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averaging-noise-level
averaging-noise-level
-1
1
0.14
0.001
1
NIL
HORIZONTAL

@#$#@#$#@
## THIS MODEL IS AN EXTENSION OF THE MODEL SUPPLIED IN THE MODELS LIBRARY OF NETLOGO
Part of the original text was left in the model description by Wilensky 2003

## WHAT IS IT?

The Belousov-Zhabotinsky reaction (or B-Z reaction for short) is an unusual chemical reaction.  Instead of steadily moving towards a single equilibrium state, it oscillates back and forth between two such states.  Before this "chemical oscillator" was discovered, it was thought that such a reaction could not exist.

If you do the reaction in a mixed beaker, the whole beaker regularly changes color from yellow to clear and back again, over and over.  In this case, we say that the reaction is oscillating in time.  However, if you do the reaction in a thin layer of fluid trapped, then a beautiful pattern emerges of concentric or spiral waves of color change passing through the fluid.  

The B-Z reaction is a redox reaction that periodically moves between an oxidized and a reduced state, and has been demonstrated for various chemicals.  The chemical reaction is quite complex (including 18 reactions and 21 species, according to the Fields-Koros-Noyes model). There is little hope that a model with so many chemical reactions and spatial distribution will be fully quantitatively modelled. 

As it turned out from the experiments (Stys et al. submitted), there are two independent processes of which one includes "ignition" which probably has the character of local phase transition and "reaction-diffusion" process - the long known chemical reaction. 

The abstract features shared by the real reaction and this model include:

1. Two end states.  
2. A positive feedback mechanism.  
3. A negative feedback mechanism.

The model included in the NetLogo Models Library was originally Dewndey published in Scientific American in 1988. Presented model is the expansion of the implementation of for Netlogo by Uri Wilensky in 2003. The model is a multsitate cellular automaton (or CA) that, in its original implementation, produced spiral waves.  Similar spiral waves have also been observed in biological systems, such as slime molds. However, such spiral waves are seldom observed in the B-Z reaction. There are mostly observed a few stages until the final stage of rather irregular swirls is achieved. 

Thus, the original model of Dewney calls for extensions. As we show, it is posible to extent the models to cover more of the features of the real B-Z reaction. 

Already in the original Dewndey model, certain number of allowed state levels is needed - minimally 80, dependent on the level of the constant g. It looks that DEWNDEY´S MODEL IS INHERENTLY A MODEL OF SEMI-DISCRETE SYSTEM. And, with critical view, any natural objects are semi-discrete, i.e. discrete but interacting. 

The model as it is now is a specific case of multistate celular automaton which is also cyclic (i.e. when a treshold is reached, he value is re-set to zero) and totalistic, although quite unusual since two different rules for averaging are applied. We inluded also stochastic element to each of the processes. By a genial intuition of Dewndey included the two totalistics processes, one for "healthy" cells - the phase transition and the other for "infected" cells - the reaction-diffusion. 

The process in the cell in the state 0 is ignited by the infection from neigbouring "infected" cell. This is similar to the "initial" process which we assume to be of the PHASE TRANSITION character.

In any other cases we assume the MOST PRIMITIVE REACTION-DIFFUSION PROCESS. The diffusion process is simulated by averaging. The reaction process is simulated by the reaction of the first order. By considering 200 levels and precision of state value to 10 decimal points we are within 12 orderes of magnitude of the definition of the state - i.e. we have smoothened our simulation. 

To some extent the model now explores TECHNICAL LIMITS OF THE REACTION - DIFFUSION MODELLING. I.e. in each case when the "diffusion" and the "reaction" process occur with different rate constant, the faster process must be modelled as stepwise. The "smoothness" in contrast to the inherently discrete character of computer simulation is usually solved by addition of the noise, in our case we add gaussian noise to each process.  

The interesting observation in the given combination of constants is that while the "chemical" noise - (g-noise-level) may be increased up to 30% (!!), the system behaviour is critical to "diffusive noise" (averaging-noise-level) which within the range of maximal levels 12% to 14.5 % discriminates three different types of behaviour: (1) formation of large circular waves at averaging-noise-level at noise level below 12%, (2) formation of a few centres of dense circular waves in the window 12% to 14.5% and (3) quick prevalence of circular waves above 14.5%. The critical behaviour leading to swirl formation is is merging of two dense waves. 

## HOW IT WORKS

The original Wilensky 2003 model is described as follows: 

Each cell has a state which is an integer from 0 to max-state.  We choose to show state 0 as black, max-state as white, and intermediate states as shades of blue.

Suppose we call state 0 "healthy", max-state "sick", and anything in between "infected".  Then the rules for how each cell changes at each step can be described as follows:

1. A cell that is sick becomes healthy 
% In the terminology of the article (Stys et al. submitted) the primitive object underwent phase transition to the activated phase state - its Gibbs energy of phase transition was released / consumed.

2. A cell that is healthy may become infected, if enough of its eight neighbors are infected or sick.  Whether this happens is affected by the k1 and k2 sliders.  (Lower k1 means higher tendency to be infected by infected neighbors; lower k2 means higher tendency to be infected by sick neighbors.)
% In the terminology of the article (Stys et al. submitted) the phase transition progresses from cells in activated phase state to neighbouring cells in ground phase state. 

3. A cell that is infected computes its new state by averaging the states of itself and its eight neighbors, then adding the value of the g slider.  (Higher g means infected cells get sicker more rapidly.)
% In the terminology of the article (Stys et al. submitted) the averaging of states is the most primitive model of the DIFFUSION PROCESS and the addition of g value is the most primitive model of the CHEMICAL REACTION. The Gibbs energy released / consumed by the phase transition is equalibrated by the reaction-diffusion process. 

1 is the negative feedback; 2 and 3 are the positive feedbacks

The features different form the Wilensky 2003 model act in the following way:

Initial state is calculated by the exponential distribution. Excess of the maximal state is allowed which, upon the use of proper parameters, results in obtaining of just a few points from which waves evolve.
 
The k1, k2 and g constants may be decaying, which should resemble the change in chemical composition of the solution.

Noise is added to each of the processes. 

Level of state is averaged to a given number of decimal places - not necessarily to integer. 

Images may be printed. It is assumed that you create a img4 folder in the same folder in which the model is stored. 

These are only qualitative descriptions.  To see the actual math used, look at the FIND-NEW-STATE procedure in the Code tab.


## HOW TO USE IT

Press SETUP to initialize each cell in the grid to a random state.

Press GO to run the model.

## THINGS TO NOTICE

Run the model with the default slider settings. This procedure best resembles real B-Z reaction course. (Actually, quite a bit better then the original Dewndey model.) But it still lacks some of the important features and namely the early period is significantly different.

In the beginning of the run the first 5 timeframes are "preparatory" i.e. may cells get infected and only a finite number becomes infected. If sufficiently low number is infected at the timeframe 5, we observe evolution from a small number of centres which resembles well the behaviour of obesrved in the experiment. 

In the next stage (frame 18) there is already observed rounding of internal objects. In a while the shape of emanating centers becomes circular and eventually the space is filled by waves emanating from circular objects similar to those observed in the experiment. 

The model also very well simulates the initialisation of swirls. Form swirls emanate elliptical waves which, after a really long simulations, result in interchanging swir-wave behaviour.   

Equally well as in the original Dewndey´s model, this model is based to a big extent on intuition rather than serious numerical analysis. As well, we have so far done only a few basic tests and have no complete view what happens and why. 

We would be happy for any feedback at stys@jcu.cz, zhyrova@frov.jcu.cz or nahlik@frov.jcu.cz . 

## THINGS TO TRY

What is the effect of varying the different sliders?  You can think of k1 and k2 as affecting the tendency for healthy cells to become infected, and g as affecting the speed with which the infection gets worse.

Other types of noise may be used.

Other types of neighbourhood, i.e. circular, may be used - although we may see that circular object occur naturally although the square Moore neighbourood is used.

And, if you like, let us know what you found at stys@jcu.cz, zhyrova@frov.jcu.cz or nahlik@frov.jcu.cz . 

## NETLOGO FEATURES

`find-new-state` is a long and rather complicated procedure.  It could be clearer if it were split into subprocedures, but then the model wouldn't run quite as fast.  Since this particular CA takes so many iterations to settle into its characteristic pattern, we decided that speed was important.

## RELATED MODELS

Boiling, in the Physics/Heat section, is another cellular automaton that uses similar, though simpler, rules.  The early stages of the Boiling model resemble the early stages of this model.

Fireflies, in the Biology section, is analogous to the B-Z reaction in a stirred beaker (the whole beaker "synchronizes" so it's switching back and forth all at once, like the fireflies).

Many models in the NetLogo models library can be thought as systems composed of positive and/or negative feedback mechanisms.

## CREDITS AND REFERENCES

The B-Z reaction is named after Boris Belousov and Anatol Zhabotinsky, the Russian scientists who discovered it in the 1950's.

A discussion of the chemistry behind the reaction, plus a movie and some pictures, are available at http:http://www.faidherbe.org/site/cours/dupuis/oscil.htm.

The cellular automaton was presented by A.K. Dewdney in his "Computer Recreations" column in the August 1988 of Scientific American.

See http://www.hermetic.ch/pca/bz.htm for a pretty screen shot of the cellular automaton running on a very large grid (using custom software for Windows, not NetLogo).

In the Bachelor Thesis of Saskia Hiltemann you my find more information about the behaviour of multistate cellular automata: Saskia Hiltemann, Multi-coloured Cellular Automata http://www.liacs.nl/assets/Bachelorscripties/16-SaskiaHiltemann.pdf


## HOW TO CITE

If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software:

* Stys, D. (2014). Extended B-Z Reaction model in NetLogo !!! Include web page !!!!
* Dewdney, A. K. 1988 The hodgepodge machine makes waves. Scientific American, 259(2), 104–107.
* Wilensky, U. (2003).  NetLogo B-Z Reaction model.  http://ccl.northwestern.edu/netlogo/models/B-ZReaction.  Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.
* Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.

## COPYRIGHT AND LICENSE

For part of the model applies the following:

Copyright 2003 Uri Wilensky.

![CC BY-NC-SA 3.0](http://i.creativecommons.org/l/by-nc-sa/3.0/88x31.png)

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.

This model was created as part of the projects: PARTICIPATORY SIMULATIONS: NETWORK-BASED DESIGN FOR SYSTEMS LEARNING IN CLASSROOMS and/or INTEGRATED SIMULATION AND MODELING ENVIRONMENT. The project gratefully acknowledges the support of the National Science Foundation (REPP & ROLE programs) -- grant numbers REC #9814682 and REC-0126227.
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Polygon -7500403 true true 165 255 210 210 255 195 225 255 165 285
Polygon -7500403 true true 135 180 90 135 45 120 75 180 135 210
Polygon -7500403 true true 165 180 165 210 225 180 255 120 210 135
Polygon -7500403 true true 135 105 90 60 45 45 75 105 135 135
Polygon -7500403 true true 165 105 165 135 225 105 255 45 210 60
Polygon -7500403 true true 135 90 120 45 150 15 180 45 165 90

square
false
0
Rectangle -7500403 true true 30 30 270 270

square 2
false
0
Rectangle -7500403 true true 30 30 270 270
Rectangle -16777216 true false 60 60 240 240

star
false
0
Polygon -7500403 true true 151 1 185 108 298 108 207 175 242 282 151 216 59 282 94 175 3 108 116 108

target
false
0
Circle -7500403 true true 0 0 300
Circle -16777216 true false 30 30 240
Circle -7500403 true true 60 60 180
Circle -16777216 true false 90 90 120
Circle -7500403 true true 120 120 60

tree
false
0
Circle -7500403 true true 118 3 94
Rectangle -6459832 true false 120 195 180 300
Circle -7500403 true true 65 21 108
Circle -7500403 true true 116 41 127
Circle -7500403 true true 45 90 120
Circle -7500403 true true 104 74 152

triangle
false
0
Polygon -7500403 true true 150 30 15 255 285 255

triangle 2
false
0
Polygon -7500403 true true 150 30 15 255 285 255
Polygon -16777216 true false 151 99 225 223 75 224

truck
false
0
Rectangle -7500403 true true 4 45 195 187
Polygon -7500403 true true 296 193 296 150 259 134 244 104 208 104 207 194
Rectangle -1 true false 195 60 195 105
Polygon -16777216 true false 238 112 252 141 219 141 218 112
Circle -16777216 true false 234 174 42
Rectangle -7500403 true true 181 185 214 194
Circle -16777216 true false 144 174 42
Circle -16777216 true false 24 174 42
Circle -7500403 false true 24 174 42
Circle -7500403 false true 144 174 42
Circle -7500403 false true 234 174 42

turtle
true
0
Polygon -10899396 true false 215 204 240 233 246 254 228 266 215 252 193 210
Polygon -10899396 true false 195 90 225 75 245 75 260 89 269 108 261 124 240 105 225 105 210 105
Polygon -10899396 true false 105 90 75 75 55 75 40 89 31 108 39 124 60 105 75 105 90 105
Polygon -10899396 true false 132 85 134 64 107 51 108 17 150 2 192 18 192 52 169 65 172 87
Polygon -10899396 true false 85 204 60 233 54 254 72 266 85 252 107 210
Polygon -7500403 true true 119 75 179 75 209 101 224 135 220 225 175 261 128 261 81 224 74 135 88 99

wheel
false
0
Circle -7500403 true true 3 3 294
Circle -16777216 true false 30 30 240
Line -7500403 true 150 285 150 15
Line -7500403 true 15 150 285 150
Circle -7500403 true true 120 120 60
Line -7500403 true 216 40 79 269
Line -7500403 true 40 84 269 221
Line -7500403 true 40 216 269 79
Line -7500403 true 84 40 221 269

x
false
0
Polygon -7500403 true true 270 75 225 30 30 225 75 270
Polygon -7500403 true true 30 75 75 30 270 225 225 270

@#$#@#$#@
NetLogo 5.0.4
@#$#@#$#@
setup
repeat 200 [ go ]
@#$#@#$#@
@#$#@#$#@
@#$#@#$#@
@#$#@#$#@
default
0.0
-0.2 0 0.0 1.0
0.0 1 1.0 0.0
0.2 0 0.0 1.0
link direction
true
0
Line -7500403 true 150 150 90 180
Line -7500403 true 150 150 210 180

@#$#@#$#@
0
@#$#@#$#@