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Phyllotaxis

2017-01-17T22:44:27Z

Bedn00: Phyllotaxis is the repeated production of similar elements (leaves, petals) on the tip of a plant, producing a regular pattern. Goal of the simulation in NetLogo is to create a model of phyllotaxis using different approaches.

Project name: Phyllotaxis

Class: 4IT496 Simulation of Systems (WS 2016/2017)

Author: Bc. Naďa Bednárová

Software used: NetLogo

== Introduction ==

Phyllotaxis is the repeated production of similar elements (leaves, petals) on the tip of a plant, producing a regular pattern. <ref name = "Douady"> S. Douady, Y. Couder. "Phyllotaxis as a Dynamical Self Organizing Process. Part I." J. Theor. Biol 178 (1996): 255-274. </ref> These arrangements of elements fall into two categories - creating one leaf at a time in distichous or spiral mode (following Fibonacci series or golden mean) or creating two leaves simultaneously at the same height. In this simulation I will study the first category.

According to Douady and Couder<ref name = "Douady"/>, there are three main approaches to the study of the origin of leaves arrangement.

=== Geometrical Approach ===

This approach studies the geometry behind leaves arrangement without trying to explain why or how exactly these patterns are formed.

The simplest model of the formation of phyllotactic patterns (for example in sunflowers or daisies) uses a special form of Fermat’s spiral equation. The formula was originally proposed by Vogel <ref>Vogel, Helmut. "A better way to construct the sunflower head." Mathematical biosciences 44.3-4 (1979): 179-189.</ref> in the following format :

<math> \phi = n \times 137.5^\circ </math>

<math> r = c \times \sqrt{n} </math>

Where <math display="inline"> \phi </math> represents the angle between the reference direction (direction of the first floret) and ''n''-th floret and is calculated as a product of ''n'' (denoting the number of a floret in given order) and angle between two subsequent florets. In order to get the Fibonacci sequences, the golden angle of approximately 137.5 degrees is used. Finally, ''r'' is the distance between the middle (capitulum) and the center of given floret, giving the parameter ''c'', so called "growing factor" <ref name="ch4"> Prusinkiewicz, Przemyslaw, and Aristid Lindenmayer. The algorithmic beauty of plants. Springer Science & Business Media, 2012 : 99-105. </ref>.

Fibonacci sequence (FS) is the following sequence of numbers : 1, 1, 2, 3, 5, 8, 13, 21, 34 ... , where each number is a sum of two previous numbers. Using simulation we can verify that the spiral patterns formed on a flower using golden angle as a distance of two subsequent petals create Fibonacci sequence, such that the number of spirals formed by leaves/petals in one direction is a number from the Fibonacci sequence; the number of spirals in the other direction is the number that follows in FS and the approximation of the intersections of these two spirals (another set of spirals) can form their sum, thus the third number of FS <ref> Jean, Roger V. Phyllotaxis: a systemic study in plant morphogenesis. Cambridge University Press, 2009. </ref>. See the comparison of model vs. reality for examples.

=== Dynamics of Growth ===

This approach studies the dynamic principles behind morphogenesis forming phyllotactic structures. A basic dynamic model of leaves formation from the meristem was first proposed by Hofmeister in 1868. He observed that small protrusions, called primordia, are formed in the apical region of a plant, where each primordium is created in a place where it has the most space available for its growth (e.g. to be able to catch the most of sun rays) <ref name="Douady"/>.

=== Physiology of The Apical Meristem ===

In order to determine the physical processes responsible for the creation of phyllotactic patterns in plant's leaves or flower petals, different theories about interaction of primordia have been studied. These theories are based on either a diffusion of inhibitors, a reaction diffusion process (Turing), or a contact pressure. The interaction between primordia creates a process of repulsion or inhibition and it uses complicated dynamic equations. However, none of the theories gives a 100% correct model. <ref name="Douady"/>

=== Model of Douady and Couder ===

This model is based on the experiment done by Douady and Couder on a horizontal teflon dish filled with silicone oil placed in a vertical magnetic field <ref name="Douady"/>. The primordia are simulated by drops of ferrofluids that are periodically placed in the middle of the dish on given intervals T. The drops spontaneously move in the magnetic field away from the center of the dish. Every new drop is repeled from both the existing and the previous ones, causing it to move in the direction with minimum repulsive energy. The model of phyllotaxis inspired by this experiment combines previous dynamical and physiological models and creates patterns that can be studied by geometrical approach. It also allows various modifications, for example of the function of repulsive energy.

== Goals ==

Goal of the simulation in NetLogo is to create a model of formation of the arrangement of leaves based on different parameters configuration (number and size of the leaves, starting angle, distance etc.). Furthermore, some of above mentioned theories about morphogenesis / meristem growth will be simulated and Fibonacci sequence creation in phyllotactic patterns will be tested.

== Simulation ==

I created 3 simulations in NetLogo-5.3.1. First based on geometrical approach, second is a simple diffusion model with the same adjustable parameters as in the first one (also with a 3D model) and the third one is a dynamic model based on Douady and Couder approach.

=== Spiral Model ===

This model follows the geometrical approach described above, which means that it starts with the first leaf in the middle and then creates a new leaf at position <math display="inline"> x = r \times \cos{\phi} </math> and <math display="inline"> y = r \times \sin{\phi} </math>. We can then observe how different starting angles and distance parameters c change the pattern of the flower <ref name="ch4"/>.

==== Key Parameters ====

There are two functions, '''setup_spiral''' and '''do_spiral'''. The first one sets the environment and variables, while the other one periodically creates the leaves using equations described above.

Parameters that we can set are:
* '''angle''' - the distance between two following leaves
* '''leaf-size''' - the size of each leaf
* '''dist''' - the distance parameter c
* '''dynamic-leaf-size''' - in case dynamic leaf size is turned on, the size of leaves gradually increases with the distance from the center

[[File:2017BednarovaSpiral.jpg|800px]]

=== Diffusion Model ===

The diffusion model is based on a principle similar to a spiral, which means that similar parameters are set up, except that the leaves are created in the middle/center and they gradually move away in the direction of the defined angle. This model is a preliminary for the next model, where the original angle is not defined, but rather created dynamically.

==== Key Parameters ====

There are two functions, '''setup_diffuse''' and '''do_diffuse'''. On every tick, the main function '''do_diffuse''' creates one new leaf in the center with the orientation '''angle''' multiplied by ticks, which means that the first leaf will have the orientation equal to the original angle and every next leaf will have the same angle distance from the previous one. Furthermore, in every step, all existing leaves move in their direction by the distance set in '''dist'''.

Parameters that we can set are:
* '''angle''' - the distance between two following leaves
* '''leaf-size''' - the size of each leaf
* '''dist''' - the distance by which a leaf moves in every step in a given direction
* '''dynamic-leaf-size''' - same as in the previous model

[[File:2017BednarovaDiffuse.jpg|800px]]

=== Dynamic Model ===

According to the study by Douady and Couder <ref name="Douady"/>, unlike in the experiment, where the drops are placed in the middle of the dish, the model contains a central region representing the apex of a plant stem with radius rc (in our case same as r0). Primordia are created on the edge of this region, specifically on the place with the lowest repulsive energy E, which is the sum of repulsive energies of all already existing leaves. To calculate the energy, we need to create a potential primordia around the edge of the central region, called '''primordia''' also in the simulation code. These potential leaves can have different '''density''' around the apex, depending on how precise calculation is required. Then the energy E is calculated for each potential primordium and the one with minimal repulsive energy is transformed into real leaf.

==== Key Parameters ====

There are again two functions, '''setup_dynamic''' and '''do_dynamic'''. The important thing is that in '''setup_dynamic''' we set the parameter '''converg''' that is being plotted in order to study the angle representing the variable distance between two following leaves. As described in the model, the angle should gradually stabilise itself and converge to a value. In the '''do_dynamic''' method <math display="inline"> 360 \times density </math> the potential primordia are created around the central circle with radius r0 and heading in the direction of the vector connecting the point (0,0) - center of the circle - with the center of given primordia. Afterwards the minimum energy is calculated using two different equations:

<math> E = \sum_{i=1}^N 1 / d_i </math>

<math> E = \sum_{i=1}^N 1 / d_i^3 </math>

Where <math display="inline">d_i, i=1..N</math> is the distance of the given primordium from i-th leaf.

After calculating the energy affecting each primordium, the one with a minimum energy is selected and transformed into an actual primordium/leaf, as described above. All leaves then move (on every tick) with a gradual acceleration in the given direction using velocity equation <math display="inline"> V(r) = V_0 \times r / r_0 </math>, which means that the step of a particle with distance r from the center will be '''dist''' (the original size of one step) times r, divided by r0 (radius of the center). To make the model copy real processes, any further reorganization between particles based on repulsive energy isn't allowed. <ref name="Douady"/>

We can set the following parameters:
* '''leaf-size''' - the size of each leaf
* '''dist''' - the distance parameter, can be understood as original velocity
* '''density''' - is the precision of calculation, meaning how many potential primordia there are around the central apex (360 up to 3600, thus density 1 up to 10; where for the density of 10 each degree contains 10 possible primordia)
* '''rc''' or '''r0''' is the radius of central apex and the starting distance from the center for all leaves
* '''energy''' - total energy affecting given primordia - options <math display="inline"> 1 / d </math> or <math display="inline"> 1 / d^3 </math>

As we can see in the following model, the pattern formed created Fibonacci sequence of 5 and 8 spirals and the angle that it converges towards is oscilating around 137.5 degrees (cca. +/- 0.2).

[[File:2017Bednarova-Dynamic1.png|800px]]

=== Model vs. Reality ===

As explained in the previous chapter, we can use the simulation to verify the hypothesis of Fibonacci sequences in spiral patterns on a flower/plant. First, we can see on the first random picture of Dahlia that a number of purple spirals (8) + a number of black spirals in the other direction (13) give us the sum of 21 equal to number of white spirals. All three numbers are subsequent Fibonacci numbers.

Second, we can test the same hypothesis in NetLogo, using our models. The following two patterns (2D and 3D) were created while playing with parameters to make the picture as similar to the real flower as possible.

[[File:2017BednarovaDahlia.jpg]]

By modifying the original angle of 137.5 degrees, similar spiral patterns were formed, but the numbers now did not follow the Fibonacci sequence.

== Conclusion ==

Three different approaches have been described and implemented to test the formation of the phyllotactic patterns. I tested different parameters to create petals/leaves arrangements that resemble real flowers and follow the same Fibonacci numbers. However, in order to create a simulation, we have to neglect many parameters participating in the real process. Since there are many different theories about the formation of different phyllotactic patterns, the scope of this work was not enough to cover more of them. Nevertheless, the topic is very interesting and definitely a good subject of research for scientists, as well as for general public.

=== Possible Extensions ===
Few possible future model extensions:
* Energy could depend only on certain modifiable number of neighbouring leaves.
* Different energy equations could be tested.
* Different velocity functions can be tested.
* Other approaches can be modeled.

== Code ==

[[File:2017BednarovaPhyllotaxis.nlogo]]

== Resources ==
<references />

File:2017BednarovaPhyllotaxis.nlogo

2017-01-17T21:40:29Z

Bedn00:

File:2017Bednarova-Dynamic1.png

2017-01-17T19:54:23Z

Bedn00:

File:2017BednarovaSpiral.jpg

2017-01-17T18:31:04Z

Bedn00: 2017 Nada Bednarova - Phyllotaxis

2017 Nada Bednarova - Phyllotaxis

File:2017BednarovaDiffuse.jpg

2017-01-17T18:02:00Z

Bedn00: 2017 Nada Bednarova - Phyllotaxis

2017 Nada Bednarova - Phyllotaxis

File:2017BednarovaDahlia.jpg

2017-01-16T21:10:57Z

Bedn00: Used in WS 2016/2017 Phyllotaxis by Nada Bednarova

Used in WS 2016/2017 Phyllotaxis by Nada Bednarova

WS 2016/2017

2017-01-14T12:51:57Z

Bedn00:

Semestral papers from winter term 2016/2017. Please, put here links to the pages with your paper. First you need to have your [[Assignments WS 2016/2017|assignment approved]].

==Simulations==

--[[User:Kellistefl|Kellistefl]] ([[User talk:Kellistefl|talk]]) 01:23, 18 December 2016 (CET) [[Poison and Insect Population]]

[[User:Xushk00|Xushk00]] ([[User talk:Xushk00|talk]]) Kateryna Ushanova [[Optimal schedule of Moscow metro]]

--[[User:Bedn00|Bedn00]] ([[User talk:Bedn00|talk]]) 13:51, 14 January 2017 (CET) Naďa Bednárová [[Phyllotaxis]]

Assignments WS 2016/2017

2016-12-14T23:08:25Z

Bedn00:

{{DISPLAYTITLE:Assignments WS 2015/2016}}

{{Ambox
| text = <div>
Please, put here your assignments. Do not forget to sign them. You can use <nowiki>~~~~</nowiki> (four tildas) for an automatic signature. Use Show preview in order to check the result before your final sumbition.
</div>
}}

{{Ambox
| text = <div>
Please, strive to formulate your assignment carefully. We expect an adequate effort to formulate the assignment as it is your semestral paper. Do not forget that your main goal is a research paper. It means your simulation model must generate the results that are specific, measurable and verifiable. Think twice how you will develop your model, which entities you will use, draw a model diagram, consider what you will measure. No sooner than when you have a good idea about the model, submit your assignment. And of course, read [[How to deal with the simulation assignment|How to deal with the simulation assignment]].
</div>
}}

{{Ambox
| type = content
| text = <div>
In order to avoid possible confusion, please, check if you have added '''approved''' in bold somewhere in our comment under your submission. If there is no '''approved''', it means the assignment was not approved yet.
</div>
}}

== Brownian motion Stock simulation (Willem Van de Velde, wilv00) ==

The simulation will contain the evolution of a portfolio of stocks. Each stock has a certain drift, volatility and initial market price.
Over time the value of a stock change dependent on a brownian motion and the time. A person wants to invest a certain amount of money in stocks.

The goal of this simulation is to see how the persons initial investment will evolve over time, what portfolio structure would be ideal for the initial investment?

Method: Monte carlo Simulation;
Program: Excel

[[User:Wilv00|Wilv00]] ([[User talk:Wilv00|talk]]) 09:52, 13 December 2016 (CET)

Hello, could you be more specific? How many stocks? what will be the options for the simulation user to setup his own portfolio? how will you implement the periodically returning depressions? I would expect that the portfolio should consist from stocks, bonds and deposits and the user would choose between the split of the portfolio among these three financial instruments, rather than choosing particular stocks (as it is done in reality). What will be the source of the data?

[[User:Oleg.Svatos|Oleg.Svatos]] ([[User talk:Oleg.Svatos|talk]]) 20:48, 13 December 2016 (CET)

I would use the stock market of my home country (Belgium) so the BEL-20. I would do the simulation with the option of selecting a portfolio that consists indeed of bonds, deposits and stocks taking into account with current market intrest from Belgium and returns on these instruments. I would involve 5 different stocks from the BEL-20 and combine these into a portfolio. So the total simulation would involve a part that is invested in bonds, deposits and a portfolio of 5 stocks (i haven't looked which specific stocks i would use). The goal will than be determing which percentage of his initial investment he would have to invest in bonds, stocks and deposists to maximize his return after a certain period of years (taken into account the different risk measures associated with the different portfolio's).

I wasn't thinking about involving periodically returning depressions. This would mean that after a certain period of time I would have to change the volatility and drift of the stocks and the market intrest right?

[[User:Wilv00|Wilv00]] ([[User talk:Wilv00|talk]]) 12:09, 14 December 2016 (CET)

Yes, you have to play with it little bit as the value of the instruments/portfolios changes differently during the depression as you can see for instance in graph [http://www.diffen.com/difference/Bond_vs_Stock here]. These business cycles is what makes the portfolio value prediction so hard, so it is necessary to incorporate them in the simulation. Simulation '''approved'''.

[[User:Oleg.Svatos|Oleg.Svatos]] ([[User talk:Oleg.Svatos|talk]]) 13:43, 14 December 2016 (CET)

== Simulation proposal (Jakub Esterka, xestj00) ==

Simulation (in NETLOGO) will be set up as a FPS computer game (e.g. Counter Strike) with two teams and set number of players in each team (between 1 and 64). Goal of a simulation will be to find out how much more members can other team have for winning rate to be still 50 % in average. Second goal will be to find out how big advantage randomly selected player has to have for his team to dominate (after 10 games). And the third goal is to find the average rate of domination – how many games is needed for one team to start to dominate the other in every game (team members have almost maximum accuracy of 100).

- Each team can have 1 – 10 players

- Each team begins opposite of each other

- Each player moves randomly and have random accuracy (1-20)

- One player from each team can have an advantage (accuracy of 80)

- When two players from opposite teams meet, they shoot at each other with random accuracy

- If one player shoots the other one, he has its shooting accuracy increased

- When all players from one team die, game resets but players keep their accuracy (except for the first goal)

First goal is obligatory and in this research players will not keep their accuracy. If the simulation will work as planned and it will be possible for players to keep their accuracy, other goals are viable as well.

--[[User:Xestj00|Xestj00]] ([[User talk:Xestj00|talk]]) 18:42, 9 December 2016 (CET)

== Software Developers (xvatj00) ==

=== Description ===
There is a small company for software development. The company takes small orders (around 3 months long) for software. The software development process consists of 3 parts - analysis, coding and testing. For each project, there is a set number of analysts, programmers and testers, who work on the given project (and no other) until it’s finished. There are 2 main types of orders - well defined and very generally defined. For well defined orders (30% of all orders), the average time of the process of analysis is shorter than in case of the generally defined order, and requires less analysts. The company knows the average times of all parts of the development as well as the average salaries of the three positions. Each part of the development also has its probability of delay and average delay (in case delay occurs). There are different scenarios in cases of delays.
Alongside specific positions, there are also “ultimate” workers, who work on the project from its beginning until the end - meaning they do all 3 parts of the development themselves. These workers have higher salaries (all are seniors), but take less time to finish the work.
Method: Simprocess

=== Goal of the Simulation ===
The goal is to see, if it’s more convenient to have specialized workers, or to hire “ultimate” workers instead, and to have an overview of the projects’ employee expenses. To reach the goal, the plan is to create 1 simulation which takes into consideration only the process with the positions divided, and a second simulation which shows the process of “ultimate” workers. In both cases, the process will be taking all the incoming orders - there will be only this one type of process.

Jana Vataščinová, [[User:Xvatj00|Xvatj00]] ([[User talk:Xvatj00|talk]]) 20:29, 10 December 2016 (CET)

== Police System (Kateryna Ushanova, xushk00) ==

This simulation will show police system and how this system will impact the crime rate of the city. Criminals will try to destroy the city's infrostructre and to make crime against civilians.
In this simulation I will use three strategies that maybe used to fight crime:

1) applying medium-sized police force

2) a strategy in which a small police force attempt to recruit civilians into the police force

3) the police react on the crime in a brief period of time

The most important factor that influence skill of the police to fight the crime is work experience. The amount of experience is propotional to the cost of fighting crime.

The criminals will be walking randomly around the visual area. During their walking there is a random chance that they will commit a crime. When they do the crime, the civilians around them become victims, and the area they occupy becomes a bad zone. Bad zone make people more likely to turn to crime.
Two main factors influence whether or not civilians will turn to crime:

1)the level of the city (user controls)

2)the simulated factors (the result of the police work; the state of the zone; the effects of random events that the user controls)

'''The goal of this simulation is:
'''

- to see how changing the work experience level effects each strategy.

- to test out different strategies and factors to see how this effects the crime rate.

- to observe possible outcomes by changing techniques and factors in a different ways.

'''Method - NetLogo

Kateryna Ushanova, [[User:Xushk00|Xushk00]] ([[User talk:Xushk00|talk]]) 13:35, 11 December 2016 (CET)

== (Poison and Insect Population, Kellianne Stefl) ==

NetLogo simulation of the steady decline in insect population in Germany from 1989 to now, also future predictions. Many other factors contribute but I will use pesticides in my visual representation.

Over the last 25 years, etymologists have set up tents all over German woodlands and meadows to observe natures insects. The average biomass of insects sampled in a 6-month period (May to October) has steadily decreased from 1.6 kilograms (3.5 pounds) (56 ounces) per trap in 1989 to just 300 grams (10.6 ounces) in 2014. 18.9% of the average biomass 25 years ago. This simulation will illuminate the devastating effects of damage to the bottom of the food chain.

-adjustable poison release

Goal:
Predict hypothetical decline of insect population: how many years until there are none left.

[[User:Kellistefl|Kellistefl]] ([[User talk:Kellistefl|talk]]) 12:47, 12 December 2016 (CET)

== Phyllotaxis (Nada Bednarova, bedn00) ==

Phyllotaxis is the process of arrangement of leaves on a plant stem. While growing, leaves on a plant/flower/cone usually follow Fibonacci sequences or use golden ratio to form the most convenient angles. There are several theories on where the arrangement comes from.

Simulation in NetLogo would simulate the arrangement of leaves based on different parameters configuration (number and size of the leaves, starting angle etc.). Furthermore chemical or physiological theories about morphogenesis / meristem growth can be simulated (for example Turing's diffusion equations).

--[[User:Bedn00|Bedn00]] ([[User talk:Bedn00|talk]]) 00:07, 15 December 2016 (CET)