View source for Tutorial: Dynamic system model

[[Category:Analytica Tutorial]]
[[Category: Foxes and Hares model]]
<breadcrumbs>Analytica Tutorial > {{PAGENAME}}</breadcrumbs>
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This chapter shows you how to:
*Use the [[Dynamic]]() function and the system variable [[Time]]
*Customize the appearance of variable nodes
*Organize a model using modules
*Make duplicates of nodes and modules
*Create aliases of variable nodes
*View the results of two variables simultaneously

Before starting this chapter you should know how to perform the following actions which will no longer be described in detail:

Create a new model; Open an existing model; Save; Save As... (See [[Tutorial: Open a model to browse]])

Create and define new variables; Enter attributes in Attribute or Object windows; Draw influence arrows between nodes. (See [[Tutorial: Create a model]])

You should also be familiar with Array variables and Index variables. (See [[Tutorial: Arrays]])

==Using the Time index==
In this model you will set up a dynamic system to model a predator/prey relationship. The populations of foxes and hares will be set up as state variables: one-dimensional arrays indexed by [[Time]].

Start by opening a new model and titling it '''''Foxes and Hares Tutorial'''''. Create a general variable titled '''''Hare population'''''.

Using either the [[Attribute panel|Attribute window]] or the [[Object window]], place the cursor in the definition field and choose [[Table]] from [[the Expression popup menu]]. The Indexes window will appear.

Notice that '''''Time '''''is listed as one of the indexes in the left hand window. You did not have to define this index before using it. [[Time]] is a [[System_variables#System_Indexes|system index]] in Analytica that is always available to you. Note that it is distinguished by the system variable icon ([[File:Chapter_7.1-updated.png]]). Although it is a system variable, the values of ''Time ''can be edited just like any other index.
{{Release|4.6|4.6|
Select the ''Time ''index and click the transfer button ([[File:Chapter_7.1-updated.png]]) to select it as an active index in the right-hand window. Click '''OK '''to exit the Indexes window.}}

{{Release|4.6|4.6|:[[File:Chapter_7.3-updated.png]]}}
{{Release|5.0||
{{CalloutAnnotationBlock|[[File:Chapter 7.01-5.4.png]]|{{CalloutAnnotation|Create a new general variable titled '''Hare population'''.|n=1|v=30|pt=60,132}}{{CalloutAnnotation|Open the  Attribute pane (or Object window), select '''Definition''' in the attribute popup window then select '''Table''' in the Expression popup menu.|n=+|v=150|pt=410,393}}}}}}
{{Release|5.0||
Select the ''Time ''index and click the transfer button ([[File:Chapter_7.1-updated.png]]) to select it as an active index in the right-hand window. Click '''OK '''to exit the Indexes window.
{{CalloutAnnotationBlock|[[File:Chapter 7.02-5.4.png]]|{{CalloutAnnotation|Select ''Time'' in the indexes dialog box'.|n=+|v=50|pt=135,100}}}}
{{CalloutAnnotationBlock|[[File:Chapter 7.03-5.4.png]]|{{CalloutAnnotation|Click the transfer button to move ''Time'' to the Selected indexes window.|n=1|v=50|pt=398,208}}}}
}}

Analytica will show an [[To edit a table|edit table]] for the Hare population variable. Notice that the default index values for [[Time]] are 0,1, and 2. You will need to have more than just three time points in your [[dynamic]] simulation.

{{Release|4.6|4.6:[[File:Chapter 7.4-updated.png]]}}{{Release|5.0||[[File:Chapter 7.04-5.4.png]]}}

Go to the [[Definition menu]] and select '''Edit time'''. Click the [[Expression popup menu]] and select '''Sequence'''. Analytica will prompt to make sure you want to change the [[Time]] index to a [[Sequence]]. Click '''OK'''.

{{Release|4.6|4.6:[[File:Chapter 7.5-updated.png]]}}{{Release|5.0||{{CalloutAnnotationBlock|[[File:Chapter 7.05-5.4.png]]|{{CalloutAnnotation|Select '''Edit Time''' from the Definition menu.|n=1|v=20}}{{CalloutAnnotation|Select '''Sequence''' from the Expression popup menu (just above the Definition field.|n=+|v=100|pt=150,325}}}}}}

Press OK when asked to replace the current definition with a sequence.

Edit the sequence by keeping 0 for the start value and entering 10 for the end value. Click '''OK''' to exit the [[Object Finder dialog]].

{{Release|4.6|4.6|:[[File:Chapter 7.6-updated.png]]}}{{Release|5.0||{{CalloutAnnotationBlock|[[File:Chapter 7.06-5.4.png]]|{{CalloutAnnotation|Select sequence to span from 0 to 10|n=3|v=120|pt=220,330}}{{CalloutAnnotation|Select sequence to span from 0 to 10|n=3|v=120|pt=360,330}}}}}}

If the edit table for Hare population is not already open select the Hare population node and click on the '''Result''' button ([[File:Chapter 4.54-updated.png]]). Notice that our time index now goes from 0 to 10. We will leave the values of the Hare population zero the moment.

{{Release|4.6|4.6|:[[File:Chapter 7.8-updated.png]]}}{{Release|5.0||[[File:Chapter 7.08-5.4.png]]}}

==Setting up the Hare sub-model==
Create three new general variables titled '''''Initial hare population''''', '''''Hare fertility''''', and '''''Hare births'''''. [[Draw arrows|Draw influence arrows]] as follows:

* from '''''Hare fertility '''''to '''''Hare births'''''
* from '''''Initial hare population '''''to '''''Hare population'''''
* from '''''Hare population '''''to '''''Hare births'''''

Define ''Initial hare population ''as '''''500'''''. Change its [[Identifier]] to '''''Hare_zero'''''.

Define ''Hare fertility ''as '''''60%'''''. We will take this to mean that the number of newborn hares will be equal to 60% of the total population during the previous time period!

Define ''Hare births ''as <code>Hare_population*Hare_fertility</code>. This is the total number of births during a time period.

{{Release|4.6|4.6|:[[File:Chapter 7.9-updated.png]]}}{{Release|5.0||{{CalloutAnnotationBlock|[[File:Chapter 7.09-5.4.png]]|{{CalloutAnnotation|Create, define and draw influence arrows to 3 new variables as instructed above.|v=170|pt= 50,120}}{{CalloutAnnotation|Create, define and draw influence arrows to 3 new variables as instructed above.|v=170|pt= 50,220}}{{CalloutAnnotation|Create, define and draw influence arrows to 3 new variables as instructed above.|v=170|pt= 50,320}}}}}}


==Using the Dynamic() function==
Now we consider how the hare population will develop over time. The hare population will be influenced by the number of hare births, which will be influenced by the hare population during the previous period. Ultimately the value of ''Hare population ''depends on the value of itself during the previous time period. We can address this situation using the [[Dynamic]] function.

Using either the Attribute window or the [[Object window]], open the [[Definition]] field of the ''Hare population ''variable. Click on the [[Expression popup menu]] and select [[File:Chapter_5.6-updated.png]].

Delete the existing table expression and replace it with the following:

:[[File:Chapter_7.10-updated.png]]

The [[Dynamic]]() function can be entered directly as an expression or accessed as a built-in function (by selecting '''Other... '''in the [[Expression popup menu]]). It is listed under the '''Special '''library heading.

Note that the [[Dynamic]] function automatically incorporates the [[Time]] system variable. It was not necessary to initially define our variable as a table indexed by ''Time'', although this step was included in the first section of this chapter to illustrate the ''Time ''index.

Select the ''Hare population ''node and click the '''Result''' button ([[File:Chapter_4.54-updated.png]]).

{{Release|4.6|4.6|:[[File:Chapter_7.11-updated.png]]}}{{Release|5.0||:[[File:Chapter 7.11-6.0.png]]}}

This is a simple exponential growth curve. It represents a rapidly multiplying population of immortal hares with no natural predators and an unlimited food supply. You will need to introduce a limiting factor to avoid this buried-in-bunnies scenario.

==Completing the Hare sub-model==
Quickly before it’s too late, create four new general variables as follows:

[[File:Chapter_7.12-updated.png]]

Finally you need to change the [[definition]] of Hare population as follows:

<code>Dynamic(hare_zero, self[time -1] + hare_births[time - 1]- hare_deaths[time - 1])</code>

{{Release|4.6|4.6|:[[File:Chapter_7.13-updated.png]]}}{{Release|5.0||:[[File:Chapter 7.13-6.0.png]]}}

Now we have accounted for mortality and limited food supply by introducing a crowding factor. When the crowding factor is equal to one, the hare population has reached the carrying capacity for the habitat. At this point the death rate will be equal to the birth rate.

Select the Hare population node and click the Result button ([[File:Chapter_4.54-updated.png]]), then switch to the graph result view.

{{Release|4.6|4.6|:[[File:Chapter_7.14-updated.png]]}}
{{Release|5.0||
{{CalloutAnnotationBlock|[[File:Chapter 7.14-6.0.png]]|{{CalloutAnnotation|Press the Graph icon to display the result graph|pt=13,105}}}}
{{CalloutAnnotationBlock|[[File:Chapter 7.14.2-6.0.png]]|{{CalloutAnnotation|Introducing a crowding factor limits the population to the carrying capacity}}}}}}

==Using the color palette==
In order to make an influence diagram easy to understand it is often useful to color code nodes according to function. The [[Color in influence diagrams|color palette]] is used for this purpose. You can access it in two ways:
*Right-click the node and select '''Show/hide color palette''' from the context menu
*Go to the [[Diagram menu]] and select '''Show/hide color palette'''
In the color palette you can select from standard colors or click '''Other...''' to select from the entire 24-bit RGB color space.

{{Release|4.6|4.6|:[[File:Chapter_7.15-updated.png]]}}{{Release|5.0||:[[File:Chapter 7.15-6.0.png]]}}

<tip>
The [[Node Style dialog|node style]] and [[Diagram Style dialog|diagram style]] windows are also useful for customizing the appearance of your diagram. They can be accessed in the same manner as the [[Color in influence diagrams|color palette]].
</tip>

==Creating a module==
[[Modules and Libraries|Modules]] are an important feature of Analytica. They allow you to place a collection of nodes inside a single object, a module, which has the appearance of a single node. In this way, the top level of your diagram can be a simplified view showing influence arrows between modules. Each module can be opened as its own [[diagram window]] to show detailed interactions of the variables inside. Modules can even be placed within modules to create a nested hierarchy of detail.


{{Release|4.6|4.6|
<tip>
If you have several diagram windows open at once you can click the Diagram window button ([[File:Chapter 7.16-updated.png]]) to return to the top level diagram at any time.
</tip>}}
{{Release|5.0||
<tip>
If you have several diagram windows open at once you can click the Diagram window button ([[File:Chapter 4.18-updated.png|28px]]) to return to the top level diagram at any time.
</tip>}}

Creating a new module is just like creating a variable node. Drag the Module node from the node palette to the [[diagram window]]. A module looks similar to a node but it has a thick border around it and displays the title in bold print. Whereas double-clicking a variable node will open an [[Object window]], double-clicking a module will open a new [[diagram window]]. Attributes for the module can be entered using the [[Attribute panel|Attribute window]] at the bottom of the diagram. Your new module will have the following [[attributes]]:

* Title: '''''Hare module'''''
* Identifier: '''''Hare_module'''''
* Description: '''''A dynamic sub-model of the hare population'''''

To populate your new module, start by making a multiple selection of all the nodes except for the module itself.

Reminder: To make a multiple selection hold down the Control or Shift keys, or drag a rectangular frame around the nodes you want to select.

{{Release|4.6|4.6|:[[File:Chapter_7.17-updated.png]]}}
{{Release|5.0||
{{CalloutAnnotationBlock|[[File:Chapter 7.17-6.0.png]]|{{CalloutAnnotation|Create a new module by dragging the Module icon down to the diagram space.|v=110|n=1|pt=645,160}}{{CalloutAnnotation|Make a multiple selection of all variable nodes by dragging diagonally across the space. (Make sure that the module is not also selected).|v=250|n=2|pt=30,183}}{{CalloutAnnotation|Make a multiple selection of all variable nodes by dragging diagonally across the space. (Make sure that the module is not also selected).|n=2|v=250|pt=619,585}}{{CalloutAnnotation|Drag one of the selected nodes to the module and release).|v=450|n=3|}}}}}}

Your top level diagram should now contain just the module by itself. Double-click the module to see the variables in a separate window.

{{Release|4.6|4.6|:[[File:Chapter_7.18-updated.png]]}}
{{Release|5.0||
{{CalloutAnnotationBlock|[[File:Chapter 7.18-6.0.png]]|{{CalloutAnnotation|Double-click on the module to open a new diagram window.|v=72|n=+|pt=37,112}}}}}}


<tip>
If a single [[diagram window]] takes up the entire workspace press the maximize/un-maximize toggle ([[File:Chapter 7.19-6.0.png]]) near the upper right corner of the window.
</tip>

==Duplicating a module==
Now you will make a similar sub-model for the fox population. Since the basic structure will be similar to the hare sub-model, you can save time by duplicating it automatically.

Select the hare [[Modules and Libraries|module]] in the top level diagram window. Select '''Duplicate nodes '''from the [[Edit menu]]. A [[Create and edit nodes|duplicate]] module will appear.

Edit the Title and Identifier of the duplicate to read '''''Fox module'''''

Edit the Description attribute to read '''''A dynamic sub-model of the Fox population'''''

{{Release|4.6|4.6|:[[File:Chapter_7.20-updated.png]]}}
{{Release|5.0||
{{CalloutAnnotationBlock|[[File:Chapter 7.20-6.0.png]]|{{CalloutAnnotation|After selecting the Hare module, go to the '''edit''' menu and select '''Duplicate nodes'''.|v=0|n=1}}{{CalloutAnnotation|Retitle the duplicate as '''Fox module'''.|v=130|n=+}}}}}}

Double-click the ''Fox module ''to open its [[diagram window]]. {{Release|5.0||(Or click on the Module's Diagram window [[Pop-up_node_controls|Hover Icon]] [[File:DiagramWindowHoverIcon.png]])}}.

The diagram appears to be identical to the original but you will notice some important differences in the Identifiers. Open the [[Attribute panel|Attribute window]] and check the identifier for the Hare population node. Notice it has been changed to '''''Hare_population1'''''. Analytica has appended the '''1 '''to ensure that the identity of the new variable is distinct from the original. Different variables are allowed to have identical titles but they must always have different identifiers

[[Identifier]]s in functional expressions are modified accordingly. This ensures that the duplicate variables belong to a self-contained group that does not interfere with the original variables.

Drag one of the selected nodes to the [[Modules and Libraries|module]] and release.

:[[File:Chapter 7.21-updated.png]]

==Completing the Fox sub-model==
Edit the titles and identifiers in the fox module as follows:
(These edits mostly involve substituting the word ''Fox'' for ''Hare''.)

:[[File:Chapter 7.22-updated.png]]

Change the definition of input variables as follows:
*Initial Fox population: '''''120'''''
*Fox fertility rate: '''''35%'''''
*Fox carrying capacity: '''''200'''''
The fox population is now a self-contained sub-model. Select the Fox population variable and click on the '''Result''' button ([[File:Chapter 4.54-updated.png|24px]]). Or you can hover over the node and click on the result [[Pop-up_node_controls|hover icon]].

{{CalloutAnnotationBlock|[[File:Chapter 7.23-6.0.png]]|{{CalloutAnnotation|Due to a lower fertility rate, the fox population  approaches it's carrying capacity more slowly than the hare population.}}}}

==Creating aliases==
Although it is useful to organize a model with modular structure, accessing variables inside modules can be less convenient than having them all in the same window. To address this problem Analytica allows you make an [[Alias nodes|alias]] of a variable and place it anywhere you wish. The alias provides local access to the variable but it is merely a representation, not a duplicate, of the original node.

In this case it will be convenient to access the ''Hare population ''and ''Fox population ''nodes on the parent diagram without having to open the [[Modules and Libraries|modules]].

Click the Diagram button ([[File:Diagrambutton5.4.png|24px]]) to open the top level diagram. Open the ''Fox module ''and arrange the windows so that both are visible.

Select the ''Fox population ''variable. Go to the [[Object window]] and select '''Make alias'''. The [[Alias nodes|alias]] looks similar to the original node but the title will be shown in italics. Drag the ''Fox population ''alias to the parent diagram.

{{Release|4.6|4.6|:[[File:Chapter_7.24-updated.png]]}}
{{Release|5.0||
{{CalloutAnnotationBlock|[[File:Chapter 7.24-6.0.png]]|{{CalloutAnnotation|Arrange windows so that the ''Fox module'' diagram and it's parent diagram are both visible.|v=0|n=1}}{{CalloutAnnotation|Select the ''Fox population'' variable and select '''make alias'''  from the  Object menu.|v=130|n=+}}{{CalloutAnnotation|Drag the ''Fox population'' alias to the parent diagram window.|v=260|n=+}}{{CalloutAnnotation|Note that alias nodes always show titles in italics.|v=375|pt=268,402}}}}}}

Repeat the same procedure to create a ''Hare population'' alias on the top diagram.

{{Release|4.6|4.6|:[[File:Chapter_7.25-updated.png]]}}
{{Release|5.0||
{{CalloutAnnotationBlock|[[File:Chapter 7.25-6.0.png]]|{{CalloutAnnotation|Create an alias for the Hare population  variable in the same manner.|v=0|n=4}}{{CalloutAnnotation|Influence arrows are automatically drawn.|v=100|pt=112,180}}{{CalloutAnnotation|Influence arrows are automatically drawn.|v=100|pt=392,180}}}}}}

==Drawing influence arrows across modules==
Now you are ready to introduce predatory interaction! Create a new general variable titled '''''Predations'''''. This will represent the number of hares eaten by foxes in a given time period. The predation level is assumed to be proportional to the product of fox and hare populations.

Draw influence arrows from the ''Hare population ''alias to ''Predations'', and from the ''Fox population'' alias to ''Predations''. Enter the following expression to define the new variable: <code>Hare_population*Fox_population / 600</code>'

{{CalloutAnnotationBlock|[[File:Chapter 7.26-updated.png]]|{{CalloutAnnotation|Create a new general variable titled '''predations.'''|v=5|n=1|pt=291,189}}{{CalloutAnnotation|Draw influence arrows from the ''Hare population'' and ''Fox population'' aliases to the ''Predations'' variable.|v=120|n=2|pt=173,310}}{{CalloutAnnotation|Draw influence arrows from the ''Hare population'' and ''Fox population'' aliases to the ''Predations'' variable.|v=120|n=2|pt=381,310}}{{CalloutAnnotation|Define ''Predations'' as '''Hare_population''' *  '''Fox_population/600'''|n=3|v=400|pt=12,465}}}}


Unfortunately for the hares, each predation will count as a death for their team. Open the ''Hare module'' and arrange windows so that the ''Hare module'' diagram and its parent diagram are both visible. [[Draw arrows between modules|Draw an influence arrow]] from ''Predations'' to ''Hare deaths''.

{{CalloutAnnotationBlock|[[File:Chapter 7.27-updated.png]]|{{CalloutAnnotation|Open the ''Hare module'' and arrange window so that the ''Hare module'' diagram and it's parent diagram are both visible.|v=5|n=4}}{{CalloutAnnotation|Draw an influence arrow from the ''Predations'' variable to the ''Hare deaths'' variable.|v=180|n=5}}}}

Re-define Hare deaths as: <code>Hare_population*Hare_ndr + Predations</code>

{{CalloutAnnotationBlock|[[File:Chapter 7.28-updated.png]]|{{CalloutAnnotation|Edit definition of ''Hare deaths'' to read '''Hare_population * Ndr + Predations'''.|v=0|n=6}}}}

If you look closely at the '''Hare population''' and ''Hare deaths'' nodes you will notice small arrowheads on the margins. These indicate influence relationships with other nodes outside the module.

{{CalloutAnnotationBlock|[[File:Chapter 7.29-updated.png]]|{{CalloutAnnotation|Arrowheads indicate influences directed outside the module.|v=110|pt=444,128|path=r(396,*)}}{{CalloutAnnotation|Arrowheads indicate influences directed outside the module.|v=110|pt=401,208|path=r(396,*)}}}}

Predations are a food source for fox population and have a positive effect on their numbers. This can be modeled by making the ''Fox carrying capacity'' proportional to the ''Predations'' variable.

Re-define ''Fox carrying capacity'' as: '''Predations*1.6'''

{{CalloutAnnotationBlock|[[File:Chapter 7.30-updated.png]]|{{CalloutAnnotation|Arrowheads indicate influences directed outside the module.|n=+|pt=12,160|path=r}}}}

<tip>
Analytica will allow you to use a variable in an expression before drawing its influence arrow. The variable will not be listed in the Inputs popup menu so you must type the exact identity correctly. Influence arrows will automatically be drawn after you finish editing the expression.
</tip>

==Viewing multiple results simultaneously==
Your model is now complete. Click the Diagram button {{Release|4.6|4.6|([[[File:Chapter_7.16-updated.png|24px]])}}{{Release|5.0||([[File:Diagrambutton5.4.png|24px]])}} to open the top diagram. As you already know, you can view the results of the fox and hare populations separately by selecting either one of the population aliases and clicking the Results button ([[File:Chapter 4.54-updated.png|24px]]). But there is an easy way to combine both variables on the same graph. Select both [[alias nodes]] simultaneously (holding down the Control or Shift keys while selecting). Now when you click the Results button a dialog box will appear asking if you want to [[Compare Results|compare]] more than one result. Click '''OK'''. Analytica will automatically create a comparison node. A comparison node is essentially a list variable with the variables to be [[Compare Results|compared]] as elements of the list. 

The default title in this case will be ''Compare1''.

:[[File:Chapter 7.31-updated.png]]

The result window for ''Compare1'' will contain information from both of the selected nodes. Choose Time as the horizontal axis in the [[Result window]] of ''Compare1''. You can now see the result of the predatory interaction.

[[File:Chapter 7.32-updated.png]]

==Foxes and Hares Act III==
How does the Fox/Hare drama end? You can extend the simulation by simply adding more values to the [[Time]] index.

Go to the [[Definition menu]] and select '''Edit time'''. Extend the sequence to go from '''''0''''' to '''''100'''''. Click '''OK''' to exit the [[Object Finder dialog]].

:[[File:Chapter 7.33-updated.png]]

Select the ''Compare1'' node and click the '''Results''' button ([[File:Chapter 4.54-updated.png]]) again.

:[[File:Chapter 7.34-updated.png]]

Now you can experiment by changing input values to see how they impact the result.

If you would like to see a more complete Foxes and Hares example, open '''Foxes''' and '''Hares.ana''' in the '''Tutorial Models''' folder.

==Summary==
In this chapter you have:
*Used the [[Dynamic]]() function to model a feedback loop in a dynamic system.
*Edited the system variable [[Time]] which is used as an index in the [[Dynamic]]() function
*Customized the appearance of nodes using the [[Color in influence diagrams|Color palette]]
*Grouped nodes into [[Modules and Libraries|Modules]]
*Used the [[Create and edit nodes|Duplicate command]] to create a new module with similar structure to the original
*Created [[Alias nodes|aliases]] for convenient access to nodes contained within modules
*Established [[Draw arrows between modules|influence relationships]] between modes in different diagram windows
*Created a [[Compare Results|comparison node]] to view the results of two variables simultaneously.

==See Also==
<div style="column-count:2;-moz-column-count:2;-webkit-column-count:2">
*[https://www.analyticacloud.com/acp/Client/AcpClient.aspx?inviteId=5&inviteCode=730213&subName=acp%20demos Play the Foxes and Hares model in Analytica Cloud Player]
* [[Creating the Foxes and Hares model (based on chapter 7 of tutorial)]]
* [[Time]]
* [[Dynamic]]
* [[Dynamic Simulation]]
* [[Expand your model]]
* [[Combining models into an integrated model]]
* [[Example Models]]
* [[Example Models and Libraries]]
* [[Modules and Libraries]]
* [[Draw arrows between modules]]
* [[Drawing arrows between variables in different modules]]
* [[Create and edit nodes]]
* [[User input nodes and user output nodes]]
* [[Alias nodes]]
* [[Node Style dialog]]
* [[Diagram Style dialog]]
* [[Color in influence diagrams]]
* [[Comparing results]]
* [[Tutorial videos]]
</div>


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