Density

Density Rules generate attraction for a community and size class based on the density of the same community and size class. The attraction layer generated from a density rule can be added to the attraction strength from a forage or density rule. This rule was designed for intra-species forage, but with the attraction strength based on density instead of a forage unit.

Density Rules tab

The Density Rules tab allows you to configure the attraction of a community and size class based on its density. The strength of attraction may also be adjusted based on seasonality. The strength of attraction also dissipates from the elements with maximum attraction strength.

Multiple density rules may be created for a community and size class (Figure).

Figure. The Density Rules interface.

Rule Name: displays name of selected Density rule as part of dropdown list of all Density rules in the study.

Add, Rename, Copy, and Delete Rule: described above in the common features section (Figure x).

Community: dropdown list of communities in the study.

Size Class: this dropdown field lists the size class members of the selected community.

Strength (function of density and season) Plot: this plot displays the configured look-up table for the Strength (function of density and season). See Strength (Function of Season).

Strength (function of density and season) Table: this table defines the strength of attraction based on a given season. See Strength (Function of Season).

Weight: this field applies a multiplier to the attraction strength. This is set to 1 by default.

Dissipate: this dropdown field determines the interpolation method used when the attraction strength dissipates from the source of maximum strength.

Radius: this radio button dictates that the attraction strength dissipates using a radius value with the map units. This is set to 1 by default.

Slope: this radio button dictates that the attraction strength dissipates using a slope value with the unit of strength/map unit. This is set to 1 by default.

Apply: clicking this button saves the density rule.

Strength (Function of density and season)

The Strength (Function of density and season) has a look-up table and plot for the density rule. The look-up table determines the attraction strength based on the density values of the community and size class that the density rule is configured for. The strength value is limited from -1 to 1. Entering a value outside of this range produces a warning. Entering a value of 1 defines that the attraction in the source elements is at full strength. Entering a value of -1 causes repulsion from the source elements. The interpolation can be set as Linear or Step.

By default, the Strength (Function of density and season) has a single column for a season beginning on January 1. More seasons may be added to the table using the Seasonal Variation dialog. The beginning of a season is defined by entering the day and month in the format of “ddMmm”. For example, a season that starts on March 5 would be entered as “05Mar”. This dialog has controls for whether the strength values in between seasons are determined using Linear, Step, or Cubic interpolation. Each defined season is added as a column. The attraction strength must be configured for each season.

Configuring the Strength (Function of density and season) table also updates the curve in the plot.

Dissipation Options

Attraction strength dissipates from the source elements. These elements are recruited by the community and size class configured for the density rule. The dissipation options configure the rate that the attraction strength dissipates.

The Weight modifies the overall strength of attraction during dissipation. This is set to 1 by default.

The Dissipate dropdown configures whether it follows a linear or sigmoidal dissipation. By default, this is set to Linear. When the Sigmoid option is selected, the additional field to configure the Stop at % strength is displayed.

The attraction strength may dissipate from the source location using a Radius or a Slope. By default, the option for Radius is selected with a value of 1 map unit.

Configuring a Density Rule

When creating a density rule, you must:

Assign a unique rule name,
Select the community and size class to configure density rule for,
Configure additional seasons as needed,
Configure the strength, and
Configure the dissipation options.

Creating a New Rule Name

The first step in creating a new density rule is to assign a rule name. Because multiple density rules may be created in a study and for the same community and size class combinations, each density rule must have a unique name. When attempting to create a density rule that duplicates an existing rule’s name, an error is produced.

To create a new density rule:

Click the Add Rule This loads the New Rule dialog (Figure).
Enter a unique rule name for the new density rule in the New Rule
Click OK in the New Rule
After the New Rule dialog closes, the new rule name is displayed in the Rule Name

Figure. Interface for creating a new density rule.

Select Community and Size Class

When creating a density rule, the community and size class is selected from the Community and Size Class dropdown fields. The content in the Size Class dropdown depends on the selected community.

To select the Community and Size Class:

Expand the Community drop down. The drop down lists all the available communities in the study.
Select the community to create the density rule for. This updates the content in the Size Class
Expand the Size Class The dropdown lists all the available size classes for the selected community.
Select the size class to create the density rule for.

Configuring Strength (Function of density and season)

When configuring the strength for the density rule, identify whether there is seasonal variation. Configuring the seasonal variation adds the columns to the Strength (function of density and season) table and makes it more efficient for entering all the strength values in the table.

To configure the seasonal variation for the strength (function of density and season):

In the Strength (function of density and season), click the Edit button for the Seasonal Variation. This loads the Seasonal Variation
Select whether the Interpolation Type is Linear, Cubic, or Step.
Add the beginning of seasons following the format of “ddMmm”.
Click the OK This closes the Seasonal Variation dialog.

To configure the strength (function of density and season):

In the Strength (function of density and season) table, add the rows needed for each season. To add more rows, access the context menu from the table and select the Append Row or Insert Row(s)
Select whether to use Linear or Step interpolation in the Interpolation
Under the Variable Value column, enter variable values.
Under the Strength column, configure the strength value.

Configuring Dissipation Options

To configure the attraction dissipation for a density rule:

Configure the Weight of the dissipation of the attraction strength.
Configure whether to use Linear or Sigmoidal dissipation for the attraction strength.
Select whether to use Radius or Slope.
- If Radius is selected, configure the Map Units for the attraction strength to dissipate.
- If Slope is selected, configure the Strength/Map Unit for the attraction strength to dissipate.
Click Apply button to save rule (Figure).

Figure. Click the Apply button to save the Density rule.

Simulation

Instinctual, forage, road, and density are horizontal rules that generate attractions. Simulation with a density rule in place generates an attraction map with attraction strength dissipating from the elements populated by the selected community and size class in the density rule. This strength then dissipates across the study area depending on the following conditions:

Whether the rule uses Linear or Sigmoidal dissipation, and
Whether the rule uses a Radius or Slope

The attraction strength dissipates to zero or to its weakest value based on the Radius or Slope value. The attraction strength in the source elements, where attraction has maximum strength, to where it reaches minimum strength is then computed using Linear or Sigmoidal dissipation.

Dissipation is detailed in the Instinctual rules section.

Attraction Strength with Seasonal Factor

Each density rule has a season configured. By default, a density rule uses a single season that begins on January 1. The value configured under the Factor is the maximum attraction strength for that season. A season for the density rule begins on the date configured in the table. Between seasons, the factors are linearly interpolated.

Application of Weight to the Attraction Strength

After a density rule’s attraction strength is computed and seasonal factor is applied, the weight is applied. Applying weight to the attraction strength depends on 2 factors:

The setting for the Interaction of density and instinctual/forage potential fields in the Movement tab of the Communities
The number of density rules available.

When the option selected in the Interaction of density and instinctual/forage potential fields in the Movement tab is set to use the Linear combination using rule weights option, the weight is applied to the attraction strength. This is regardless of the number of available density rules.

When the other options of Maximum of potential gradients, Weighted average (density given priority), or Maximum enforcing density rules are selected for the Interaction of density and instinctual/forage potential fields in the Movement tab, then the weight is multiplied to the attraction strength of a given density rule then the product divided by the total density weight. See the following formula:

$\begin{array}{l}\displaystyle strength(e)_{with weight}=strength(e)*weight_{rule}/total density weight\end{array}$

Where:

strength(e)_{with weight}	is the attraction strength applied with weight
strength(e)	is the attraction strength before weight is applied
weight_rule	is the weight for a given density rule
total density weight	is the sum of all the weights of all available density rules

The Total Density Weight is computed using the weight and seasonal factor per density rule.

$\begin{array}{l}\displaystyle total density weight= weight_{r1}*seasonal factor_{r1}+weight_{r2}*seasonal factor_{r2}+...+weight_{rn}*seasonal factor_{rn}\end{array}$

Where:

weight_rn	is the attraction strength applied with weight
seasonal factor_rn	is the attraction strength before weight is applied

Aggregated Attraction Strength with Multiple Density rules

Multiple density rules may be created for the same community and size class. When there are multiple density rules, these are summed together. The total attraction strength for all available density rules for the same community and size class is computed using the following formula:

$\begin{array}{l}\displaystyle strength(e)_{total}=(strength(e)_{r1}+strength(e)_{r2}+ ... + strength(e)_{rn}\end{array}$

Where:

strength(e)_total	is the weight for a density rule
strength(e)_total	is the seasonal factor for a density rule

Aggregated Attraction Strength with Rules that Generate Attraction

In addition to density rules, instinctual forage, and road rules also generate attraction strength. The total attraction strength is tracked in a simulation with these multiple rules present. The total attraction strength is the summation of all the total attraction strength for each rule that generates attraction.

Handling Undefined Density Values

Because density rules reference the density values of the community and size class, undefined values may be present in the data. When there are undefined values, either:

The strength value at the lowest positive density value is used, or
The strength value when density is 0 is used.

When there are undefined values using strength at the lowest density value, the Strength (function of density and season) table is configured with the following values (Table).

Table. Density and strength values when there are undefined values using strength at the lowest density value.

Density	Strength (-1 to 1)
1	0.20
5	0.75
10	0.90

Undefined values of density use the strength of 0.20, which is the value for the lowest density value of 1 in the table.

When there are undefined values using strength at 0 density value, the Strength (function of density and season) table is configured with the following values (Table).

Table. Density and strength values when there are undefined values using strength at 0 density value.

Density	Strength (-1 to 1)
-1	0.50
0	0.25
0.001	0.40
5	0.10
10	0.90

The undefined values use the strength of 0.25, which is the value when the density value is 0. Note that it does not use the strength of 0.5 when the density value is -1, even though this is the lowest value of density in the table.

Simulation Output

After a simulation is computed with density rules, the results can be observed during animation in the Simulation Map. The attraction strengths from density rules are generated in the results Simulation HDF tables.

When a study only has density rules, an attraction layer is not generated in the results folder.

The Simulation Map displays the attraction strength from density rules in a simulation.

When a simulation with density rules is successfully computed and the checkboxes are checked for Density in the Spatial Output Controls, the Simulation HDF file has tables tracking the attraction strength values.

Under the Rule folder of each size class folder of a community, the DensityAttraction folder is generated. The DensityAttraction folder organizes the computed attraction strength of each density rule for a community and size class. A folder for each density rule contains the DensityAttraction table that holds the computed attraction strength for that given density rule.

Under the size class’s folder of a community, the DensityAttraction table tracks the weighted average of the attraction strength of all the density rules for that community and size class.

Under the size class’s folder of a community, the TotalAttraction table tracks the weighted average of the attraction strength of all the rules that generate attraction for that community and size class.

Configuring Spatial Output Controls

The Spatial Output Controls, which are accessed from the Project Properties dialog, control whether additional tables are generated in the simulation HDF file after computing a simulation with configured density rules.

The Spatial Output Controls have checkboxes for Density under the Rules that generate attraction section. By default, the Per Rule checkbox is unchecked and the Total checkbox is checked.

When the Per Rule checkbox is checked, the DensityAttraction tables for the individual density rules are generated in the simulation HDF file.

When the Total checkbox is checked, the table with aggregated attraction strength from all density rules are generated in the simulation HDF file.

Configuring the Animation Properties

In the Animation Properties, a simulation with a computed density rule has options to display the attraction strength from either a single density rule or all density rules in a simulation.

The Attraction Layer radio button is available when a density rule is configured for a community and size class combination. The dropdown that corresponds to the Attraction Layer radio button has the following options specific to density rules.

Density Rule Total: Aggregated attraction strength for all density rules in a simulation.
Density Rule: When this option is selected, the Rule dropdown is enabled. The selected rules’ attraction strength is displayed on the Simulation Map.
Total Attraction: When this option is selected, the aggregated attraction strength from all the rules that generate attraction are displayed on the simulation map.

In addition to the Attraction Layer radio button and its dropdown, the Rule dropdown controls which specific rule to display the attraction strength for on the Simulation Map.

Animating the Density Rule

To animate the attraction strength from a specific density rule on the Simulation Map, the Animation Properties may be configured by:

Select the Community-Size Class to animate on the Simulation Map.
Select the Attraction Layer radio button.
Select Density Rule in the corresponding dropdown.
Select the specific density rule to display the attraction strength for.
Configure the Tic Interval and Contour Limits as needed.
Click the Apply

To animate the aggregated attraction strength for all density rules for a community and size class in the Simulation Map.

Select the Community-Size Class to animate on the Simulation Map.
Select the Attraction Layer radio button.
Select the Density Rule Total in the corresponding dropdown.
Configure the Tic Interval and Contour Limits as needed.
Click the Apply

To animate the aggregated attraction strength for all rules that generate attraction for a community and size class in the Simulation Map.

Select the Community-Size Class to animate on the Simulation Map.
Select the Attraction Layer radio button.
Select the Total Attraction in the corresponding dropdown.
Configure the Tic Interval and Contour Limits as needed.
Click the Apply