This post is inspired by Matt Asher's "quick-and-dirty" R simulation work on Population Growth. Matt uses it to create aRt. I felt that his core idea provided a very good framework to introduce various concepts of simulation.

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**The Basic Idea behind this Simulation**

We take an area (it could be a country, state, city) that is made up of units (cells) that can be settled (occupied).We allow the initial 'pioneers' to settle first. (Seeding the cells)

We then establish some rules for future residents to settle. (Settling Rules)

Plot the area, and collect some stats to study the growth of the population under different starting conditions and different settling rules.

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**Pseudo-Code for the Simulation**

Define parameters: Area Width, Height

Number of Pioneers P and future settlers S

for each pioneer p{

settle p according to seeding rules in a valid cell in the area

}

for each settler S{

Find an unoccupied Cell.

Settle S in that cell, if the settling criteria permits it

}

Print Iterations Stats

Print out the Plot of the Area

Print out the Plot of the Area

###
**A Few Changes from Matt's R code**

- I switched from his matrix to a dataframe. I also switched from his image to ggplot.
- Delineated the initial parameters, seeding and settling clearly so that anyone who downloads the code can experiment
- Created a few different Seeding Functions for the pioneers (beyond random seeding)
- Matt allowed settling if the new cell was adjacent to an occupied cell. I played around with a few different population settling functions. (rectilinear, diagonal etc).
- Added a function to collect a few statistics regarding population growth
- Ran the simulation a few 100 times (without plotting) to compare different growth and settling schemes

### Color Coding the Population

- Black cells : Unoccupied
- Blue cells: Pioneers
- Orange cells: Settlers who found a cell in 1-2 attempts
- Yellow cells: Settlers who found a cell after more than 2 scouting attempts

Here are some runs with a 30x30 area.

Here's what one run looked like. (Blue cells are the initial pioneers. The other colors represent the number of steps taken for them to find a home cell.)

### Trying Out Different Seeding Schemes (for the pioneers)

As opposed to starting out with randomly strewn cells, what if we started out with a city downtown (a densely populated central rectangle) and let the settlers follow them?

I tried a rectangluar ring (annular cells for pioneers) starting point to see how the population grew inside and outside. So I used this seeding function:

One final seeding scheme I tried was to have two thick bands (columns of pioneer cells) and to let the population grow from there. Seeding function:

### Different Population Growth Schemes

The default settling rule that Matt Asher uses is a very good one, one that I like. If a given cell is adjacent to any occupied cell (one of its 8 neighbors) then it becomes a valid cell to settle down in.Just for experimentation, I tried a few other schemes.

#### 1. Rectilinear Growth

Rule: You can only settle down in the 4 cells that are due N, S, East or West of an occupied cell.#### 2. Diagonal Growth

Rule: A settler can occupy any cell that is diagonally adjacent to another occupied cell.The code can be found here.

### Collecting Stats

Now that we have all the functionality that we wanted, it is time to start collecting some statistics for each run ('replication'). There are many possibilities for what we could collect in a population simulation.Here are a couple:

- How many settlers found a 'home cell?'
- How many scouting attempts did each settler make before they found a home?

### Making Multiple Replications

>st

Iter FoundHome NumSettlers Percent

1 1 243 350 69.42857

2 2 214 350 61.14286

3 3 213 350 60.85714

4 4 258 350 73.71429

5 5 235 350 67.14286

Iter FoundHome NumSettlers Percent

1 1 243 350 69.42857

2 2 214 350 61.14286

3 3 213 350 60.85714

4 4 258 350 73.71429

5 5 235 350 67.14286

The full R code I used for these runs can be found here. Feel free to give it a spin and try out your own settling/growth schemes.

Reference: Matt Asher's post on his StatisticsBlog