A Cellular Automata Approach to Population Modeling

Alexa M. Silverman

Purpose To use cellular automata to model

population growth and change To observe the effects of temperature

on the behavior of populations To determine whether a cellular-based

model of population is valid and realistically predicts the behavior of individuals in a population

Cellular Automata

A cellular automaton is a ‘cell’ on a grid which determines its state (‘live’ or ‘dead’) based on the states of neighboring cells.

2D cellular automata consider all eight neighboring cells.

2D cells with live neighbor counts

Agent-Based Modeling

• Uses individual ‘agents’ with simple instructions to observe emergent behavior

• Based on interactions between agents• More variable results than mathematical models

Agent-Based ModelingSome well known models, in NetLogo:

DaisyworldLovelock & Watson

Rabbits Grass Weeds (Predator/Prey)U. Wilensky

AltruismMitteldorf & Wilson

Life

“Life” notation is written s/b, where numbers on the right side of the slash represent neighbor counts needed to survive and numbers on the left side of the slash represent neighbor counts needed for cell birth.

“34 Life” (34/34) “Conway’s Game of Life” (23/3)

CA Modeling

The field of cellular automata modeling is still relatively new, but several models have been created.

“Rumor Mill” models spread of a rumor

“Urban Suite – Cells”models growth of cities

14/3 Population Model

For this model, the rule 14/3 was chosen because it causes cells to grow and move in a pattern resembling the spread of a species (starts localized and becomes more widespread).

14/3 automata suggest two types of individuals: “antisocial” (survives with 1 neighbor) and “social” (survives with 4 neighbors).

Code and Testing

This program was created in NetLogo because the NetLogo graphics window allows for ‘real time’ view of population growth and change. NetLogo code is object-based, which is ideal for a model where each cell must “know” its state, and the states of its eight surrounding cells.

In test runs, cell population and temperature are graphed. Cell ‘birth rate’ (likelihood of a ‘dead’ cell to come to life) varies quadratically with temperature and temperature varies linearly with changes in population. A highly variable population, therefore, will cause frequent changes in temperature.

Code and Testing

Population varies as 100 - .25(temperature - 21)^2

Temperature varies as current temp + (1/20)(pop. - last pop.)

made using equationgrapher.com

NetLogo Interface

Walls

The “make-wall” button allows the user to draw “walls” around segments of the population. The cells cannot cross these walls.

Code Sample

Methods to ‘birth’ and ‘kill’ cells

Methods to check if a cell will ‘survive’ or be ‘born’ the next turn

Results

High initial density causes quick decline in populations, isolation of groups

Results

Ideal initial density seems to be around 55%

Results

• With default settings for temperature (21 degrees Celsius) and population density (55%), population and temperature eventually drop off.

• Does this model global warming?

ResultsThe nature of cellular automata causes each test run to be slightly different even with the same initial parameters. Here are the results of five test runs with default settings:

1 9 17 25 33 41 49 57 65 73 81 89 97 105

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Java Program

• Will extend capabilities of model due to Java’s object-oriented nature

• Difficulties of Java programming with CA

• Extensions—virus spread modeling?

Java Program

CellDriver

CellPanel JPanel

extends

Cell

Timer

Grid

ActionListener

Graphics

BufferedImage

class hierarchy

Sources and Background

Individual-Based Artificial Ecosystems for Design and Optimization by Srinivasa Shivakar Vulli and Sanjeev Agarwal

Using a Genetic Algorithm to Evolve Behavior in Multi-Dimensional Cellular Automata by R. Brukelaar and Th. Back

A Hybrid Agent-Cellular Space Modeling Approach for Fire Spread and Suprression Simulation by Xiolin Hu, Alexandre Muzy, Lewis Ntaimo

Pattern-Oriented Modeling of Agent-Based Complex Systems: Lessons from Ecology by Grimm, Revilla, et al.

Seeing Around Corners by Jonathan Rauch A Brief History of Cellular Automata by Palash Sarkar

Image Sources on the Web

http://www.anl.govhttp://www.kevlindev.comhttp://ccl.northwestern.edu/netlogo/http://psoup.math.wisc.edu/mcell/http://i.cnn.net