The presentation for the project of high school students Yonatan Biel and David Hua made in the Students and Teachers As Research Scientists (STARS) program at the Missouri Estimation of Distribution Algorithms Laboratory (MEDAL). To see animations, please download the powerpoint presentation.

1. Population Dynamics in Conway’s
Game of Life and its Variants
David Hua and Yoni Biel

2. Background and Motivation
 Cellular automata (CA) as computational models
 Can simulate any algorithm (computation).
 Useful in computational theory, biology, physics,
mathematics, artificial intelligence.
 Used to study emergence of complex behavior, self-
organization, self-replication, and other aspects.
 Conway’s game of life is one of simplest yet powerful CA.
 The purpose of this project
 Study Conway’s game of life and its variants.
 Focus on population dynamics in terms of the rule set used
and initialization of the simulation.
 Learned programming in C++.

3. Outline
1. Cellular automata.
2. Conway’s game of life and its variants.
3. Population dynamics of studied CA variants.
4. Summary, conclusions, and future work.

4. What are Cellular Automata?
 Computational models
arranged on a grid of cells.
 Each cell is in a state.
 Grid changes over a number
of discrete time steps.
 Change of cell state
determined by its current Cellular automaton for
state, states of its neighbors, simulating diffusion/aggregation
and the set of rules. [http://www.hermetic.ch/pca/da.htm]

5. Differences from Most
Other Models
 Three interesting features of CA
 Parallelism: Every cell is updated at the same time.
 Localism: Every cell is updated based upon its neighbors.
 Homogeneity: Every cell is updated using the same rules.

6. Why Cellular Automata?
 Cellular automata can simulate any algorithm via
implementing universal Turing machine.
 Cellular automata can demonstrate and model
emergence of complex global behavior from simple local
rules, self-organization, self-replication, population
dynamics.
 Cellular automata useful in computational theory,
biology, physics, artificial intelligence…

7. 2D Cellular Automata
 Cells arranged in a two-dimensional grid.
 Each cell has 8 neighbors
• Opposite sides may connect so that the grid wraps around
(for top/bottom row and left/right column).

8. Conway’s Game of Life
 Conway’s game of life is 2D cellular automaton.
 Two possible states for each cell
 Alive
 Dead
 States can change
 Living cell can die (death).
 Dead cell can become alive (birth).
 Simple set of rules specifying
 Death (overcrowding, underpopulation).
 Birth (reproduction).

9. Basic Rules of
Conway’s Game of Life
1. Living cells die if they have 2. Living cells die if they have
fewer than 2 neighbors more than 3 neighbors
(underpopulation/loneliness) (overpopulation)
3. Dead cells that have 3 neighbors 4. Otherwise, there is no change
become alive (whether cell is alive or dead)
(reproduction)

10. Game of Life - Behaviors
 Wide range of common
structure types:
 Mobile groups,
spaceships
 Oscillators
 Static structures
 …
 Structures and their
interaction crucial for
simulating
computations or
processes.

11. Summarizing Rules; Game of
Life Variants
 Rules can be summarized in a simple statement defining the
rules for survival and birth (also else is just dead).
 Examples:
 B3 / S23 (Conway’s original rules)
 Born if 3 neighbors are alive.
 Survives if 2 or 3 neighbors are alive.
 B36 / S23 (high life)
 Born if 3 or 6 neighbors alive.
 Survives if 2 or 3 neighbors alive.
 B2 / S (seeds)
 Born if 2 neighbors are alive.
 Never survives.

12. Rule Sets Used for 2-State
Game of Life
 Rule sets:
 Game of life (B3 / S23)
 Reversed GOL (B23 / S3)
 Evens (B02468 / S02468)
 Day and night (B3678 / S34678)
 Maze (B3 / S12345)
 Walled cities (B45678 / S2345)

13. Our Research
• For each rule set, set a few important inputs as controls for
each simulation
• World size – 20x20 cells
• Number of time steps – 100 steps
• Number of runs of each simulation – 100 runs
• Independent variable
• Initialization percentages of living cells – 10%-90%
• Analyze behavior of various rule sets.
• For each rule set, analyze influence of controllable variables on
1. Percentages of living cell populations.
2. Percentages of changed cell states per time step (rate of change).

14. Game of Life - Dynamics
Population Level Over Time
70
 Convergence upon
60 20.00%
common asymptote.
% Living Cells
50 30.00%
40 40.00%
30 50.00%
20
10
60.00%  Initial population
0
Time Interval
decline.
 Limited range of
Rate of Change Over Time
initializations
% Change From Previous Time
60
50 20.00% achieve this.
40 30.00%
40.00%
30
20
50.00%
60.00%
 Restrictive survival
10
0
rule.
Time Interval

15. Reversed GOL (B23/S3)
Example
 Begins by expanding
very quickly
 Seems to change in
waves
 Not very many cells
stay alive from time
step to time step

16. Reversed GOL (B23/S3)
Population Dynamics
% Living % Changed
90 90
80 80
70 70
Percentage of Cells Changed
Percentage of Living Cells
60 60
50 50
40 40
30 30
20 20
10 10
0 0
1 11 21 31 41 51 61 71 81 91 101 1 11 21 31 41 51 61 71 81 91
Number of Time Steps Number of Time Steps
20% initialization 40% initialization 20% Initialization 40% Initialization
60% initialization 80% initialization 60% Initialization 80% Initialization
• Populations very stable • Maybe many live & dead cells
• Overcrowding still kills switch places

17. Evens (B02468/S02468)
Example
 No recognizable
patterns
 All regions seem to
change constantly
 All movement is
chaotic

18. Evens (B02468/S02468)
Population Dynamics
% Living % Changed
90 90
80 80
70 70
Percentage of Cells Changed
Percent of Living Cells
60 60
50 50
40 40
30 30
20 20
10 10
0 0
1 11 21 31 41 51 61 71 81 91 101 1 11 21 31 41 51 61 71 81 91
Number of Time Steps Number of Time Steps
20% Initialization 40% Initialization 20% Initialization 40% Initialization
60% Initialization 80% Initialization 60% Initialization 80% Initialization
• Populations very stable • Initial population size doesn’t matter
• Half changes and half is static

19. Day and Night (S34678/B3678)
Example
 Life and death are
symmetrical – living
and dead cells
behave the same
way
 Large regions of
living/dead cells
 Regions have
similar
activity, chaotic
boundaries

20. Day and Night (S34678/B3678)
Population Dynamics
Population Level Over Time
 No convergence in 120
10.00%
20.00%
population level or 30.00%
% Living Cells
100
80 40.00%
rate of change. 60
50.00%
60.00%
40
70.00%
20 80.00%
 Relatively stable; 0 90.00%
no significant
Time Interval
initial population
% Change From Previous Time
Rate of Change Over Time
decline. 40 10.00%
35 20.00%
30.00%
 Rule set – living 30
25
40.00%
50.00%
and dead are 20
15
60.00%
70.00%
treated 10
5
80.00%
90.00%
equally, less 0
Time Interval
survival pressures.

21. Maze (S12345/B3)
Example
 Static rule set:
stops changing
after pattern is
complete.
 Consistent maze
pattern for all
initializations.

22. Maze (S12345/B3)
Population Dynamics
Population Level Over Time
 Convergence of a
range of 100
initializations. 90
80
 Rate of change 70
10.00%
20.00%
quickly drops to
30.00%
% Living Cells 60 40.00%
50.00%
zero. 50 60.00%
70.00%
40 80.00%
 Stable, expanding 30
90.00%
population – 20
tolerant survival 10
rule. 0
Time Interval

23. Walled Cities (S2345/B45678)
Example
 Polygonal cities
filled with chaotic
activity.
 Activity continues
only within cities
after they are built.

24. Walled Cities (S2345/B45678)
Population Dynamics
 Initial population Population Level Over Time
drop. 100
90
 Limited range of 80
10.00%
initializations 70 20.00%
30.00%
converge despite lack
% Living Cells
60 40.00%
50.00%
of interaction between 50 60.00%
70.00%
cities.
40
80.00%
90.00%
30
 Restrictive survival
20
10
rule similar to game of 0
life. Time Interval

25. 3-State Game of Life
Example
 An additional living cell
type represents a second
species or group.
 Rules used are the game
of life with identity of
new species determined
by dominant neighboring
cell type.
 Cells coalesce into
homogeneous mobile
masses.
 Each region becomes
overtaken by one cell
type.

26. 3-State Game of Life
Population Dynamics
 Initializations are Population Level Over Time Population Level Over Time
based on difference
between initial Initial Difference: 16% Initial Difference: 12%
populations, with
% Living Cells
% Living Cells
20 20
2.00% 4.00%
total initial 15
10
18.00% 15
10
16.00%
population of 20%. 5 5
0 0
 Rates of change are Time Interval Time Interval
same as game of life.
 Little correlation Population Level Over Time Population Level Over Time
between populations Initial Difference: 8% Initial Difference: 4%
of each species.
% Living Cells
% Living Cells
15 15
6.00% 8.00%
10 14.00% 10 12.00%
 Higher initializations 5 5
had higher rates of 0 0
decline. Time Interval Time Interval

27. Summary
 Presented basics of CA.
 Presented basics of Conway’s game of life (simple CA).
 Explored population dynamics for several variants of the
game of life as well as concrete examples.
 Considered both 2-state and 3-state variants.

28. Conclusions
 Simple rule sets for CA can yield complex behavior.
 Small change to rule set can yield completely different results.
 Changes in the initialization of cell populations can
sometimes yield similar dynamics, but sometimes the
dynamics are dramatically affected (depends on rules).
 Rule sets can be categorized by population dynamics, which
appear to be affected by the survival rule
 Convergence upon optimum population levels/rates of change.
 Initial behavior and time for stabilization.
 Limited range of initializations achieving an optimum state.
 Additional states introduce new possibilities for simulating
competition and species-specific pressures.

29. Future Work
 Simulations of biological and ecological systems
 Example: Spreading of forest fires
 3 colors for live plants, fire, & empty space
 Rules
 Fire consumes all plant neighbors
 Fire can’t pass over empty spaces
 Plants survives with any neighbors until fire reaches it
 Space stays as space
 This can simulate a very important phenomenon rather
easily.
 Simulate ecosystems, evolutionary systems, social
systems, …

30. Acknowledgments
 STARS-2012 sponsors
 Pfizer Inc.
 LMI Aerospace Inc. / D3Technologies
 St. Louis Symphony Orchestra
 Solae
 University of Missouri in St. Louis
 Washington University in St. Louis
 Martin Pelikan (mentor) supported from NSF under grants
ECS-0547013 and IIS-1115352. Any opinions, findings, and
conclusions or recommendations expressed in this material
are those of the authors and do not necessarily reflect the
views of the National Science Foundation.