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A Cellular Automata Approach to Population Modeling

Explore the effects of temperature and virus spread on populations using cellular automata. Determine if a cellular-based model accurately predicts population behavior.

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A Cellular Automata Approach to Population Modeling

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  1. Alexa M. Silverman A Cellular Automata Approach to Population Modeling

  2. Purpose • To use cellular automata to model population growth and change • To observe the effects of temperature and virus spread 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

  3. 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

  4. Agent-Based Modeling • Uses individual ‘agents’ with simple instructions to observe emergent behavior • Based on interactions between agents • More variable results than mathematical models

  5. Agent-Based Modeling Some well known models, in NetLogo: Altruism Mitteldorf & Wilson Rabbits Grass Weeds (Predator/Prey) U. Wilensky Daisyworld Lovelock & Watson

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

  7. 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

  8. 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).

  9. 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.

  10. 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

  11. NetLogo Interface

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

  13. Results The 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:

  14. Java Program • Models the spread of a virus • Utilizes Java's object-oriented nature and Java's Graphics package • Displays cell movement in a graphics window and exports data to a file • 23/3 Rule (Conway's Game of Life) is used because it models the short-term movements of a population

  15. Code and Testing • Initial density of cells is set at 50% • Initial infected percentage increases by 1% each run from 0 to 100% • Infected cells have a 50% chance of infecting healthy neighbors • At each generation, outputs infected percentage—also seen as BufferedImage painting infected cells red

  16. Java Program class hierarchy CellDriver extends CellPanel JPanel BufferedImage Grid Timer Graphics ActionListener Cell

  17. Results Regardless of initial percent infected, infected cells approach 19-20% over time Environment appears to have a carrying capacity However, virus still propagates even with a low initial number of cases

  18. Discussion • NetLogo program • Limitations: does not account for other factors such as environmental forcings • Supports theory of global warming • Java program • Limitations: virus is dormant; has no effect • Quarantining necessary regardless of initial cases • Extensions • 3D environment • Additional factors on population behavior • Implications for further research

  19. Sources and Background • Individual-Based Artificial Ecosystems for Design and Optimization bySrinivasa 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

  20. Image Sources on the Web • http://www.anl.gov • http://www.kevlindev.com • http://ccl.northwestern.edu/netlogo/ • http://psoup.math.wisc.edu/mcell/ • http://i.cnn.net

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