1. Spatial models II (diffusion, two dimensions)

2. Overview Grid vs. polygon Random-walk models Lévy flight and fractals
Individual-based models
Cellular automata

3. Grid approach
Campbell HF & Hand AJ (1999) Modeling the spatial dynamics of the US purse-seine fleet operating in the western Pacific tuna fishery. CJFAS 56:

4. Polygon approach
Fulton EA et al. (2011) Lessons in modelling and management of marine ecosystems: the Atlantis experience. Fish and Fisheries 12:

6. A random walk Imagine an individual in a single dimension
At any time there is an equal probability the individual will move left or right
Location
Time step
19 Random walks.xlsx “Simple random”

7. A random walk Expected distribution after n time periods
Frequency
Location
19 Random walks.xlsx “Simple random”

8. Discrete jumps Equal probability of staying, jumping left or jumping right
Location
Frequency
Time step
Location
19 Random walks.xlsx “Random jump”

9. Large population size pleft = probability of moving left
pright = probability of moving right
pleft = 0.2
pright = 0.2
pleft = 0.1
pright = 0.3
Prop. of individuals
Cell number
Cell number
19 Random walks.xlsx “Jump large popn”

10. Random walk in two dimensions
Start at (0,0) with light gray, jump randomly over both x and y, with lines gradually getting darker.
Run 1
Run 2
Run 3
Run 4
19 Random walk 2D.r

11. Lévy flights
Random motion (foraging) punctuated by long-distance straight movements
(searching)
Random walk
(7000 steps)
Lévy flight (µ = 1.5)
(7000 steps)
Metzler R & Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Physics Reports 339:1-77

12. Gray seals North Sea Latitude Longitude
McClintock BT et al. (2012) A general discrete-time modeling framework for animal movement using multi-state random walks. Ecological Monographs

13. High resolution GPS data for wandering
albatrosses
Humphries NE et al. (2012) Foraging success of biological Lévy flights recorded in situ. PNAS 109:

14. Lévy flights are self-similar
The pattern looks similar at large scales and fine scales
So do rivers, coastlines, clouds, and tree branches
They are fractals
Famous question: how long is the coastline of the US?
Famous example: the Mandelbrot set
Mandelbrot set. Initial image of a zoom sequence: Mandelbrot set with continuously colored environment.
Coordinates of the center: Re(c) = -.7, Im(c) = 0
Horizontal diameter of the image: 3.076,9
Created by Wolfgang Beyer with the program Ultra Fractal 3.
Uploaded by the creator.
Mandelbrot set: picture by Wolfgang Beyer using Ultra Fractal 3

15. Mandelbrot set (explanation given for those curious, video supplied for its beauty)
Repeatedly apply. If abs(Zn) goes to infinity as n gets large then c is outside the Mandelbrot set; if abs(Zn) remains bounded then c is inside the Mandelbrot set.
Complication: c is a complex number. Real part of c is the x-coordinate, imaginary part of c is the y-coordinate, color is how many iterations before it goes to infinity.
Black = inside the Mandelbrot set, colors = outside the set
Still image from screen snapshot of video

17. Individual based model (IBM) of wildebeest
In each cell model the amount of grass and the number of wildebeest
Amount of grass is set initially in each cell
Grass can either be consumed or not consumed
Wildebeest know the amount of grass in the current cell and in the adjacent cells
They move to each adjacent cell with a probability proportional to the grass abundance

18. Cumulative probability
Movement algorithm
Habitat quality
Probability of moving
Cumulative probability
0.4
0.8
1.5
0.2
0.1
0.3
0.5
0.08
0.17
0.31
0.04
0.02
0.06
0.10
0.08
0.25
0.56
0.60
0.65
0.81
0.83
0.90
1.00
Pick a uniform random number between 0 and 1 RAND() in Excel, runif(n=1) in R
Based on cumulative probability move to:
< 0.08, top left cell
, top middle cell
, top right cell, etc.,
, stay in the center cell, etc.,
, bottom right cell
17 Random walks.xlsx, sheet Wildebeest prob

19. No consumption of grass
Habitat quality very
high in center
Habitat quality more
spread out
17 Wildebeest model.r

20. With grass consumption
Grass level resulting in half of the maximum grass consumption
Maximum consumption of grass
Grass in cell i, j
Consumption by one wildebeest
Wildebeest in
cell i, j
Proportion consumed
by many wildebeest
Grass available in next time step, t+1
17 Wildebeest model consumption.r

21. With grass consumption
17 Wildebeest model consumption.r

22. Cellular automata Each cell is in one of several possible states
Rules determine state transition
All cells identical
Loop over cells
Contrast with IBM where you loop over individuals
E.g. Conway’s Game of Life

23. Game of Life Invented by John Conway (Univ. Cambridge)
Popularized by Martin Gardner in 1970
Rules of checkerboard; cell with 8 neighbors
Survival: 2-3 neighbors
Death: 0-1 neighbors (isolation), ≥ 4 neighbors (overcrowding)
Births: empty cell with exactly 3 neighbors
Gardner M (Oct 1970) The fantastic combinations of John Conway’s new solitaire game “life”. Scientific American 223:

24. Game of Life Dies Dies Dies Stable block Stable blinker (period 2)
Time 1
Time 2
Time 3
Dies
Dies
Dies
Stable block
Stable blinker (period 2)
Gardner M (Oct 1970) The fantastic combinations of John Conway’s new solitaire game “life”. Scientific American 223:

25. Demos of Game of Life http://www.bitstorm.org/gameoflife/
Glider, gosper glider gun
a-plus, switchenginepuffer

26. Summary Alternative approaches to modeling spatial structure
Grid models
Polygon models
Methods for modeling movement
Random walk
Individual-based models (wildebeest)
Cellular automata