Spatial models II (diffusion, two dimensions)

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

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 56:1266-1277

Polygon approach Fulton EA et al. (2011) Lessons in modelling and management of marine ecosystems: the Atlantis experience. Fish and Fisheries 12:171-188

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”

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

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

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”

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

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

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

High resolution GPS data for wandering albatrosses Humphries NE et al. (2012) Foraging success of biological Lévy flights recorded in situ. PNAS 109:7169-7174

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 http://en.wikipedia.org/wiki/File:Mandel_zoom_00_mandelbrot_set.jpg 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

Mandelbrot set (explanation given for those curious, video supplied for its beauty) www.youtube.com/watch?v=9G6uO7ZHtK8 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 http://www.youtube.com/watch?v=9G6uO7ZHtK8

www.youtube.com/watch?v=9G6uO7ZHtK8

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

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 0.08-0.25, top middle cell 0.25-0.56, top right cell, etc., 0.60-0.65, stay in the center cell, etc., 0.90-1.00, bottom right cell 17 Random walks.xlsx, sheet Wildebeest prob

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

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

With grass consumption 17 Wildebeest model consumption.r

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

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 223:120-123

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 223:120-123

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

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