globals [ ;q ] ; q is the probability that a butterfly moves ; directly to the highest surrounding patch patches-own [ elevation visited? ] turtles-own [ initial-patch finished? ] to setup ca set q precision q 2 let x0 74 let y0 74 ; Assign an elevation to patches and color them by it ifelse (real-terrain) [ input-elevation set x0 110 set y0 135 ] [ build-hills set x0 71 set y0 71 ] let max-elevation max [elevation] of patches let min-elevation min [elevation] of patches ask patches [ set pcolor scale-color green elevation min-elevation max-elevation set visited? false ] ; end of "ask patches" crt 500 [ set size 2 set color red ; Set initial location of butterflies ; setxy x0 + random 10 - 5 y0 + random 10 - 5 ; setxy x0 y0 setxy random-pxcor random-pycor set initial-patch patch-here set finished? false set visited? true ; set pcolor yellow pen-down ] reset-ticks end ; of setup procedure to go ; This is the master schedule ask turtles with [not finished?] [move] if (count patches with [visited?]) != (count patches with [pcolor = yellow]) [ print "# visited patches does not match # yellow patches." ] tick ; stop when all the butterflies are at the summit, or ; 1000 ticks, whichever comes first if ticks >= 1000 or all? turtles [finished?] [ stop ] end ; of go to move ; The butterfly move procedure, in turtle context ; Decide whether to the highest ; surrounding patch with probability q ifelse random-float 1 < q [ let current-elevation elevation move-to max-one-of neighbors [elevation] ; move-to max-one-of (patch-set [ self neighbors ]) [elevation] ; uphill elevation if elevation < current-elevation [ show "Turtle is moving downhill." ] ] ; Move uphill [ move-to one-of neighbors ] ; Otherwise move randomly set visited? true set pcolor yellow end ; of move procedure to build-hills ; Assign an elevation to patches and color them by it ask patches [ ; Elevation is a sine function of X, Y coordinates ; with maximum elevation of 400 when sin is 1.0 set elevation 200 + (100 * (sin (pxcor * 3.8) + sin (pycor * 3.8))) ]; end of "ask patches" end to input-elevation file-open "ElevationData.txt" while [not file-at-end?] [ let next-x file-read let next-y file-read let next-elevation file-read ask patch next-x next-y [set elevation next-elevation] ] file-close end ; Corridor width is the number of patches visited by butterflies ; divided by the average distance a butterfly is from its initial position ; ; If a butterfly moves in a straight line, the number of patches visited ; equals the distance from its initial patch, so corridor-width = 1 ; The more a butterfly wanders off a straight line, the more patches ; it will visit, so the wider the corridor will be. ; to-report corridor-width let pcount count patches with [visited?] let dist mean [distance initial-patch] of turtles ifelse dist = 0 [report 1] [report pcount / dist] end @#$#@#$#@ GRAPHICS-WINDOW 220 10 678 469 -1 -1 3.0 1 10 1 1 1 0 0 0 1 0 149 0 149 0 0 1 ticks 30.0 BUTTON 15 20 86 53 NIL setup NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 15 60 79 94 NIL go T 1 T OBSERVER NIL NIL NIL NIL 1 SLIDER 15 105 187 138 q q 0 1 0.4 0.01 1 NIL HORIZONTAL MONITOR 15 150 109 195 Corridor Width corridor-width 2 1 11 SWITCH 95 20 205 53 real-terrain real-terrain 1 1 -1000 BUTTON 90 60 153 93 step go NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 15 240 107 273 Erase trails clear-drawing NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 15 200 112 233 Increment q set q q + 0.1\nset q precision q 2\nif q > 1 [ set q 1 ] NIL 1 T OBSERVER NIL NIL NIL NIL 1 @#$#@#$#@ # Butterfly Model ODD Description This file describes the model of Pe’er et al. (2005). The description is taken from Section 3.4 of Railsback and Grimm (2012). ## Purpose The model was designed to explore questions about virtual corridors. Under what conditions do the interactions of butterfly hilltopping behavior and landscape topography lead to the emergence of virtual corridors, that is, relatively narrow paths along which many butterflies move? How does variability in the butterflies’ tendency to move uphill affect the emergence of virtual corridors? ## Entities, State Variables, and Scales The model has two kinds of entities: butterflies and square patches of land. The patches make up a square grid landscape of 150 × 150 patches, and each patch has one state variable: its elevation. Butterflies are characterized only by their location, described as the patch they are on. Therefore, butterfly locations are in discrete units, the x- and y- coordinates of the center of their patch. Patch size and the length of one time step in the simulation are not specified because the model is generic, but when real landscapes are used, a patch corresponds to 25 × 25 m2. Simulations last for 1000 time steps; the length of one time step is not specified but should be about the time it takes a butterfly to move 25–35 m (the distance from one cell to one of its neighbor cells). ## Process Overview and Scheduling There is only one process in the model: movement of the butterflies. On each time step, each butterfly moves once. The order in which the butterflies execute this action is unimportant because there are no interactions among the butterflies. ## Design Concepts The _basic principle_ addressed by this model is the concept of virtual corridors—pathways used by many individuals when there is nothing particularly beneficial about the habitat in them. This concept is addressed by seeing when corridors _emerge_ from two parts of the model: the adaptive movement behavior of butterflies and the landscape they move through. This _adaptive behavior_ is modeled via a simple empirical rule that reproduces the behavior observed in real butterflies: moving uphill. This behavior is based on the understanding (not included in the model) that moving uphill leads to mating, which conveys fitness (success at passing on genes, the presumed ultimate objective of organisms). Because the hilltopping behavior is assumed a priori to be the objective of the butterflies, the concepts of _Objectives_ and _Prediction_ are not explicitly considered. There is no _learning_ in the model. _Sensing_ is important in this model: butterflies are assumed able to identify which of the surrounding patches has the highest elevation, but to use no information about elevation at further distances. (The field studies of Pe’er 2003 addressed this question of how far butterflies sense elevation differences.) The model does not include _interaction_ among butterflies; in field studies, Pe’er (2003) found that real butterflies do interact (they sometimes stop to visit each other on the way uphill) but decided it is not important to include interaction in a model of virtual corridors. _Stochasticity_ is used to represent two sources of variability in movement that are too complex to represent mechanistically. Real butterflies do not always move directly uphill, likely because of (1) limits in the ability of the butterflies to sense the highest area in their neighborhood, and (2) factors other than topography (e.g., flowers that need investigation along the way) that influence movement direction. This variability is represented by assuming butterflies do not move uphill every time step; sometimes they move randomly instead. Whether a butterfly moves directly uphill or randomly at any time step is modeled stochastically, using a parameter _q_ that is the probability of an individual moving directly uphillinstead of randomly. To allow _observation_ of virtual corridors, we will define a specific “corridor width” measure that characterizes the width of a butterfly’s path from its starting patch to a hilltop. ## Initialization The topography of the landscape (the elevation of each patch) is initialized when the model starts. Two kinds of landscapes are used in different versions of the model: (1) a simple artificial topography, and (2) the topography of a real study site, imported from a file containing elevation values for each patch. The butterflies are initialized by creating five hundred of them and setting their initial location to a single patch or small region. ## Input Data The environment is assumed to be constant, so the model has no input data. ## Submodels The movement submodel defines exactly how butterflies decide whether to move uphill or randomly. First, to “move uphill” is defined specifically as moving to the neighbor patch that has the highest elevation; if two patches have the same elevation, one is chosen randomly. “Move randomly” is defined as moving to one of the neighboring patches, with equal probability of choosing any patch. “Neighbor patches” are the eight patches surrounding the butterfly’s current patch. The decision of whether to move uphill or randomly is controlled by the parameter _q_, which ranges from 0.0 to 1.0 (_q_ is a global variable: all butterflies use the same value). On each time step, each butterfly draws a random number from a uniform distribution between 0.0 and 1.0. If this random number is less than _q_, the butterfly moves uphill; otherwise, the butterfly moves randomly. ## CREDITS AND REFERENCES Pe’er, G., Saltz, D. & Frank, K. 2005. Virtual corridors for conservation management. _Conservation Biology_, 19, 1997–2003. Pe’er, G. 2003. Spatial and behavioral determinants of butterfly movement patterns in topographically complex landscapes. Ph.D. thesis, Ben-Gurion University of the Negev. Railsback, S. & Grimm, V. 2019. _Agent-based and individual-based modeling: A practical introduction_, 2nd edition. 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