Home

Modeling Tools Version 2.0 User`s Manual

1. The value entered for starting generation specifies the generation you want to start plotting values of N for i e the program simulates generations up to this value but does not plot them to the screen In the case illustrated in Figure A15 for example we will start plotting values at generation 25 and then 50 more generations of population growth will be simulated each generation will be plotted to the screen for a given value of rj The dialog entry Initial N is the starting population density K is the Carrying Capacity and C is the minimum sustainable population density for each rj According to the values in the dialog box we will start each population with a population density or size of 50 a Carrying Capacity of 100 and a minimum density or number of 0 Blowups and Computational Time in r vs N Plots Setting minimum and maximum values for N and r allows you to blow up any region of interest to look at it in more detail The coordinates of the region can be identified by clicking with the mouse as described in the get N utility and recorded and commented on in the run record using the record coordinates option and the text editing capabilities of the program In this way you can do successive explorations at greater and greater magnification Note however that if you blow up the region vertically choosing N min and N max closer in order to keep the same visual density of points you will have to increase
2. to stop 10 Is Pipi Rane k Figure A27 Dialog box for entering parameters for binomial distribution model of blending inheritance f X Binomial Distribution BD 0 55 X 0 f X 0 0 0 5 1 P 0 5 Q 0 5 Start Gen 1 Stop Gen 5 Figure A28 Example of the binomial distribution for f x 0 5 from generation 1 to 5 Modeling Tools 41 There are again two legends associated with Blending Inheritance simulation plots The bottom legend gives the values of p and q for the Oth generation Start Gen and Stop Gen are the starting and stopping generations respectively The legend on the right of a Blending Inheritance plot gives the point color or pattern associated with each generation The default pattern is solid for all generations but you can change this by selecting patterns in the Point Size option under the Simulation menu Interpreting Plots j File Edit BU Eaouee Window E Initialize amp Plot ro ReDraw ass Point Size Show Grid LINBS Graph Coordinates lt Binomial UVariance sum of Variance 5 oe ol Sum of Binomial Horizontal Legend Vertical Legend Record Coordinates Figure A29 Simulation menu options Binomial As can be seen from Figure A29 when Blending Inheritance is initially started up the Binomial option under the Simulation menu is selected as indicated by a check mark This will produce plots similar to the one depicted in Fi
3. 0 Ore Logistic Growth 2 0 aoe Logistic Growth 2 0 eae Logistic Growth 2 0 Kind application Kind application Size 97 516 bytes used 36K on disk Size 97 516 bytes used 96E on disk Where Frank SCSI O Where Frank SCSI O Created Mon Jul z3 1989 4 44 PM Created Mon Jul z3 1989 4 44 PM Modified Thu Mar 12 1992 9 36 PM Modified Thu Mar 12 1992 9 36 PM Version Logistic Growth Version 2 0 Version Logistic Growth Version 2 0 Jeff Schank amp Bill Wimsatt Jeff Schank amp Bill Wimsatt Suggested Memory Size K TOG Suggested Memory Size KJ 700 Application Memory Size K Application Memory Size KJ 1200 Before After Figure A16 Changing the amount of memory allocated to a program Running a Simulation Figure A17 is a bifurcation graph that resulted from running the parameter settings in Figure A15 except that the starting generation was set at 100 and then 100 additional generations were simulated and plotted for each value of rj using Logistic Growth These plots do take some time to generate a minute to many minutes depending on the numbers you specified for the initializing and plotting generations so the program allows you to interrupt plotting by simultaneously typing the Apple key and period When you do an r vs N plot all menu items except Graph Coordinates Initialize amp Plot and of course r vs N plot will be grayed out This is because these grayed out options
4. As mentioned in the last section points can be displayed in patterns colors or both patterns and colors There are five types of patterns which are determined by the percentage of pixels used to generate the pattern i e for a black and white monitor the order in which patterns will appear in plots in this program is black 100 of the pixels are filled dark gray 75 gray 50 light gray 25 and white 0 If you have a color monitor there are seven possible colors including black that can be displayed by this program These colors are in the order in which they appear in plots black red magenta yellow green cyan and blue This is our best attempt to order the standard colors to fit the color spectrum ordered from low frequencies to high If you have the patterns option on when you are using a color 20 Modeling Tools monitor then patterns will be filled with the color selected e g red with the gray pattern i e 50 pixels are filled will appear pink ReDraw The ReDraw option active for N vs t and N t vs N t m plot modes only allows the user to change the speed at which the plot is drawn to the window as illustrated in Figure A12 You can choose a plot delay of between 0 and 10 where 0 produces the fastest and 10 the slowest redraw The default value is set at 3 The value z entered causes a delay which is essentially machine independent of approximately z 60ths of a second between the plotting of e
5. Darwin 2 13 1867 or a shorter version of this since this is too long for a filename and take it with you NOTE that if the system crashes goes down or is deliberately shut down before you save this file it is gone so save soon and often Note Pads Note pads are additional text windows that you can make or open to record comments and information about simulations up to 8 note pads can be open at the same time rvs N Plots The r vs N menu option allows you to among other things generate bifurcation plots see the section above on simple population growth models and also to plot where the population is as a function of the starting value of N and the value of r for a small number of generations at a time Indeed you can do this for as little as 1 generation Modeling Tools 25 Entering Parameters When you select this option you will see a dialog box for entering parameters like that in Figure A15 If we use generations used or number of generations we can use the same dialog box for N vs r and Net vs r rus N Parameters Begin r at End r at Minimum M Maximum M Plotted Plotted Starting Generations to Generation be Plotted Initial N K 100 C Do Color PICT C Round N he Figure A15 r vs N parameters dialog box The r vs N option generates a graph with population density N plotted on the vertical axis and values of the growth rate parameter r plotted on the horizontal axis
6. Microsoft Corporation Helvetica Times and Palatino are registered trademarks of Linotype Hell The BioQUEST Library and BioQUEST Curriculum Consortium are trademarks of Beloit College Each BioOQUEST module is a trademark of its respective institutions authors All other company and product names are trademarks or registered trademarks of their respective owners Portions of some modules software were created using Extender GrafPak by Invention Software Corporation Some modules software use the BioQUEST Toolkit licensed from Project BioQUEST Table of Contents Modeling Tools User s Manitid livsdssscdeiiaccecsieti atiassinetishiae een aw iiatares 7 How To Use Logistic Growth and Blending Inheritance ccccsseseeeseeeeeeseeeeeeeseneees 7 What You Need To Know To Use These Programs sccisiscnsisccscrsscissassiaccessaaranaassvasarasiciia 7 How To Use the Simulation Programs for Logistic Growth and Blending HIVE TUEATACE ccassavsicccvanisncenters A EEEE EE AEE AREER REEERE EE 7 Starting the FP ROCUAUNS sxicaienssavaccsirsnnacnctona ao Stansaaa eel etaayesueaaaeleeeiae Manstan sued eaooreneaandannsameaninn 7 Running Simulation Experiments e seseeesesesesesesesesesesesesesesesssssssssssssnsssssesssesesesesesesene 7 Logistic Growth gaits ect cctiprcas ecb eccat pas emeizenoteran ybivaeape e aaaea Eae aia 8 Entering Parameters for the N vs t and N t vs N t m Plot Modes in the Initialize amp Plot Dialog BOX ss siscos
7. a MAC Plus or SE Thus the maximum time a run should take you to plot is about 2 minutes 400 points per run x 10 runs 4000 Modeling Tools 13 Running a Simulation When you are finished click on the OK button and a plot similar to the one illustrated in Figure A5 will appear after a few seconds of calculation a spinning beach ball cursor will appear while the actual computation is occurring N N vs time plot 0000 CESEN N t 0 25 r 2 K 100 C 0 At 1 Plots 5 Ar 0 2 Figure A5 Example simulation output In Figure A5 the horizontal axis is time in generation units and the vertical axis is the population size or density The horizontal line about 3 4 of the way up this graph is the Carrying Capacity K set here at N 100 This figure shows 5 runs all starting at No of 25 and lasting 25 generations for which the growth rate r starts at 2 0 and is incremented by 0 2 in each successive run All of these values can be read off the legend at the bottom of the graph Each point on the graph represents the density of a population at a given generation time t Different simulation experiments can be distinguished in this case because different patterns filling the points listed as run 1 5 in the legend on the right represent different simulation experiments Thus there are five patterns black through 3 gray scales to white and five simulation experiments See section on Patterns and Colors in Plot Display and Interactio
8. calculating effective population size Modeling Tools 9 Entering Parameters for the N vs t and N t vs N t m Plot Modes in the Initialize amp Plot Dialog Box Parameters Model Parameters Simulation Parameters N 0 2 CO Change N by 5S change r by Numerical precision L Round N 1 iterations 10 S At gt 0025 Generations 0 lt begin end 400 At begin at jo end at k Figure A2 Dialog box for entering parameters for logistic growth model After selecting the Initialize amp Plot option see Figure A8 below a dialog box will appear through which parameters and a specification of the simulation experiments to be run can be entered Upon selecting Initialize amp Plot default parameters will appear as shown in Figure A2 On the left side of the dialog box illustrated in Figure A2 the model parameters of the minimum density limited logistic growth equation are listed the initial population size N at the start of a simulation population growth rate r the Carrying Capacity k of the environment for the population and the minimum size C of the population which can sustain growth The fifth parameter of this model is the change in time parameter called At the default value of which is set at 1 The parameter At must be set so that At 0 0025 since one can t do less than one generation this value will generate 400 points in that one generation When At 1 thi
9. gt i fm We gey e amp Modeling Tools h Version 2 0 Ss p Library VH Users Manual Jeff Schank Indiana University William Wimsatt University of Chicago A BioQUEST Library VII Online module published by the BioQUEST Curriculum Consortium The BioQUEST Curriculum Consortium 1986 actively supports educators interested in the reform of undergraduate biology and engages in the collaborative development of curricula We encourage the use of simulations databases and tools to construct learning environments where students are able to engage in activities like those of practicing scientists Email bioquest beloit edu Website http bioquest org Editorial Staff Editor John R Jungck Beloit College Managing Editor Ethel D Stanley Beloit College BioOQUEST Curriculum Consortium Associate Editors Sam Donovan University of Pittsburgh Stephen Everse University of Vermont Marion Fass Beloit College Margaret Waterman Southeast Missouri State University Ethel D Stanley Beloit College BioQUEST Curriculum Consortium Online Editor Amanda Everse Beloit College BioOQUEST Curriculum Consortium Editorial Assistant Sue Risseeuw Beloit College BioQUEST Curriculum Consortium Editorial Board Ken Brown University of Technology Sydney AU Peter Lockhart Massey University NZ Joyce Cadwallader St Mary of the Woods College Ed Louis The University of Nottingham UK Eloise Carter Oxford College Claudia Neuhauser University of Min
10. have no function inr vs N plots Note r vs The time for a run is approximately proportional to right left initialization generations plot generations with initialization generations taking slightly less time than plot generations See estimates for times in footnote 2 above For the blending inheritance simulation the number of points plotted approximately doubles each generation so e g generation 10 takes as long as all of generations 1 9 You should be able to figure out why after reading the discussion of the blending simulation in the full Modeling manual 28 Modeling Tools N plots are not automatically redrawn when you call up a dialog box so make sure that you copy the graphs if you want to save them immediately after the program is finished plotting See the section on Transferring Plots to Other Programs below for details on copying graphs You can use the Graph Coordinates option to resize the graph but you will then have to choose the r vs N menu option again to replot the graph from scratch Therefore it is best to decide before plotting how big you want your graph to be revs N plot Nit 03 50 r 1 K 100 C 0 At 1 4r 0 006154 start Gen 50 MN plotted 30 Figure A17 r vs N bifurcation graph resulting from parameters in Figure A15 r vs Ne Plots The r vs Ne plotting option is a first simple step in connecting the population dynamics generated by the logistic growth and minimum densit
11. i acnismioavanasadiieasddvnanenvinansmanesnnaaasdeiunminsiavnies 39 Interpreting PIOUS scietecaeiaanicaidchiedatinieinniiied E earners 41 Plot Display and Interaction Options s s ssssssessessesssesisrerisisssiniresesteseseneenese 43 Saying WINAGOW Seiser a a E E E 44 Opemnge WINdOW oisein enesenn enan r nE EEES EA 44 Transferring Plots to Other ProgramS ssesssssssieresesesisisterssisesisestereneseseneenene 44 Printing VV UE OW 8 svsiresssessorcesosatanoncantaniccsnie wats tinditannanditoalwavaeeytiasnasstanesdageanaa isiin 44 Quitting and Closing VV OW S sarc ens cats iat iad cataete ectopic d atew estan otammeines 44 Modeling Tools User s Manual The Modeling Tools module contains the simulation programs Logistic Growth and Blending Inheritance and is intended to accompany the Modeling module The Modeling module can be found in the BioQUEST Collection folder on The BioQUEST Library CD How To Use Logistic Growth and Blending Inheritance What You Need To Know To Use These Programs We will assume that you are already acquainted with the Macintosh interface and the most basic terms and operations of the Macintosh operating system If you are not familiar with these basic terms and operations you should either refer to the Macintosh Owner s Guide for help in using this manual or find somebody to help you who is already familiar with the Macintosh The terms that you will need to be familiar with are menu menu option
12. the Advanced Topics section of the chapter on the logistic in the full Modeling manual found in the BioQUEST Collection folder This kind of plot is normally drawn without the lines option which produces a confusing welter of complexity It is used here to indicate the one thing the lines option is good for in this mode since and this is generally true the line between two points is drawn after the first point and before the second the part of it that covers over a point circle gives a vector representation of where the trajectory is going next The patterns of migration are absolutely deterministic for a given set of parameter settings and lag time m but can change significantly for changes in these settings so using lines to give transition directions is often useful and especially so in this plotting mode where the transition directions are often unintuitive Note on 16 Modeling Tools phase plots and abstract art While each of the plotting modes can produce a variety of informative and intriguing graphs our own experience suggests that this plot mode with lines can produce a variety of quite pleasing and unfamiliar graphic effects You might try experimenting with these Plot Display and Interaction Options There are several options available for altering the display of a plot and interacting with it If you click and hold on the Simulation menu Figure A8 you will see that several options are available All of thes
13. the color N vs r and Ne t vs r plot modes Something approaching motion picture speeds are attainable only for black and white PICTs in the r vs N mode which in trials appear to present about 15 frames second independent of how many points are in each plot a product of the fact that these unlike the others are stored as bitmaps Here the data presentation rate can be as high as 50 000 points sec One caveat virtual memory must be turned off for usable motion pictures or maybe I should say for pictures with usable motion For the size of the graphs to guarantee total visibility of all points within the boxed area the differences between the appropriate graph coordinates should be no larger than Bottom Top 160 and Right Left 331 Since the default settings of Top and Left are 30 and 25 respectively this implies settings of 190 for Bottom and 356 for Right If you want to see the labels at the bottom and right subtract 70 and 40 from these figures It is also worth noting that the scrapbook can save larger graphs than you can see in its window so you don t have to crop them small enough to see them unless you want to as for a movie A warning is in order however Graphs in PICT format can take a long time to print out on a LaserWriter that uses Postscript because of the time it takes to convert the PICT QuickDraw commands into Postscript commands have actually gotten presentation rates as high as 30 frames sec fo
14. the number of generations plotted proportional to the magnification so large blowups in the vertical dimension add computational time There is no such slowdown for blowups in the horizontal dimension making r min and r max closer since the program does 1 r value for every pixel on the horizontal axis though of course since the program does one r value for each pixel of graph width changing the graph coordinates left and right affects run length So shorter graphs take proportionately less time By checking the do color PICT option a color picture of will be generated using 8 colors one color for each generation plotted If more than 7 generations are plotted then the program will cycle through the same 7 colors i e black red magenta yellow green cyan and blue Note that this option is very memory intensive and that your computer may not have enough memory to complete it If there is not enough memory available to create a color graph try quitting and changing the memory available to the application by choosing Get Info and increasing the amount of memory as indicated in Figure A15 A color picture will be calculated without drawing it to the screen first Only after it is calculated will it be drawn to the screen Finally by checking Round N the population size or density for each generation is rounded to the nearest whole number Modeling Tools 27 Locked MOL Locked re Logistic Growth 2
15. window dialog box text box alert icon mouse point cursor click double click button radio button check box close box and desk accessory If you already know these terms you should have no trouble using these programs but even if you don t many of these terms are explained in the context in which they are used below How To Use the Simulation Programs for Logistic Growth and Blending Inheritance Starting the Programs To start either the Logistic Growth or Blending Inheritance programs double click on the application icon you want to run To double click an icon move the cursor over the icon you want to run and click the mouse twice in rapid succession The icons of these two programs are illustrated in Figure A1 1 o 10 mo Logistic Growth 2 0 Blending Inheritance 2 0 Figure A1 The Logistic Growth and Blending Inheritance program icons Running Simulation Experiments In this How To manual we will describe how to use two simulation programs to explore the ideas presented in the Modeling module We will begin by describing the Logistic Growth program because it has a slightly more complicated 8 Modeling Tools interface than Blending Inheritance and because many of the interface features in Blending Inheritance are similar to those in Logistic Growth Logistic Growth The simulation program Logistic Growth provides the user with a number of ways to graphically analyze interpret and understand the beha
16. 5 N t 1 N t vs N t 1 plot for Gens from 1 to 50 141 78 N t 0 N t 1 0 0 00000 aRoOn N t 0 70 89 141 78 N t 0 25 r 2 K 100 C 0 At 1 Plott 5 Ar 0 2 Figure A7 An example of an N t vs N t m plot The only constraints on the value of m is that m is a whole number greater than zero and that m is less than or equal to the number of generations times Af After you select a value for m click on OK and a graph similar to Figure A7 will appear Note that for best results you should change the coordinates of the graph under this option so that the horizontal and vertical axes have the same pixel lengths as described below The check box Show stability line is set to be on by default and controls whether or not the stability line is drawn In Figure A7 the horizontal axis represents the population density at generation time t and the vertical axis represents the population density at time t m In this case m 1 so the vertical axis is labeled N t 1 The additional vertical line about 3 4 of the way down the horizontal axis is the Carrying Capacity K plotted for N at time t The diagonal line running from the lower left hand corner of the graph to the upper right hand corner is the stability line so called because points which are exactly on it have the same value at tandt m See the much fuller discussion of the use and significance of this type of plot in the section on phase plots in
17. E 23 DOTS ES EPEE A TE A TT 24 FVS N Plots ereirnnrenni ee E i EER O EOT EEE 24 Entering ParametetSi sisira e e EE E 25 Blowups and Computational Time in r vs N Plots ee 26 R nning Simulaflon seisusesse anii iiini aena 27 tys Ne Plots siiwstsstinsenncicssasvsiagatasandsadaneseie E ae IEE EEA EEEa EiS 28 Perri Parameters seeing a a a a a 28 Running SIMUL ALON sig sive shsivoiastaisrantm cass shanealnanecsosangnvsbasbaiwonsehounoiiaeaeataapaiee 29 Transferring Plots to Other ProgramS sicoscssisscasientasntessustieaneonartiaastmamansaaente 31 Savine PICT Flesh sanaci i EARE E N 31 Saving Text Hl arenero aii AR N 32 Opening PICT Baneg icscaaceaniiseladacinasseeencienn R A a 32 Opening Text Files as NOt DO ONS wiccacisenesssuicnssssssetecsassnseserinoeeoistddeavsisionsaassionas 33 PPR WINdOWSassassin esse caesar E A EE AA 33 Quitting and Closing Win dOw S isacissisniusisasensuenonsestassantdanissvanneannsanisiabaraadeuinarsteelte 33 Copying and Pasting Graphs scencsieacsalusmastacrenseedetina nti chomivacntdd haeapoasenes onantaeumnes 33 A Note on PICTs and the Latent Powers of the ScrapBook 0 00 35 Copying and Pasting Text iisicacciciio tis onsnnscendniaddasnmdilaasaravinnastdenensacarteiustaneonsans 37 6 Modeling Tools Windows Menina cedas a e a aaa a aae 37 Font Mencerna ee OE EAEE 38 Blending Inheritance essed coiicess accu inhi leeincdcansspcabdasiaaadnidepcianiatnanaceeeeuaeeanee 39 Entering Parameters
18. Macintosh family of computers The other check box controls whether or not the value of N is rounded to the nearest integer value after the first generation a move which might appear to increase the biological realism of the model since we don t have fractional numbers of organisms in the real world If this box is checked values of N are rounded otherwise they are not This option has particularly interesting consequences which may not be apparent See the relevant section Roundoff Errors etc referred to above Finally the user can specify the number of generations that are plotted to the window The settings in the dialog box above will plot generations 0 through 50 The only constraint on plotting is that the number of points plotted per run must be less than or equal to 400 also see Scaling Graphs below We have for example plotted generations 50 000 through 50 400 though you will have to wait while the computer calculates the first 50 000 generations before you see anything And in that case you d probably be wise to set the number of runs to 1 only Modeling Tools 11 Data and Parameter Errors For the initial population density N and parameters K and C there are constraints placed on the numerical values which can be entered this will also be true for all dialog box interactions with this program and the Blending Inheritance program We have left it possible to put in values which don t make biological sense but t
19. STIC GROWTH 2 0 RUN RECORD A o y HHH He i i A a ba Ea i u 1 I 1 i read A AL a ee AE If PLOT PARAMETERS Mrs t Plot 1 T R ei nF ks oe RT ee ee ee ft Eo Tupe ofplot N vs t Generations O to 50 at i n c goi m o Ge i i fn i Initial poulation size N 25 Growthrate r 2 0 E F ry h A Carrying capacity K 100 Critical size C O i dh 5 ch Sampling interval Cin generations At 1 uu T J Curve family increment in Ar O 2 Plots 3 Graph Coordinates Top 30 Bottom 200 Left 40 Right 365 Point Size 4 Fill Mode Pattern 25 Comments 01 25 r 2 K 100 O at 1 Plots 3 ar 0 2 Time 24 73 N 25 52 ERY VALUABLE TO US Figure A13 Record Coordinates into the Run Record Window Other Windows 22 Modeling Tools All windows other than the plot window in both Logistic Growth and Blending Inheritance are text windows All text windows in these applications are editable but limited in two ways First they can contain at most 32 000 visible and invisible characters Second only unformatted text can be placed in these windows e g graphics cannot be put into these windows The run record is a special kind of text window to which certain kinds of data run parameters are automatically sent and to which other kinds of data may be manually sent using the record coordinates option and comments added Three of these windows the help notebook the bug comments notebook and the run record notebook are op
20. Similarly in this graph the sum of variance contributed by all terms from the left to and including the middle term at 5 0625 In this case the middle term contributes nothing which happens whenever a point is exactly at the mean of the distribution and can be detected by a flat line segment to the left of the point Modeling Tools 43 var X BD Sum of Variance 0 1375 X 0 Y var X 0 0 0 5 1 P 0 5 Q 0 5 Start Gen 1 Stop Gen 5 Figure A31 Example plot of Sum of Variance Sum of Binomial Figure A32 is an example plot of the sum of the binomial distribution resulting from selecting the Sum of Binomial option under the Simulation menu The sum of the binomial distribution is the cumulative sum starting at the left of the frequencies of the individuals having different characters in a population summed from character 0 to 1 Thus the sum of the binomial distribution at x 1 0 is the sum of all the frequencies of characters in the population and is equal to 1 0 X f X Sum of Binomial Distribution 1 1 X 0 E f X 0 P 0 5 Q 0 5 Start Gen 1 Stop Gen 5 Figure A32 Example plot of Sum of Variance Plot Display and Interaction Options 44 Modeling Tools The plot display and interaction options are the same as those for Logistic Growth except for the following We have added a Standard Deviations option which allows you to place horizontal lines in the sum of binomial distribution plot where the standard
21. This simulation works by using the logistic growth equation described above and plotting N for as many generations as you like against different values of the growth rate parameter r In Figure A15 you will notice that the first things asked for are begin r and end r starting values These values set the bounds of the horizontal axis Begin r is at the origin and end r is the right most value Thus in this case we will plot values of r from 1 0 to 3 0 The number of values plotted along the horizontal axis is determined by the width in pixels of the graph You can determine how many values will be plotted by opening the Graph Coordinates Window and subtracting left from right i e values plotted Right Left The formula for incrementing r along the horizontal axis is rj41 rj Ar where Ar r end r begin right left and i indicates the ith pixel starting from the origin of the graph For example looking at Figure A15 above we see that Left 25 Right 463 and r end r begin 3 0 1 0 2 0 Thus in this case Ar 2 0 438 0 00456621 and rj 1 ri 0 00456621 You then need to enter minimum and maximum values for N to be plotted within That is only values of N that fall within these upper and lower limits will be plotted to the screen In this case we would plot from 0 to 180 Next you are asked 26 Modeling Tools to enter values for the number of generations to simulate population growth for each value rj
22. ach point plus of course the actual computation time the computer requires to plot a point A word of warning if you enter 10 and there are several thousand points to be plotted this can take up to 10 minutes or more to complete Redraw affects only plotting speed not computation and is intended for demonstration purposes Varying plotting speed sometimes naturally leads to perception of different patterns in the data see Wimsatt 1991 and you may find it worthwhile to play with this option when you have a particularly intriguing data set Plot Delay 0 plot delay lt 10 Plot Delay CD a Figure A12 Plot Delay dialog box Get N Get N active for N vs t and N t vs N t m plot modes allows you to enter an N value from a plot as an initial population size or density value in the parameter dialog box by clicking within the plot frame This can occur when the cursor takes this form a and then by clicking the mouse the x and y coordinate values appear above the plot as can be seen in both Figures A5 and A7 You can move a value of N in the N vs t plot or a value of N t m into the parameters dialog box by first selecting the Get N option from the Simulation menu a check mark will indicate that this option is on and then moving the cursor to the point desired and pressing the mouse button If the Get N option is not checked the number selected will not Note that these are only approximate values sinc
23. and top 30 Current Window Size right 421 bottom 258 Kk Modeling Tools 19 Figure A10 Graph Coordinates dialog box Point Size The Point Size option active in all modes save r vs N allows you to change the size and pattern of points as illustrated in Figure All The size of a point can range from 1 to 100 pixels with the default size set for size 4 Larger point sizes might be useful for better visibility in a large window with a very large monitor e g for classroom viewing and also with appropriate x scaling for assessing random coverings Points can also be filled solid or with up to five patterns by selecting the corresponding radio buttons If you are working on a Macintosh with a color monitor multiple simulation trajectories will be printed in seven colors in a standard order in cyclic rotation see below Colors and patterns cycle independently so that there are 35 possible combinations more than enough for the maximum of 10 trajectories Thus the same color pattern combination never reappears on a color monitor though the patterns will repeat on a black and white monitor for trajectories 6 10 Note that point size can also be used to set the width and pattern of lines that are plotted in the r vs Ne t simulations described below Point size 1 point size lt 100 Point Size i Patterns Fill Point O Solid C Crane Figure A11 Point size dialog box Patterns and Colors
24. associated with the approximate equations for temporal effective population size Modeling Tools 31 Transferring Plots to Other Programs Logistic Growth and Blending Inheritance allow the user to save a graph as a PICT see note on PICTs below type file which can then be opened in standard graphics handling programs SuperPaint or MacDraw It also allows the user to copy and paste graphs into other programs such a MacWrite or Word Saving PICT Files To save a graph as a PICT file choose theSave PICT as option from the File menu ase Edit Simulation Window Font New Note Pad Open 07 ipg Apie R RRg e are PICT as Print Setup Print Window Figure A20 Save PICT as menu option A dialog box will appear that requests a name for the picture Type in a name in click Save as indicated in Figure A21 32 Modeling Tools O 3 0 INDIVID DIFF FIG Mm HasicPraararrn w Sot Seat SEGUE rGG TE E SiPGRYECDRYEC DPHPRGEP GT anma HE gi arming inher bee E PD fis Raha eon 4 Be Save Picture as N us t Plot 4 k Figure A21 Save as dialog box example The PICT document saved will look like the one depicted in Figure A20 My Fict File My Text File Figure A22 PICT and Text file icons Top Logistic Growth icons and bottom Blending Inheritance icons Saving Text Files When working on a text file the user can save their work just as with
25. cial names it then becomes laborious to figure out which standard graph names e g N vs t Plot 27 assigned to the sets of graph parameters when they are saved in the run record correspond to which special names given these files as informative names when they are saved This option will also give a record accessible without quitting the program of which graphs have been saved 24 Modeling Tools Run Record PLOT PARAMETERS Movs t Plot 1 Type of plot Mvs t Generations Oto 50 Initial poulation size N 25 Growth rate er 2 0 Carrying capacity K 100 Critical size C O Sampling interval Cin generations At 1 Curve family increment in ar 0 2 Plots 5 Graph Coordinates Top 30 Bottom 200 Left 40 Right 465 Point Size 4 Fil Mode Pattern Comments Figure A14 Sample Run Record with one entry This gives you a complete record of your work The data from any runs you choose can then be copied we recommend that you don t cut them so that they remain here also to any other notebook for further commentary or analysis Perhaps you want to include minimal comments here Note that if you don t save this file you won t have it when you quit the program and if you leave this file on the machine it will be modifiable or deletable by others Therefore we recommend that at the end of the day you save this file which is an important part of your data with your own name and or the date added Run record Charles
26. cketed by in sequential order reading from left to right with each menu followed by its choices bracketed by ee if they lead to a submenu or dialog box allowing or requiring further choices and bracketed by e if they do not Submenu options are preceded by complementing the following the menu option used to get to them and bracketed by e s Run Record This is the RUN RECORD notebook which automatically records all of the information you will need for a complete record of your session with this software see Figure A14 for a sample run record This includes e All parameters necessary to generate the data of a run are entered as a multiline block followed by a COMMENTS section A run occurs whenever you select Initialize and plot for plot modes N vs t or N t vs N t m r vs N or r vs N e t from the Simulation menu and close the resulting dialog box by using the OK command e Any data which change the appearance of a graph the status of the plot K amp C and lines commands and the point size show grid and graph coordinates dialog boxes are entered as an additional line modifying the data in the block e Any data collected using the record coordinates option is entered as a single line with the names and values of the x and y coordinates Planned Any save operations giving the file saved and the name it was saved under In a long session if the files are saved under spe
27. d you will get the dialog box illustrated in Figure A9 If you want to show the grid move the cursor over the show grid check box and click to turn it off click a second time The Auto Scale Grid check box indicates that you want to have the program automatically calculate a grid that fits the plot window each time you choose Initialize amp Plot When you do this the vertical scale will automatically be 10 greater than the largest y value plotted and if show grid is selected there will be 10 evenly scaled divisions between the x axis and that point or 11 divisions overall The maximum vertical scale printed to the left of the y axis will thus be 110 of the maximum value plotted and the default y axis division will be 10 of that value Thus the maximum value plotted can be easily read off from the show grid dialog box by multiplying this value by 10 If you want it as you will for some exercises in the blending inheritance lab be sure to record it before changing the y scale Entering a value in Change x axis by will cause if show Grid is checked a vertical line to be drawn every unit of the value entered along the x axis Thus according to the dialog box illustrated in Figure A9 a vertical line is drawn every 2 5 generations The Change y axis by value will cause a corresponding change in the drawing of horizontal lines starting from the y axis Thus in this case a horizontal line is drawn e
28. deviations would occur for a normal distribution reflecting the fact that binomial distributions with large m values are excellent approximations to the normal distribution These are dotted horizontal lines at s 0 3416 and 2s 0 4772 for illustrating the normal approximation see Figure A32 There are no Get N or Plot K amp C options in Blending Inheritance The auto scaling option is always on when the menu option Show Grid is selected you can of course specify your own grid for a given simulation but if you run another simulation the program will automatically recalculate a grid for you Saving Windows Follow the same procedure as described in the corresponding section for the Logistic Growth program Opening Window Follow the same procedure as described in the corresponding section for the Logistic Growth program Transferring Plots to Other Programs Follow the same procedure as described in the corresponding section for the Logistic Growth program Printing Windows Follow the same procedure as described in the corresponding section for the Logistic Growth program Quitting and Closing Windows See the corresponding section for the Logistic Growth program Apple J MENU cconicaccioncsrenns 34 Auto Scale Grid cccccccccccseesecseeeeeens 17 Pad Dafar yy eer nt ee 11 12 bad PATA TS ia a ssistuicstnsorislecnorapauccean 11 beach ball Cursot ccccceeeeeeseeees 12 14 beach ball cursor w
29. e options are available for the N vs t and N t vs N t m plot modes but only some of them are appropriate for the other 2 modes The remainder are grayed out in modes where they do not apply File Edit BEDS Window Font Initialize t Plot bet N ReDraw Point Size i Plotk amp C Show Grid Lines Graph Coordinates Horizontal Legend 1 Uertical Legend o Record Coordinates Figure A8 Simulation menu options Plot K amp C Plot K amp C active for N vs t and N t vs N t m plot modes has a check mark before it when initially opened up This means that it draws in appropriate lines for the parameters K Carrying Capacity and C minimum population density If you select this option and the check mark is on it will redraw the current plot without Carrying Capacity K and minimum density C plotted and this option will subsequently appear unchecked in the menu list If Plot K amp C is not checked then selecting this option will plot K and C and place a check mark before this menu option Modeling Tools 17 The option Lines is checked on when the program is started up To turn this option off simply select it if it has a check mark in front of it To turn this option back on reselect it Show Grid Show Grid active for N vs t and N t vs N t m plot modes does not have a check mark before it when initially opened because showing the grid in not a default option When you select Show Gri
30. e pixel coordinates are transformed into generations or population level numbers and there will be round off errors in doing this which should be no larger than 1 2p of the total range where p is the number of pixels along that axis of the graph as specified in the graph coordinates box Modeling Tools 21 be transferred to the dialog box but this can still be used effectively to read off values from the graph To record these data see the Record Coordinates option below Horizontal and Vertical Legends Both the horizontal and vertical legends are checked on by default Selecting either of these options allows the user to show or hide these legends and to include them or not in a graph which is to be saved Record Coordinates By default the record coordinates option is off Choosing this menu option allows the user to record data points into the Run Record Window see Figure A13 What is recorded is basically the current value of the x and y coordinates printed above the graph each time it is updated by pressing the mouse button This enormously simplifies collection of data from graphs and may be interspersed with commentary entered using the text editor features of the run record See example in Appendix F below lation Window Font ti N us t Plot 1 Help M vs time plot 2 0 EDITABLE HELF FILE Time 24 73 N 28 52 k comments notebook and the plot a y zte FOB paag gE Run Record y o EA rij E EG i E B LOGI
31. ecide to click on the Cancel button all values initially in the dialog box when you first selected the Initialize and Plot option will be restored This will also be true for all dialog box interactions in both Logistic Growth and Blending Inheritance Scaling Graphs At the bottom of the dialog box depicted in Figure A2 the user can specify the number of generations of population growth to be simulated The starting generation can be any reasonably large number less than the stopping generation for a simulation The only constraints are that the difference between the stopping and starting generations must be less than or equal to 400 x At but greater than or equal to 1 as mentioned above and that the maximum number of generations that can be entered into the stop box is 2 147 483 647 Warning if you enter such a large value you should be prepared to wait a long time up to half a year or more on a Macintosh Plus computer before it is done computing These run times refer to how long it will take to compute the run not to graph it You can however always interrupt a simulation when the beach ball cursor is on the screen by simultaneously pressing the Apple key and period on a MAC IL N vs t or N t vs N t m plots are calculated at approximately 500 generations per second plotted at approximately 200 generations per second with no lines and at about 130 generations per second with lines Rates are about 1 4 of this on
32. elieve to be the most likely problems for the neophyte user with more information to be found here in the Modeling Tools User s Manual The information in the Help file includes e Entries foreach menu item These are organized by major menu headings from left to right Apple File Edit Simulation Window Font and alphabetical within them All other entries are included in an alphabetical list following the Menu Index e Entries for options in each dialog box If you encounter problems with a dialog box and are unsure how to go on cancel the operation consult this help document and or the manual and then proceed Of course if you ve already encountered this situation we hope you did the right thing Modeling Tools 23 e Entries for any warnings you might encounter which are either special to this program or features of the Macintosh operating system which are of particular importance e Other topics from the index of the manual which seem important Consult it for further help We plan in the future to include probably in another NOTEBOOK a STRATEGIC HELP facility which will comment on particularly useful techniques for problem solving Capitalized terms in the text either occur as entries in this file in the MENU section or the following ALPHABETICAL section or refer to important operations in DIALOG boxes or elsewhere in the program The MENU section follows the organization of the MENU BAR giving the menus bra
33. en a simulation is running a beach ball cursor 64 will be spinning Whenever this beach ball cursor is spinning you can interrupt the simulation process by simultaneously pressing the apple key and the period key Any simulation will be stopped and cannot be recovered You can then start any new simulation you like Plotting Mode Options Size vs Time and Phase Plots If you move the cursor to the Simulation menu see Figure A8 below and click and hold you will see that there is a check mark that occurs before the option N vs t which means that this option is currently selected This is the population Size vs Time option The same data used to generate this plot can be used for another extremely useful kind of plot the variable lag phase plot Below the option N vs t you will see this plotting option here more precisely called N t vs N t m which plots values of population size at time t horizontal axis versus values of that same population s size at some later time t m vertical axis This is basically a temporal auto correlation plot of a variable N t with itself m periods later When you select this option a dialog box will appear requesting a value for m as shown in Figure A6 Nit us Wit mJ Enter a value for min an N t vs Nit m plot m lt generations delta_t Nit us N EJ Show stability line C Figure A6 Dialog box for entering the value of m Modeling Tools 1
34. ency of character x be greater than or equal to zero but less than or equal to one The frequency of the alternative character e g white is automatically calculated as q 1 p Below this box you are given two options generate a binomial distribution or model blending inheritance If you choose to generate a binomial distribution click on the radio button next to Binomial Distribution trials and then enter the number of trials represented in the binomial distribution you want to generate If you choose AUN 40 Modeling Tools to generate a blending inheritance distribution make sure that the blending inheritance radio button is selected and enter the number of generations of blending inheritance you want to depict That is you can enter the number of generations of blending inheritance you wish to observe Entering a value of 1 in the start text box and 5 in the stop text box will generate binomial distributions for generations 1 through 5 as illustrated in Figure A28 The constraints on the parameters entered here are a stop must be less than or equal to 10 b start must be greater than or equal to 1 and c start must be less than or equal to stop If you enter the same values for starting and stopping then only that generation will be plotted Parameters Frequency P of character H in the Population 0 lt P lt 1 O Binomial Distribution trials Blending Inheritance generations From start 1 ho
35. ened automatically when the program boots so that they will be immediately accessible under the Windows menu Up to eight other text only windows can be opened simultaneously as long as each is less than 32K in size Help e This basic help file is designed to be editable so that you can customize it and open and use it from within the program To edit this file save it under another name then reopen the saved file using the Open command and dialog box in the File menu and edit it using the commands in the Edit menu and the normal word processing commands select delete return to compose insert and delete text The maximum size of this file is 32K We have left you some room to add your own text even if you keep all of the standard help entries but since you will also have access to this standard help file you may delete it all if you wish If you do so then this file is no different than a standard Notepad Since you may open any standard text only document as a notebook you may also compose your personal help file in any word processor and then save it in text only mode to be opened by this program The program will usually open other text formats but the text will show additional formatting characters or be constrained in other ways e g text only plus line formatting will not word wrap appropriately for different sized windows The Help file contains minimal information to provide on line help with what we b
36. gure A29 The character x is assumed to measurable on a scale ranging from 0 to 1 Thus for example if our initial characters are white and red and we assign 0 to white and 1 to red then numbers between 0 and 1 are various shades of pink that become more like red in appearance as x approaches 1 and more white in appearance as x approaches 0 The vertical axis measures the frequency for a given value of the character x Variance A distribution of the contributions of each character class in the population to the total variance can be plotted by selecting the Variance option under 42 Modeling Tools the Simulation menu This will result in a plot that looks like Figure A30 The horizontal axis is the same as for the binomial plot but the vertical axis now indicates the variance contribution of each character class to the total variance in the population Note that classes right at the mean contribute nothing as one would expect var X BD Variance 0 06875 X 0 var X 0 P 0 5 Q 0 5 Start Gen 1 Stop Gen 5 Figure A30 Example plot of Variance contributions of characters in a population Sum of Variance Figure A31 is an example plot resulting from selecting the Sum of Variance option under the Simulation menu The Sum of Variance plot is the running sum of the individual variances summed from character 0 to 1 Thus the sum of the variance at x 1 0 is the total variance of the character in the population for a given generation
37. have sufficient memory especially for the graphs produced with r vs N plots can be reduced and printed at higher pixel resolution on standard paper If the window to be printed is a text window the text will be automatically formatted and printed out in consecutively numbered pages in either 9 or 12 point Geneva font Quitting and Closing Windows To quit Logistic Growth you can either choose Quit from the File menu or simultaneously press the Command key and Q The main graphics window cannot be closed or resized in this version However any desk accessories that you open up can be closed by clicking on their goaway boxes or choosing the Close option form the File menu All graphics windows can be closed either by clicking in their goaway boxes or by choosing Close from the File menu The Help Run Record and Bugs Windows are hidden and revert back to their default content i e the content of these windows present when you first open these programs when the goaway box is clicked on or the Close option is selected from the File menu In all cases you are warned to save any changes you have made Copying and Pasting Graphs After you have drawn a graph to the window you will notice that both the Copy and Clear options are black and the rest of the options are gray Figure A23 If you choose to copy a graph a copy will be made that will be saved exactly as it appears in the window even when you quit the program 34 Mode
38. here are mathematical and numerical constraints on the numbers that can be calculated which students should be made aware of In particular N and K should not be equal to zero If these constraints are violated a dialog box will appear which says that you have entered bad parameters and provide an appropriate message for correcting the error For example if 0 is entered for N then you will get a message which says that N cannot equal 0 as shown in Figure A3 Parameters Model Parameters Simulation Parameters k Figure A3 An example of entering bad parameters There is also another kind of error which can occur when entering parameter values This is the Bad Data error which occurs when a string of characters is entered which is not in the format of a number Figure A4 illustrates a Bad Data error for the parameter N In this case 25A was entered which does not symbolize a number and an alert box pops up with the message This string is not a number 25A Of course the far and away most common error of this type is to use the letter o for 0 and this catches it 12 Modeling Tools Parameters Model Parameters Simulation Parameters Figure A4 An example of a Bad Data error In either of these cases you need to click on the OK button to proceed after which you will be returned to the main dialog box with the offending entry highlighted as can be seen in Figures A3 and A4 If you d
39. ill be spinning DS SIE iisen 25 JOSIE a E E E E E 25 Begin Tireta e ai 25 bifurcation plots s ssesesssseses1ss1s10 8 24 binomial distribution eee eee 43 binomial distribution model 39 bitmap PIC Thrcenie nuce 37 BLENDING INHERITANCES8 12 37 41 44 Blowups and Computational Time in r vs N Plots ccccecceesssccccceseesesssssseseeees 26 bottom en 18 carrying capacity sevcesncudicncaronadsncasoaans 13 16 chang NPY iiccanscnscaniiacacsamanex 10 chang t Dyess an 10 change t DY itcninncndticiecnienas 10 Change x axis by sssessesesersssssesesssreseseres 17 Change y axis by esseseeesersesesescsrereree 17 changing the memory available to the application sae cetereatscelecincanecanedtettoeales 26 Check mark ccccccececessscseccsecceceesssssaees 14 Click the MOUSE woo eeeceeseesececceecesseseaeees 7 Close e a a 33 COPY aissis esenea aagi oaa ae 33 Copy Plot Parameters 0 cccee 37 Copying and Pasting Graphs 33 Copying and Pasting Text 37 CUTS OR aea a ERR 7 do color PICT soooeoneneneeeeenesnneeeeseesee 26 double Click cece ceeccesssssesecsesessessees 7 Edit menu ou cceeeeeececceceesssssecesesecsssseeess 34 eNd T oren e e aaa 25 File Menu eeeeoeneesesssssseseessssssrssressssse 33 Get Inf0 ccccccccccessssscsccstccecesssssesssereees 26 Get Noam aa 20 26 44 goaway boxes sassccccasounirisenscsineantonneandsinee 33 Graph Coordinates caucontctt
40. ines in that direction Of course you don t have to calculate this You can read it off the label on the y scale 141 78 in figure A8 or just eyeballing it 150 or 1000 it really is arbitrary will clearly do as well 18 Modeling Tools Change 4 axis by 12 8892 Change y axis by 12 8892 CI Show Grid E Auto Scale Grid Figure A9 Show Grid dialog box Graph Coordinates The Graph Coordinates option active in all modes allows the user to resize the graph within limits specified in the graph coordinates dialog box illustrated in Figure A10 The coordinate system for a Macintosh window has its origin i e x 0 y 0 in the upper left hand corner The x axis increases positively to the left of the origin as is customary and decreases negatively to the right But the y axis increases negatively as one move down from the origin and decreases negatively as one moves up which may not be customary to you When you initially open up Logistic Growth the numbers in the coordinate text boxes are set to a standard size window for the size of screen you are using That is the values for top and left cannot be smaller than the values which initially appear in these boxes and the values of bottom and right cannot be larger than 1024 pixels There are 72 pixels to the inch Graph Coordinates top bottom left right Maximum values are right 1024 and bottom 1024 Minimum values are left 25
41. ith PICTs 1 It can hold a large number of items we have dealt with scrapbooks with as many as 120 items in them and a size approaching 2 megabytes Thus if you are going to paste a lot of PICTs into a word processing document it is often more convenient to use the scrapbook than to save them as PICTs which have to be opened individually Be warned however that you can make and paste a PICT into the scrapbook which is too large to paste into some applications 2 You can save scrapbook files of pictures which can be loaded on other computers Duplicate them from the standard scrapbook file in the system folder give them another name and take them with you Rename the scrapbook on a computer you wish to use them with to something else and rename your desired file scrapbook file placing it in the system file When done reverse the process and take it with you 3 If you use Graph Coordinates from the Simulation menu to resize the graphs so they are totally visible in the Scrapbook then you can depress the right and left arrows in the bottom scroll bar in the scrapbook to show motion pictures of series of PICTs The speed of the motion pictures in all modes but N vs r in black and white is limited by how fast your computer can plot points to the screen and how many points you have per plot With a Macllfx for example plotting rate for points ranges from 650 850 sec in the N vs t and N t vs N t m plot modes to 3500 sec in
42. itincedinucteeciies 18 grid lines in only one direction 17 INDEX Help Mle cack cccoasisaradecincdensaneeoatmiamadniacees 22 ICON er e E EE iis 7 ImageWriter as cisceshiamelidiicastdinnsiaeuieane 37 aal Nhs ses doesadsestsccdivsndeestitceasss Scasastecees 26 Initialize amp Plot eee 8 9 17 35 Initialize amp Plotec cc eccecssssccccseseesesseees 8 Initialize and Plot cece eeeeeseeees 12 interrupt a simulation cccee ees 12 Cet it cer caeeetia ses E TA 16 lee a e N 18 25 Ti ESen e a eiae 17 LOGISTIC GROWTH 8 12 22 31 37 39 44 logistic growth equation 00 8 25 Logistic Growth equationsS 000 8 maximum value plotted 17 minimum density eee eee 16 minimum density limited logistic growth SQUAT OM sessissscssssvscsssssneenanvas 8 9 minimum population density 16 MOTOR PICHIFES cs aiastsrcaisasincsarsemnsonarens 35 IN WS iii ected ovietiv ed evedss a eased 37 INES Eae a ee cerces tebe tseees 14 PCE vs N t mM eesse 14 35 Opening PICT Files cacdcstasiceciraidntontacece 32 Opening Text Files as Notebooks 33 PESTO e a aN 34 pPATTETNS passers iceeni neliarnininieatiiniecnitines 41 patterns and COLOLS rinscassipoesdcraessasicntoess 19 personal help file scsssiaitavincuaessonicrsvasanenons 22 Phas Plots esemi 14 phase plots and abstract art 0 16 PICI ee ianentecnnaeaecbeabelevauautiennine 35 36 37 DE E E E E 18 Plot KE Crni 44 P
43. ling Tools Hle HA Simulation Font Copy aL Paste id Hi if i A Clear We mh T u l Figure A23 Edit menu It will often be convenient to use this program in conjunction with the desk accessory called the Scrapbook look under the Apple menu if it is not there consult the Macintosh Owner s Guide or somebody familiar with the use and installation of desk accessories To use the ScrapBook to save plots for later use in writing up your results e First size the graph using the Coordinates option under the Simulation menu to make the graph the size you want e Second select the ScrapBook option under the Apple menu e Third once the Scrapbook is open you can use the Paste option under the Edit menu to paste the plot into the scrapbook If you are trying to place the graphic in a particular place the graphic pasted is placed right before the graphic which is showing To place it last you will have to place it in front of the last picture and then cut and paste that in front of the new last one Using the Scrapbook in this way is illustrated in Figure A24 Copy Plot Parameters PET 46 TT pa nf 1 Modeling Tools 35 Scrapbook 14 Y gt SUS Fits Time 0 00 N 0 00 Figure A24 Scrapbook use example A Note on PICTs and the Latent Powers of the ScrapBook Graphs made under the Initialize amp Plot N t vs N t m r vs N and r vs Ne t options are saved in PICT format
44. lot WINdOWaasorinnnenns 22 plotting sp gdsisiiisiisersiriiesiiisisiness 20 Pomp SIZE a nraiaiccstaiisnrstaiiaren siaii aia 19 population density ce 15 25 Postsoript Wreed vacancies 36 Printing Windows siisesscecivssistsniacnaneaare 33 QuickDraw cass cansaantweideronanlakepsandaeenney 36 ETVS NE M unatn inea 30 EVS Niece 8 24 25 28 29 35 EVS Nean ERRE 28 46 Modeling Tools TVS INGE scans erate n E 8 35 Record Coordinates 6 21 23 26 TD FAW iecur anain 20 PoC PEE EE E 25 PNG D E E T 18 25 Round IN saxcspatineviacccrnnutoaeacuicentins 26 roundoff errors and finite automata 10 Ty Ci ECO jor casstaeaveedaaveetaveeuscinooakes 23 26 Saving Text Puls saicssasesatsaascatessianseteectasis 32 Scaling Graphs cisieineisiciminnspenncnie 12 Scrapbook ccreiricsiniiiaseias 34 Sh w Grid siismi eiere esssrieetveedan 17 44 Simulation menu 34 39 41 42 43 Simulation menu Figure A8 16 Simulation menu see Figure A8 BELOW ersero 14 Size VS TIE scasssasaseintiuanatapetborintedeanstoyies 14 stability LUNE sence anatieugicduieriedtanativabeasnsttivs 15 Standard DEVIAtOMS s ccaressssesseasacess 44 starting POMELAUIOM a icsasvesninsisassanveneatsiien 26 Sum of Binomial sser 43 Sum of Variance option under the Simulation Menu s es 42 temporal effective population size 8 30 text WTC OW Siri sihatscanncesedeaiaicnseamnnceonaannate 22 OPi iaoeia 18 using lines to give transition di
45. n Options below Points in this plot are connect by lines If you can t see points at 30 points second on a Plus assuming that the plotting has not been deliberately slowed down using the redraw option Computing it is another matter but we have found no reason so far to run single trajectories for more than 50 000 generations 800 seconds to calculate and up to 12 seconds to plot on the Plus or multiple trajectories for more than 1000 generations or around 10 000 total generations for 160 120 280 Plus seconds in one run The N vs r and Neff t vsr plots are an entirely different matter since a standard run of a bifurcation plot may though needn t always involve calculation of 120 000 generations and plotting 40 000 10 times as many as an N vs t plot with 10 trajectories each calculating to generation 800 and plotting from generation 800 to 1200 See the relevant footnote below in the discussion of N vs r plots for more details on this 14 Modeling Tools the patterns see points below under Plot Display and Interaction Options it is likely either that your points are too small or that you have set the points for solid tones rather than for patterns The two legends produced with this graph are optional and can be turned on or off independently of each other Note that if you are using a color monitor points will be in up to seven different colors see below Interrupting a Simulation You will notice that wh
46. nesota Angelo Collins Knowles Science Teaching Foundation Patti Soderberg Conserve School Terry L Derting Murray State University Daniel Udovic University of Oregon Roscoe Giles Boston University Rama Viswanathan Beloit College Louis Gross University of Tennessee Knoxville Linda Weinland Edison College Yaffa Grossman Beloit College Anton Weisstein Truman University Raquel Holmes Boston University Richard Wilson Emeritus Rockhurst College Stacey Kiser Lane Community College William Wimsatt University of Chicago Copyright 1993 2006 by Jeff Schank and William Wimsatt All rights reserved Copyright Trademark and License Acknowledgments Portions of the BioOQUEST Library are copyrighted by Annenberg CPB Apple Computer Inc Beloit College Claris Corporation Microsoft Corporation and the authors of individually titled modules All rights reserved System 6 System 7 System 8 Mac OS 8 Finder and SimpleText are trademarks of Apple Computer Incorporated HyperCard and HyperTalk MultiFinder QuickTime Apple Mac Macintosh Power Macintosh LaserWriter ImageWriter and the Apple logo are registered trademarks of Apple Computer Incorporated Claris and HyperCard Player 2 1 are registered trademarks of Claris Corporation Extend is a trademark of Imagine That Incorporated Adobe Acrobat and PageMaker are trademarks of Adobe Systems Incorporated Microsoft Windows MS DOS and Windows NT are either registered trademarks or trademarks of
47. r black and white bitmaps with the scrapbook but under conditions which are not yet reproduceable Relevant variables include RAM size cache allocation preparation by scrolling once through the pictures to load them all in RAM it ia always slower when they are being loaded from disk and whether virtual memory is activated Presentation rate is both slower and becomes highly irregular with virtual memory on Modeling Tools 37 You can minimize this effect by plotting fewer points use fewer runs or avoid many generations in Logistic Growth or later generations in Blending Inheritance Alternatively print them on a printer ImageWriter or paradoxically usually cheaper laser printer which does not have Postscript like the Apple IISC or GCC Personal Laser Printer II Finally you can convert PICTs into bitmaps which print much more quickly by pasting them into MacPaint documents at the cost of losing color and cutting resolution from 300 to 72 DPI dots per inch If you plan this latter course in advance it is probably easier to use the screen dump options supported within the Apple operating system caps lock plus shift plus plus 3 or 4 save or print the whole screen and shift plus plus 3 or 4 to save or print the active window The 4 direct print option works only with an ImageWriter not a LaserWriter and since these screen dumps print the frames of the windows in the second case and the whole screen in the first case both in
48. re is no noticeable difference in the values produced by either the exact or the approximate equations but as pointed out in the Modeling manual deviations can occur in certain circumstances The exact equation is always printed with a solid black line and if you have a color monitor it will appear black as well The default option for the approximate equation is to be plotted with the gray pattern option and it will appear red on a color monitor You can choose to plot both with solid lines and to change line widths by using the point size option described above If you choose a point size of 4 as illustrated in Figure A13 then each line will be 4 pixels in width a normal line is 1 pixel in width as illustrated in Figure A19 The pattern option is the default unless you have changed for another plotting option Choosing the solid option in Figure A13 will result in both the exact and approximate equation being plotted with solid lines There are again two legends associated with r vs Ne t plots both of which are optional The main thing to note about the horizontal axis is that start Gen means that population Ne t is calculated only starting from the specified starting generation in this case generation 25 Ne t plotted means that from the starting generation Ne t is calculated for then next 50 generations in this case The vertical legend on the right indicates which line is associated with the exact and which is
49. rections Ss E EE cucitalneyneuniisedor 15
50. s Menu The Windows menu allows the user to select which window will be active A check mark indicates which window is currently active see Figure A25 38 Modeling Tools ATLE Figure A25 Window menu Font Menu The Font Menu allows the user to change font size Currently only the Geneva font is available in font sizes 9 and 12 When the font size is changed the font for the whole window is changed to the font size selected Text files are also printed out using the font size currently selected as indicated by a check mark Modeling Tools 39 6 File Edit Simulation Window EAIA 1 lf N Ms time plot 14178 Time 0 N v Deneva f pe Wd i Figure A26 Font menu Blending Inheritance The Blending Inheritance program shares many features in common with the Logistic Growth program so only differences between these programs will be noted in this section For shared options refer to the last section on using Logistic Growth Entering Parameters The parameters used for generating a binomial distribution model of blending inheritance are entered by selecting Initialize amp Plot from the Simulation menu As with the Logistic Growth program a dialog box appears as illustrated in Figure A27 At the top of the dialog box illustrated in Figure A27 there is a box in which the frequency of character x e g red p can be entered The only constraint placed on this frequency is that the frequ
51. s can be interpreted as reproduction occurring synchronously once each generation As At decreases population growth becomes more like continuous growth over a generation Increases in At increase the time lag from when population size feeds back through density dependent effects to affect growth rate The further significance of this variable is discussed in the Modeling manual text for the minimum density limited logistic simulation Constraints on memory and the desire to do up to 10 runs limit the number of point plotted to 400 per run see below for a total of 4000 points per graph 10 Modeling Tools The right side of the dialog box in Figure A2 shows the simulation parameters You can do as many as 10 simulation runs a family of curves at the same time by iteratively incrementing the starting value of either of the two crucial parameters r or N You must first choose which parameter and specify the amount by which it is to be incremented or if the value entered is negative decremented in each successive run Then you must choose how many runs to do Select either change N by or changer by by clicking on the corresponding radio buttons When you initially open this dialog box the radio button corresponding to change r by will be selected If you select change N by then you will need to specify a value for the change in N for each successive simulation Next select the number of simulations to be run s
52. saving graphs In this case however the Save and Save as options under the File menu will be highlighted The icon of a text file is illustrated in Figure A22 Opening PICT Files PICT files that the user has saved can be reopened within the Logistic Growth and Blending Inheritance programs To do so choose Open from the File menu and select the PICT file you want to open Only one PICT file will be opened at a time PICT files can be saved under a different name or printed but they cannot in any other way be manipulated within these programs PICT files can however be edited in several commercial graphics programs Modeling Tools 33 Opening Text Files as Notebooks Text files no bigger than 32K can also be opened in Logistic Growth and Blending Inheritance To open a text file again go to Open from the File menu and select a text file Text files can be edited in these programs but if you prefer a different type font or size or fancier formatting you should use a commercial word processing or text editing program Printing Windows To print windows to a LaserWriter or ImageWriter choose the Print Window option under the File menu Note that you may first want to choose Print Setup in order to set the orientation of the graph on paper vertical or horizontal or to set the scaling factor to a different value The default is 100 but we have deliberately allowed for the production of graphs up to 1024 pixels wide which if you
53. scribed in the last section Similarly as with the r vs N plots you can specify the vertical range within which values of effective population size for a given value of r is to be plotted As described in the full Modeling manual the effective population size is calculated over 1 or more generations and you can calculate the effective population size starting from any generation Therefore we allow you to specify the starting generation and the number of generations to be used for calculating effective population size In Figure A18 we have decided to calculate effective population size from the initial generation 0 to generation 50 Finally as with the r vs N plots we need to specify an initial value for N for K and for C Running a Simulation Figure A19 is an example of a plot produced using the parameter values specified in Figure A18 We should set the default point size for this option to 1 rather than 4 30 Modeling Tools Melt r ws Melt plot 180 Exact App SR Ae o l 3 F N t 0 50 r 1 K 100 C 0 At 1 ar 000615385 start Gen 25 Melt plotted 50 Figure A19 rvs Ne t plot using parameters specified in Figure A18 using default point size of 4 this would be better changed to 1 from the Simulation menu As discussed in the Modeling manual section on effective population size and the logistic equation both an exact and an approximate equation for effective population size are plotted In Figure A19 the
54. sscativaeshoein abacabvoiasmeudensnnsanasshonssinnaasduatenactasmantoes 9 Data and Pararmetet Errors si ecacuivce wwacsantnncteeducechiiccaticigewendenstvusacapeedivonnndaadeteamavneuse 11 Scaling Graphs ssvaswrssicnnnna satay nnsnaitnlway asin ejudemadiedeane teajudsuaustnnnanateanaasbaneamesenieaneaieantes 12 R nning a Simulan sussniaeniirnsieri iei e a A E K 13 Interrupting a Sim la tOn irin N EA 14 Plotting Mode Options Size vs Time and Phase Plots s s sssnesessssesesesseses 14 Plot Display and Interaction Options asiasiaietotissciasssseisiadssatniaesverteasa ce aeeaganatientaiets 16 Plot K amp Ccisiesirieeesia ae lowed cag a a a eiaa E AAE EE ieee 16 Show Grider cil entessauunencedaetauatedinasaidantanadevearaedastennelasatlaabacenmatets 17 Graph CoordinateS eisissspess nin edasi 18 Point Size scnictarionienpiaansescaainachatersmseanpnandsroninsinceatndaladnnuasiedignecvatanganisiceynenteaapnecnditayns 19 Patterns and CO O18 taiicisisenieticntessatuinenssanenasseunanaacdeumAsioanyrniseingiamiuniasenacteilss 19 RV acca hatha on cd on a aia ela nana ions S 20 GEEN a a E E E A E 20 Horizontal and Vertical Lemons sisssiicssnsssiseacdstecssesiwsnsanstianeenatenaielanienisaernss 21 Record Coordinates iniciei iriiria aeie iaek inince iasi a enia irinna 21 Other WY UMNO WS 525 tasractils sca dae ao stitasaa n RANEE E EEA E EEA 21 Helpen e R E ana noice 22 R n Recordi eedeceereentreoien ta londarenne oeeaewrmnne caida NENNE ET
55. uccessively This is a whole number between 1 and 10 If you enter 1 then only one simulation will be run for the value of N entered i e the initial value of N will not be incremented by the change in N value if 1 is entered The same points hold if you select the change r by option There are two additional check boxes in the list of simulation parameters Both of these are intended to allow you to study the effects of roundoff errors in calculations though the latter option also allows one to do another interesting kind of simulation For further discussion see the section on roundoff errors and finite automata in the second chapter in the Modeling manual found in the BioQUEST Collection folder By checking the box Numerical Precision you can set the numerical accuracy of the calculations made during a simulation This has the effect of rounding all quantities used in the calculation to the number of decimal places indicated and reapplying this procedure each time an arithmetic operation actually only for x and which are the only two that can increase the number of places is applied in the calculation of the equation It is not applied for housekeeping functions and thus is not used for example in the graphics routines Numerical accuracy ranges from 0 to 6 decimal points If numerical precision is not checked the numerical precision of calculations is the numerical precision of the REAL numerical type in the
56. very 12 8892 units of population size This strange value is a result of autoscaling from which you can infer that the highest value reached was 128 892 Figure A9 shows the result of autoscaling a grid for the phase plot option for only in this case will the maximum x and y values be the same producing the same default values for grid increments in the x and y directions When autoscaling a grid with an N vs t plot the program automatically divides the x axis into 10 evenly sized intervals each 10 of the range from x min to x max You can choose any values you like for the grid increments in x and y directions and this can be very useful for detecting periodicities in the N vs t plot mode See the Advanced Topics section of the chapter on the logistic in the full Modeling manual You can also have grid lines in only one direction horizontal or vertical Although the program will not let you set the x or y axis increments to zero if you set the grid increment for the lines you wish to suppress to a value which is off scale e g if y max is 128 892 as depicted here then any number which is greater than 11 times the default value for the grid increment in the y direction or if they are different greater than 10 times the default value in the x direction will not plot on the graph Thus given the default value in A9 of 12 8892 any number of more than 11 x 12 8892 141 7812 entered in the appropriate box will eliminate visible grid l
57. vior of the logistic growth equation i e dN dt rN K N K and minimum density limited logistic growth equation i e dN dt rN K N K N C N There are three basic simulation options available under the Menu option Simulation described below These are Initialize amp Plot r vs N and r vs Ne t Data produced with the Initialize amp Plot option can be viewed either as N vs t plots or as N t vs N t m phase plots These two different suboptions will be described further below and in the Modeling manual The second major simulation option is r vs N plots This option allows you to plot the behavior of population growth with respect to different values of the growth rate parameter r Using the r vs N option bifurcation plots can be generated the normal use of programs of this type as well as a variety of other things not so commonly studied The third major simulation option is r vs Ne t plots This option allows the user to plot temporal effective population size Nefit with respect to different values of r and connects logistic population growth to issues in population genetics see the section on Logistic Growth in the Modeling manual These plots are also usefully compared with the r vs N plots for some purposes In addition as discussed in the why to section on the Logistic Growth equations the user can compare exact versus approximate equations for
58. volve graphic material which you might want to erase in a paint program if you care The 3 option saves up to 10 images at a time without exiting the program numbered screen 0 screen 1 screen 9 to disk as MacPaint documents which can be opened and worked on by most paint programs Graphs made under the N vs r option in Logistic Growth are saved as bitmap PICTs so the resolution you see on the screen is the same resolution you will see when you print a bitmap PICT on an ImageWriter or LaserWriter Note also that while you can observe runs with this option in color using a color monitor and in a color screen mode saves in this mode must be done in black and white All other saves may be done in color and in fact the PICTs are in color whether you produce them on a color or on a black and white Macintosh Thus you can do them on a black and white monitor and view them later in color by copying the scrapbook or copying them into a Word Processing document where you can watch them go Copying and Pasting Text Text can be copied or pasted from one text window to another by selecting the text to be copied and then choosing the Copy option from the Edit Menu and then moving the text insertion point to the place in the window you want to past the text see Figure A23 When the Copy Plot Parameters option is highlighted you can copy the plot parameters from the graph window and paste them into a text window such as a notebook Window
59. which preserves color attributes so you can make pictures to watch later on a color Macintosh or to print on a color printer even if the Macintosh or monitor you made them on was only black and white These will print out much more crisply on a LaserWriter than on an ImageWriter or the screen because they are constructed from mathematical functions rather than stored as bitmaps and printed at 300 dots per inch resolution The r vs N option is an exception here because there are usually far too many points often 10 000 to 80 000 to store all of their locations easily and they are stored as bitmaps which saves on memory space Indeed this fact makes it possible to see a slideshow particularly impressive in color every time you open the scrapbook or a document which has these PICTs in it It takes some time to draw the images so the running of the program is recreated each time you view the PICT This can be used to your advantage in writing a report if someone is going to read it on a computer Unfortunately hard copy does not yet have this capability If you are going to make a slideshow designed for the scrapbook be sure to size your graphics to fit in the scrapbook window If the size is left constant they will all be aligned in the same way Motion pictures are also most usefully done for a series of PICTs which are the same in most variables and varying in just one 36 Modeling Tools The scrapbook has several advantages for use w
60. y limited logistic growth equations with their underlying genetic consequences The Why To aspect for generating plots of growth rate r versus the temporal effective population size Netra or Ne is described in detail in the section on the logistic growth equation in the full Modeling manual In general the features of use and the computational time for calculating an r vs Ne plot are very much like that of the r vs N plots the r vs Ne plots take somewhat longer although you can learn quite different things from them they are in fact quite complementary Entering Parameters When you select this option you will see a dialog box similar to that in Figure A18 Modeling Tools 29 rus Neit Parameters Begin r at End r at Minimum Neit Maximum Neiti Plotted Plotted Starting Number of generation Generations for for Ne t calculating Ne t Initial N k Figure A18 Dialog box for r vs Ne t plots As you can see by comparing this dialog box with that for r vs N Figure A15 above you need much the same data in fact exactly the same data for the two graph types You are just going to do slightly different things with it With the r vs N plots you need to specify the range of r values for which you want to calculate the temporal effective population size Nerf In Figure A18 effective population size will be calculated for values of r between 0 0 and 3 0 The values of r are calculated as in the r vs N plots de

Modeling Tools Version 2.0 User`s Manual

Contents

Download Pdf Manuals

Related Search

Related Contents