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# SMNR IN BIOL OCEAN OCEAN 539

UW

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This 6 page Class Notes was uploaded by Aurore MacGyver Sr. on Wednesday September 9, 2015. The Class Notes belongs to OCEAN 539 at University of Washington taught by Staff in Fall. Since its upload, it has received 41 views. For similar materials see /class/192148/ocean-539-university-of-washington in Oceanography at University of Washington.

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Date Created: 09/09/15

Cobwebbing for discrete time single population mod els OCEAN 539B April 6 2005 These notes reiterate the cobwebbing procedure 1 demonstrated in class Cobwebbing is a useful graphical method for understanding the dynamics of a single population when that population is approximated by a discrete time demographic process that is when we can account for changes between one generation and the next rather than keeping track of continuous changes in time An equation of that type looks like pr1 prfpr where 739 is the time increment eg generations in the Daphnia case and fp speci es the population in the next generation 1041 as a function of the current population 1 The right hand side of this equation pfp is sometimes called the recurrence relation there are also other terms for it The example we used in class was the discrete logistic type equation P PT1TPT1 fT 2 To study this equation we rst rescaled it by de ning a non dimensional population p PTK From this we determined that P K107 and substituted this expression for P into 2 to get 1071 T 107 7 107 Equation 3 can be written explicitly in the form of 1 pr1 107 739 7 fltpr 71 P7 1 Cobwebbing mechanics The mechanics of cobwebbing are simple First you make a plot of the recurrence relation as a function of p on the horizontal axis and 1041 on the vertical axis You also need to add a diagonal line passing through the origin with a slope of one An example is shown in Figure 1 To advance a population through generations using cobwebbing start on the horizontal axis at the current population level that is at the point 0p Then repeat the following two steps 1 Recurrence relation for r 31 i i i H Next generation P 1 i i i i i i i i i 04 05 06 07 08 09 1 Current generation P Recurrence relation for r 31 i i i i o as i Next generation P 1 07 08 09 1 04 0 5 06 Current generation P Figure 1 First steps in cobwebbing Top d he recurrence relation7 Equation 17 plotted along with a line ofslope one Bottom the initial step in cobwebbing7 indicated by line segment 1 see text for details Recurrence relation for r 31 i i i H Next generation P 1 i i i i i 04 05 06 07 08 09 1 Current generation P Recurrence relation for r 31 i i i i o as i Next generation P 1 07 08 09 1 04 0 5 06 Current generation P Figure 2 Iterating through generations uso ing cobwebbing Top The second step in cobwebbing7 indicated by line segment 2 see text for details Bot tom The third step in cobwebbing7 indicated by line segment 3 see text for details 1 Draw a vertical line to the intersection point with the line of slope one 2 Draw a horizontal line to the recurrence relation curve Figures 1 and 2 illustrate the rst two iterations of this process 2 Cobwebbing and Matlab l have included in the Appendix the Matlab script that I used to generate these plots It is also available from the course website I tried to include some examples of useful Matlab commands such as saying graphics les and putting text onto plots It would be a good idea to scan through this code so that you remember where to nd these examples if you want to do something similar in one of your models As an extra challenge you could rewrite this code so that it loops through an arbitrary number of steps or enables you to compare the results from two or more growth rates or initial populations A Matlab code for cobwebbing XXX XXX Matlab script to perform cobwebbing OCEAN539B Spring 2005 XXX Z Parameters r 31 X Define the reproduction rate Pinitial 03 X Specify the initial population 1 Z Set up two curves to create a plot for cobwebbing Pcurrent O 05 1 Z This will be the horizontal axis Pnext r Pcurrent 1 Pcurrent Z This will be one curve on the vertical axis 0 figure Z This creates a new graphics window plotPcurrentPnext k Z Make the plot xlabel Current generation Pi Z ylabel Next generation Pi1 stringfortitle Recurrence relation for r num2strr titlestringfortitle Add some labels Z Now add a line with slope 1 hold on X This means don t erase the old plot when drawing the new plot plot0 10 1 g 0 o X This is how you write the contents of a graphics window to a file type K quothelp printquot for additional details print depsc cobO eps Z If you want a different format you could comment out the above and use Z one of the below instead Z print dpng cob1png Z print djpeg100 cob1jpg 1 Z Put in the first cobweb line This is a line that goes from the point Z Pinitial on the bottom of the plot up to the corresponding point on X the recurrence relation xcoords Pinitial Pinitial Psecond rPinitial 1 Pinitial ycoords 0 Psecond plotxcoordsycoords b Z This is how you place text labels on the plot textmeanxcoords 0035meanycoords 1 Z Save the current graphics window to a file print depsc cob1eps 0 Z Put in the second cobweb line This is a line that goes horizontally X from the previous point to the line of slope 1 xcoords Pinitial Psecond ycoords Psecond Psecond plotxcoordsycoords b Z Place text label on the plot textmeanxcoordsmeanycoords 0035 2 Z Save the current graphics window to a file print depsc cob2 eps 0 Z Put in the third cobweb line This is a line that goes vertically X from the previous point up to the recurrence relation xcoords Psecond Psecond Pthird rPsecond 1 Psecond ycoords Psecond Pthird plotxcoordsycoords b Z Place text label on the plot and save the current graphics window to a file textmeanxcoords 0035meanycoords 3 print depsc cob3eps

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