Graphing the Von Bertalanffy Growth Equation


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1 file: d:\b \von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and regression on a graph. We will coninue oday wih he nex imporan sep: drawing a nonlinear funcion on a graph. In paricular, we will plo he von Beralanffy growh equaion on a graph. Review of wha we did Jus so we are sure wha we did las ime, le's go over i. We sared wih a se of Sandard Lengh (SL) and Toal Lengh (TL) daa for fish specimens. We suspec ha here is a linear relaionship beween hese wo, i.e., as SL increases so oo does TL. This does no mean ha he relaionship will be perfec. [Why no? Think of one good reason why he TL of a fish migh no be quie as large as i "should" be, given he SL?] Therefore, if we plo TL versus SL, he poins will fall roughly in a line, bu no exacly in a line. We wan o know wha he "bes" line hrough hose poins would be, and o ge ha line, we calculae he simple regression, ofen called he leassquares regression. These calculaions (easily done in Excel), give us he imporan values o define a line, namely he yinercep and he slope, i.e., m and b in he formula y = mx + b. However, we canno plo ha funcion direcly on he graph. So we need o find anoher way o do i. Here is wha we did: we said, for each value of x ha we have, i.e., he various values of SL, wha would he value of TL be ha falls on ha line? This was he Prediced TL column ha we calculaed. We made some imporan decisions in presening he graph: daa poins were represened by individual symbols  ha is because each symbol represens a piece of daa ha was colleced  while he relaionship was represened by a line  i heoreically applies o any value wihin he range of he daa. Imporan: daa is represened by poins, relaionships are represened by lines. DO NOT connec daa poins wih lines. This is an imporan poin ha people ofen ignore in an effor o make heir graphs "look nicer" or some such hing. In some cases, no only is i wrong, bu i is absurd o connec daa poins. For example, le us say ha you are fond of juggling balls. You plo he ime you can do his wihou dropping a ball agains he number of balls you are juggling. The daa migh look like his: Number of balls juggled Time wihou dropping (min) If you were o plo hese poins, you would see a seady decline in ime wihou dropping as he number of balls juggled increases. This makes sense. However, if you connec he poins, you are in essence saying ha if you were o juggle some inermediae number of balls, say 3.5, hen you would be able o juggle hose for 12.5 minues, i.e., halfway beween he ime for 3 and 4 balls. Is his rue? Mos likely i isn'. Firs, i is hard o know wha 3.5 balls would look like and secondly, i may very well be ha juggling a half ball would hrow off your paern and migh in fac be quie a bi harder han juggling 4 mached balls. Remember: Lines are no jus here o make hings look prey: hey mean somehing so be sure ha hey mean wha you inend hem o mean. 1
2 Ploing an arbirary funcion Jus as we can plo a sraigh line using he echniques we've learned, we can also plo any oher shape of line, provided we know he equaion for i, and a se of inpu (x) values. Recall ha he van Beralanffy growh equaion illusraes a relaionship beween lengh of a fish and is age. I shows ha fish keep growing wih age, bu slow down in growh rae as hey ge older and older. Exercise 1: To plo growh a age daa Ener he following agesize daa. In order o help me roubleshoo, please ener he ormaion in he cells I sugges. The following daa should be enered wih he firs ile in cell A5 and hen proceeding down (leave a space beween he ile and he firs value). Pu he Lengh column saring in B5. A B C... 4: Age Lengh 5: L 6: 7: : : : : : : : : : :... Now, consruc an Excel graph of lengh versus age, ploing he daa as given in his able. As you can clearly see from he graph, here is no linear relaionship beween lengh and age, and in fac, we would no expec one. I will ell you ha a von Beralanffy growh equaion wih he following key parameers acually fis he daa very well, K(  0) L = L (1  e ) L = 213 K = = 0.53 Exercise 2: To plo a von Beralanffy growh equaion on he same graph as he daa in Exercise 1. Hins: we need o fiddle wih he equaion a lile o ge i ino he compuer. The way ha Excel somehing 5 undersands e is as EXP(somehing). In oher words EXP(5) is he same as e Be sure you know wha "e" is  i is he base of he naural logarihms and is equal o approximaely Pu he label "L " in cell B23 and pu he value for L in C23 Pu he label K in cell B22 and pu he value for K in C22 Pu he label 0 in cell E22 and pu he value for 0 in F22 Assuming he curren value for (age) is in A7, hen he formula for he von Beralanffy equaion would be somehing like his: =$C$23*(1EXP($C$22*(A7$F$22))) 2
3 Pu his formula in column H (i should be in H7 hrough H16). Be sure you know why. Label ha column Von B. If all is righ, hen he value in H7 should be somehing like 83.8 and he value in H16 should be Recall why we use $ signs. [And yes, you are responsible for knowing his equaion and how o use i on he lab exam.] Now plo his daa series as a conneced series (do no plo poin markers) on he same graph as he original daa se. The curve should pass hrough or close o many of he poins. Hold on o your socks, here we go... So far you have ploed a von Beralanffy growh equaion, bu he nex quesion you are no doub wondering is: where did he ge hose values for L, K and 0 from? And how can I come up wih hose values myself o impress my friends and luence people? Exercise 3: In oher words, if I jus gave you he lengh a age daa (he firs wo columns), how would you calculae he parameers needed o fi a von Beralanffy growh equaion o he daa, i.e., how would you calculae K, L and 0? In he nex series of seps we are going o do his by doing a series of regressionns and ploing wo graphs. I srongly sugges ha you look a he sample prinou o undersand where we are going wih all of his. There are various ways of doing wha we need o do, bu one of he ried and rue mehods is known as consrucing a FordWalford plo. A FordWalford plo is a graph of l +1 versus l. If you consruc such a plo and draw a regression hrough he daa, you will ge a line wih a slope. And here is he good par: K for he von Beralanffy equaion is equal o he negaive naural logarihm of ha slope, i.e., K = ln (slope of FordWalford plo) o And i ges beer. If you hen plo a 45 line on he same graph, where ha line inersecs he regression will ell you L. Bu wai, here's more... Using he L you jus obained, if you plo a graph of ln(l  l ) versus, he resuling plo should be a sraigh line. The inercep of ha line wih he yaxis (i.e., where =0) is a very useful value, i.e., 0 = inercep  ln L K No, I'm no kidding. I's rue!!! o a. Consruc he FordWalford plo for his daa and plo he 45 line Here is how you do his: Label Column C as "L+1" In Column C, pu he values for L +1 [simply copy he values from he B column bu shif hem up one. The firs row of column B will no be used and here will no be a value for age 11. This is okay.] In A19, pu he word "slope", In A20, pu he word "inercep" In B19, calculae he slope of he line of L +1 vs L (i should be 0.71) [Recall: use he SLOPE funcion you learned previously] In B20, calculae he inercep of he line of L vs L (i should be 61.52) 3 +1
4 [Recall: use INTERCEPT funcion you learned previously] Label Column D as "L+1vsL" In Column D, inser he calculaed values for he regression of L +1 vs L [Using he slope and inercep you jus calculaed] On he same spreadshee, consruc a new graph. On his graph, plo he poins for L +1 vs L Then plo he line ha goes hrough hem (from he regression you jus performed, i.e. Column D versus Column B) Make column E a column iled (x=y). Copy he values from column B o column E Then Plo lengh agains iself as a line (i.e., column E agains column B) Verical gridlines are useful for finding L especially if you change he "Scale minor uni" seing o 10 (You can add hese from he Layou Gridlines menu) Now you will have wo lines ha almos inersec. Somehow you need o exend he lines so ha you can see where hey acually cross. Think abou his... Imagine if here were a fish age 15. No such fish exiss and you have no daa for i, bu IF such a fish exised, we could say how long i would be using our regression, righ? So, in A17, pu "15" and give i a lengh of 250 (i doesn' really maer wha value you pu here, as long as i is roughly abou figure ou why). Now exend he regression calculaions o cell D17. You should deermine ha a 15 year old fish should be Similarly exend Column E. Now exend he lengh of he lines on he graph by incorporaing hese "dummy" daa poins a age 15] Add Axes iles and prin ou he graph. Using your eye and he gridlines, you should now be able o deermine where he wo lines cross. I should be abou 213. Calculae he value of K (i is he negaive ln of he slope of he regression line) Wrie on your prinou he values of K and L. b. Plo he graph of ln(l L) versus.... Turn in he hree graphs Do his by creaing a Column F of ln(l  L) Do a regression of ln(ll) versus. You will need o calculae SLOPE and INTERCEPT for his regression. Pu hose values in F19 and F20 respecively. Pu he regression values in Column G Plo he daa and regression on a new graph. [Hin: As before, you need o Exend he line by adding a dummy daa poin of age 0, so you can see where he regression line inersecs he yaxis, i.e., age 0] Calculae he value of 0 using he formula above. See how i is very close o he value of 0 ha I originally gave you? Wrie on he graph he value of 0. Do you see ha i isn' acually necessary o plo his graph? The regression gives you he necessary inercep, hough he graph confirms ha i is a reasonable value. 1. he von Beralanffy graph 2. he FordWalford Plo 3. he graph of ln(l L) versus 4
5 along wih a prinou of your spreadshee. For he spreadshee prinou, urn on he feaure o show he column and row iles. Congraulaions Grasshopper... you have now aken your firs seps owards acually using Excel, no jus playing wih i. Be sure you undersand his exercise; you will be asked o do i on he lab exam, wih no insrucions.  END 5
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