Hi Tutor, I am currently working on my Maths Internal Assessment for my IB course. Focusing on 3 models to explain the bacteria growth rate. (Gompertz Model, Logistic Model and Richards Model).

I have already tabulated the data for Gompertz and Logistic Model. But I have difficulty inputting raw data into the Richards Model. The screenshots below explain my models and some of my problems with understanding and inputting data into the Richards Model. Specifically, I am unable to understand what the 'shape parameter' of the Richards Model means.

Maths AI (HL) research project: My topic is to model bacterial growth using 3 different models (Gompertz Model, Logistic Model and Richards Model) and compare these 3 models and conclude which can better model the growth of bacteria. My raw data found from online is attached.: Bacteria Raw Data: Pseudomonas putida is a bacterium found in most soil and water habitats https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0230582 Time (h) Optical density of cell mass 0.216 0.220 W N H 0.240 0.250 0.260 0.270 0.280 0.290 0.330 0.360 0.380 0.400 0.460 0.482 0.492 0.592 0.544 0.584 0.594 20 0.620 ,644 22 0.648 23 0.648 Caption1) I have already input these data into the Gompertz models equation seen below; where parameters a, b, c were adjusted manually in spreadsheet to minimise the sum of square error at 0.068714: Initial Gompertz Model Used: f(t) = a * e -e^(b-ct) T (hrs) Optical Density N/No In (N/No) From a * e -"(b-cty Sq error a * e -6"/b-ct -elo-ct) (b-ct) 0.205 0.216 1.054 0.05227 0.10888 0.003205 0.108881 -2.718280 1.000000 0.220 1.073 0.07062 0.14102 0.004957 0.141024 -2.459602 0.900000 W N 0.240 1.171 0.15763 0.17822 0.000424 0.178215 -2.225540 0.800000 0.250 1.220 0.19845 .22025 0.000475 0.220254 -2.013752 0.700000 0.260 1.268 0.23767 0.26678 0.000847 0.266777 -1.822118 0.600000 0.270 1.317 0.27541 0.31729 0.001754 0.317288 -1.648721 0.500000 0.280 1.366 0.31178 0.37119 0.003529 0.371187 1.491824 0.400000 0.290 .415 0.34687 0.42781 0.006551 0.427807 1.349859 0.300000 0.330 .610 0.47608 .48645 0.000107 0.486447 1.221403 0.200000 0.360 1.756 0.56309 0.54641 0.000279 0.546405 1.105171 0.100000 0.380 1.854 0.61716 0.60700 0.000103 0.607001 1.000000 0.000000 0.400 1.951 .66845 0.66760 0.000001 0.667603 -0.904837 0.100000 0.460 2.244 0.80822 0.72764 0.006493 0.727636 -0.818731 0.200000 0.482 2.351 0.85493 0.78659 0.004670 0.786594 -0.740818 0.300000 0.492 2.400 0.87547 0.84405 0.000987 0.844049 -0.670320 0.400000 0.592 2.888 .06050 .89964 0.025873 0.899645 -0.606531 .0.500000 0.544 2.654 0.97594 .95310 0.000522 0.953099 -0.548812 .0.600000 0.584 2.849 .0468 .00420 0.001823 1.004199 -0.496586 0.700000 0.594 2.898 . 0638 .05279 0.000123 .052793 -0.449329 0.800000 0.620 3.024 .10671 .09879 0.000063 1.098786 -0.406570 0.900000 0.644 3.141 .14469 .14213 0.000007 1.142131 -0.367880 1.000000 0 . 648 3.161 1.15088 1.18282 0.001020 1.182823 0.332871 -1.100000 0.648 3.161 .15088 .22089 0.004901 1.22089 -0.301 194 1.200000 SOS 0.068714 SOS w/ot= 0.068714 1.65000 1.10000 0.10000 SOS= Sum of Sq Error Tabulated data from Gompertz Model2) I have also tabulated the data for the Logistic Equation with formula: = a / [1 + e^(b-cx), Tabulated data for logistic model below. https://edepot.wur.nl/202335 TABLE 2. Models used an their modified forms Equation (y=) Modified equation" (y=) Logistic: a [1 + exp(b - cx) ] 1 + exp "= (x -1)+2]) Gompertz: aexp[- exp(b - cx ) ] Aexp(- exp Hm " (x-1)+ 1 ] ) Richards: a{1 + vexp[k(t-x)])-1/ A 1 + vexp( 1 +U)exp "m(1+v)("?(x->]" Stannard: a 1 + exp - (1+kx) p A 1 + vexp(1 +v)exp "m(1+v)("(x-t>} Schnute: 1 - exp[- a (t - t, ) ] 140 1 - expl-a(72- 71)]) 1-6 1 -bexp(a) + 1-b-at) ]1/ 1-b * e= exp(1); u = shape parameter. Caption*You can only change the figures in yellow boxes Optical Density In (N/No Calc Sq error a/ 1+e^(b-cx) 1+e^(b-ct) e^(b-ct) (b-ct) T (hrs) N/No NO 0.205 - 0.216 1.054 0.0522 0.07583 0.000555 0.075825 40.224086 3.669293 1.300000 3.320114 1.200000 0.220 1.073 0.07062 .10641 0.001281 0.106407 28.663449 0.240 1.171 0.15763 0.14408 0.000184 0.144077 21.169303 3.004164 1.100000 0.250 1.220 0.19845 0.18881 0.000093 0.188805 16.154207 2.718280 1.000000 0.240155 12.700130 2.459602 0.900000 5 0.260 1.268 0.23767 0.24016 0.000006 0.000480 0.297315 0.800000 6 0.270 1.317 0.27541 0.29732 10.258465 2.225540 0.280 1.366 0.31178 0.35919 0.002248 0.359189 8.491360 2.013752 0.700000 0.290 0.3468 0.42450 0.00602 0.424499 7.184937 1.822118 0.600000 1.415 0.47608 0.49191 0.00025 0.491910 .200317 1.648721 0.500000 0.330 1.610 0.360 1.756 0.56309 0.56013 0.00000 0.560127 5.445193 1.491824 0.400000 1.854 0.61716 0.62798 0.000117 0.627976 4.856876 1.349859 0.300000 0.380 0.400 1.951 0.66845 0.69445 0.000676 0.694454 4.391939 1.221403 0.200000 0.460 0.758756 0.100000 2.244 0.80822 0.75876 0.002446 4.019738 1.105171 0.482 0.85493 0.82027 0.00120 0.820272 3.718280 1.000000 0.000000 2.351 0.492 2.400 0.87547 0.87858 0.000010 0.878575 3.471529 0.904837 -0.100000 1.06050 0.93340 0.016153 0.933401 3.267619 0.818731 -0.200000 0.592 2.888 0.544 2.654 0.97594 0.98462 0.00007 0.984617 3.097650 0.740818 0.300000 0.584 .03220 0.000216 1.032197 2.954862 0.670320 2.849 1.04689 0.400000 0.594 2.898 1.0638 1.07620 0.000152 1.076196 2.834057 0.606531 0.500000 0.620 3.024 1.10671 1.11673 0.000100 .116728 2.731194 0.548812 -0.600000 1.14469 1.15395 0.000086 1.153948 2.643101 0.496586 -0.700000 0.644 3.141 3.161 1.15088 1.18804 0.00138 1.188037 2.567260 0.449329 -0.800000 0.648 0. 648 3.161 1.15088 1.21919 0.004666 1.219192 2.501658 0.406570 -0.900000 SOS 0.038411 SOS w/o t=0 0.038411 3.05000 1.40000 0.10000 SOS= Sum of Sq Error Logistic Eq: f(t) = a/ [1+ e"(b-cx)] Euler's Num, e= 2.71828 Tabulated Data for Logistic Model3) NOW I am having PROBLEM with inputting the data for the Richards Model. I do not know what is the [u= shape perimeter] , what is [K], What is [1] and what is [v] as seen in this equation below: Richards: a{] + uexp[k(I-x)]}'\"\" Caption O Appreciate some advice. A big Thank you