Comment Notes from Excel File: NonIsothermGenLogistic.xls

Current version: February 4, 2008

On Sheet1:

In worksheet column A:
NonIsothermGenLogistic.xls [A1] <--This means "NonIsothermGenLogistic.xls" is in cell A1. The comment note below pops up when you point to cell A1.
NonIsothermGenLogistic.xls: This Excel workbook simulates the increasing logarithmic growth of a targeted microorganism during temperature controlled storage in real time. This version of the program simulates the growth of organisms whose isothermal survival curve follows the generalized Verhulst model: dN(t)/dt =k*N^a*(1-N/Na)^b. The temperature dependence of the growth parameters a(Temp), b(Temp) and k(Temp) obey an exponential relationship while Na(Temp) follows a logistic relationship. For more information on the concept, its various applications in calculating the microbial growth during storage at variable temperatures and its implementation with more complicated growth models, contact the authors (see References).

Background [A2]
Background: For generating a growth curve with this version of the program, the user needs to generate a temperature profile first by selecting a temperature model and its parameters. The default model included here, Temp(t)=TempInit+t/(Tpar1+Tpar2*t), has three parameters: TempInit (the initial temperature), Tpar1, and Tpar2 (e.g., TempInit=15, Tpar1=0.25, Tpar2=0.06).

The growth parameters are: apar1, apar2, bpar1, bpar2, kpar1, kpar2, Napar1 and Napar2 (e.g., apar1=0.4, apar2=0.022, bpar1=0.98, bpar2=0.02, kpar1=1.55, kpar2=0.04, Napar1=0.5 , Napar2=14.5).

Other models have also been proposed to characterize non-isothermal growth curves with a long lag period, MicrobialGrowthModelA.xls and curves with a short lag period, MicrobialGrowthModelB.xls (modified logistic model).

How to run [A3]
How to run: This workbook needs to run some macros. Therefore, you must click the Enable Macros button when opening the workbook. If you are using a recent version of Excel for Windows, you may need to reduce the security level in Tools>Macro>Security to Medium, then close and reopen the file to enable the macros.

Make any desired changes to the Chart Options in column A of Sheet1 and Sheet2.

Enter an initial value for each model variable in column B. The pop-up comment note that appears when the name cell above a variable's cell is pointed to with the mouse lists any restrictions on the legal range of values for that variable.

Once legal values have been entered for each variable, you may click the Clear button in column A to have the Clear macro initialize the calculation cells in column C, clear the graphical data columns D through G on Sheet1 and B through F on Sheet2 and delete any existing charts on both sheets.

Clicking the Solve button in column A first calls Clear then calls the Solve macro. The macro will solve the growth equation and update the model and graphical data cells on both sheets. It will also draw three charts on Sheet1 (Temperature vs Time, growth equation N vs Time and Log10(N) vs Time) and four charts on Sheet2 (a, b, k and Na vs Temp).

Modifications [A4]
Modifications: If you wish to see or modify the macros they are accessed from the Tools>Macro>Visual Basic Editor menu item. Checking the legality of user entries is done by the Init function which calls the NumOK and TextOK functions. All numeric cells in columns B and C are assigned names by the Init function. Those names are then used in the formulas in column C to refer to a cell in place of normal Excel cell references (such as B4 or R4C2). Unless absolutely necessary, changing the Visual Basic code should be avoided because an error can produce incorrect values or cause the program to stop working. This workbook runs properly in Windows Excel 97, 2000, XP, 2003 and 2007 and in Excel 2001 for the Mac.

Speed: With all versions of Excel the speed of the computations can be greatly improved (by 20 times or more) by changing Application.ScreenUpdating from True to False in the Solve macro. In that case, however, you will not be able to watch the charts being drawn. They will only appear after the final OK button is clicked.

Mac Problems: If you are using a pre-OS X version of MacOS running Excel 98 and receive a Run-time error saying "Method 'MaxChange' of object '_Application' failed" you need to comment out the line .MaxChange=0.000001 in the Clear subroutine by inserting a single-quote (') as the first character on the line. This is due to a bug in Excel 98. You should then enter 0.000001 manually in the Maximum change: edit box of Tools>Preferences...>Calculation.

Continuous ScreenUpdating is off by default in Excel X and Excel 2004 for MacOS X and we don't know how to reenable it. Therefore, you cannot watch the data and charts update in real time in Excel X and 2004. However, they will appear once the solution computations have finished and the final OK button is clicked. The change described above under Speed should be made with Mac Excel X and 2004.

References [A5]
References:
Peleg, M., Corradini, M.G. and Normand, M.D. 2007. The logistic (Verhulst) model for sigmoid microbial growth curves revisited. Food Research Internl. 40:808-818.

Peleg, M. and Penchina, C.M. 2000. Modeling microbial survival during exposure to a lethal agent with varying intensity. Crit. Rev. Food Sci. 40:159-172.

Peleg, M. 2003. Microbial survival curves: Interpretation, mathematical modeling and utilization. Comments on Theoretical Biology 8:357-387.

Peleg, M., Corradini, M.G. and Normand, M.D. 2004 Kinetic models of complex biochemical reactions and biological processes. Chemie Ingenieur Technik 76:413-423.

Peleg, M., Normand, M.D. and Corradini, M.G. 2005. Generating microbial survival curves during thermal processing in real time. Journal of Applied Microbiology 98:406-417.

Corradini, M.G. and Peleg, M. 2005. Estimating non-isothermal bacterial growth in foods from isothermal experimental data. Journal of Applied Microbiology 99:187-200.

Model by: Professor Micha Peleg and Maria G. Corradini, PhD.
Program by: Mark D. Normand, EDP Programmer II, and Maria G. Corradini, PhD.

Model by: Professor Micha Peleg and Maria G. Corradini, PhD.
Program by: Mark D. Normand, EDP Programmer II, and Maria G. Corradini, PhD.
Peleg & Normand located at: UMass Department of Food Science
Chenoweth Lab.
Amherst, MA 01003
Tel. (Peleg): 413-545-5852
Tel. (Normand): 413-545-2365
Fax: 413-545-1262
Email & Web (Peleg), see: http://www-unix.oit.umass.edu/~aew2000/
Email & Web (Normand), see: http://www-unix.oit.umass.edu/~adva000/
Email (Corradini): mariagcorradini@gmail.com

Last modified: February 4, 2008

Sheet1 Chart Options: [A13]
Sheet1 Chart Options: The following cells allow the user to set the values of certain attributes of the charts on this worksheet. In some cases restrictions are placed on the values that may be entered.

TimeAxisMin [A14]
TimeAxisMin: The cell below contains the minimum value for the Time axis on the charts. TimeAxisMin must be greater than or equal to 0.

TimeAxisMax [A16]
TimeAxisMax: The cell below contains the maximum value for the Time axis on the charts. TimeAxisMax must be greater than 0. and greater than TimeAxisMin.

TimeAxisTitle [A18]
TimeAxisTitle: The cell below contains a title for the Time axis on the charts. TimeAxisTitle can contain from 0 to 255 characters.

TempAxisMin [A20]
TempAxisMin: The cell below contains the minimum value for the Temperature axis on the Temp vs Time chart. TempAxisMin must be greater than or equal to 0.

TempAxisMax [A22]
TempAxisMax: The cell below contains the maximum value for the Temperature axis on the Temp vs Time chart. TempAxisMax must be greater than 0. and greater than TempAxisMin.

TempAxisTitle [A24]
TempAxisTitle: The cell below contains the title for the Temperature axis on the Temp vs Time chart. TempAxisTitle can contain from 0 to 255 characters.

TempMainTitle [A26]
TempMainTitle: The cell below contains the main title for the Temp vs Time chart. TempMainTitle can contain from 0 to 255 characters.

TempLineStyle [A28]
TempLineStyle: The cell below contains an index for the LineStyle of the Temp vs Time curve. TempLineStyle may be either 1=Continuous, 2=Dash, 3=Dot, 4=DashDot or 5=DashDotDot.

TempLineWeight [A30]
TempLineWeight: The cell below contains an index for the LineWeight of the Temp vs Time curve. TempLineWeight may be either 1=Thin, 2=Medium or 3=Heavy.

TempLineColor [A32]
TempLineColor: The cell below contains an index for the LineColor of the Temp vs Time curve. TempLineColor may be either 1=Black, 2=White (Not allowed), 3=Red, 4=Bright Green, 5=Blue, 6=Yellow, 7=Pink, 8=Turquoise, 9=Dark Red, 10=Green, 11=Dark Blue, 12=Dark Yellow, 13=Violet, 14=Teal, 15=25% Gray or 16=50% Gray.

NAxisMin [A34]
NAxisMin: The cell below contains the minimum value for the N (Growth) axis on the N vs Time chart. NAxisMin must be greater than or equal to 0.

NAxisMax [A36]
NAxisMax: The cell below contains the maximum value for the N (Growth) axis on the N vs Time chart. NAxisMax must be greater than 0. and greater than NAxisMin.

NAxisTitle [A38]
NAxisTitle: The cell below contains the title for the N (Growth) axis on the N vs Time chart. NAxisTitle can contain from 0 to 255 characters.

NMainTitle [A40]
NMainTitle: The cell below contains the main title for the N vs Time chart. NMainTitle can contain from 0 to 255 characters.

NLineStyle [A42]
NLineStyle: The cell below contains an index for the LineStyle of the N vs Time curve. NLineStyle may be either 1=Continuous, 2=Dash, 3=Dot, 4=DashDot or 5=DashDotDot.

NLineWeight [A44]
NLineWeight: The cell below contains an index for the LineWeight of the N vs Time curve. NLineWeight may be either 1=Thin, 2=Medium or 3=Heavy.

NLineColor [A46]
NLineColor: The cell below contains an index for the LineColor of the N vs Time curve. NLineColor may be either 1=Black, 2=White (Not allowed), 3=Red, 4=Bright Green, 5=Blue, 6=Yellow, 7=Pink, 8=Turquoise, 9=Dark Red, 10=Green, 11=Dark Blue, 12=Dark Yellow, 13=Violet, 14=Teal, 15=25% Gray or 16=50% Gray.

Log10NAxisMin [A34]
Log10NAxisMin: The cell below contains the minimum value for the Log10(N) axis on the Log10(N) vs Time chart. Log10NAxisMin must be greater than or equal to 0.

Log10NAxisMax [A36]
Log10NAxisMax: The cell below contains the maximum value for the Log10(N) axis on the Log10(N) vs Time chart. Log10NAxisMax must be greater than 0. and greater than Log10NAxisMin.

Log10NAxisTitle [A38]
NAxisTitle: The cell below contains the title for the Log10(N) (Growth) axis on the Log10(N) vs Time chart. Log10NAxisTitle can contain from 0 to 255 characters.

Log10NMainTitle [A40]
Log10NMainTitle: The cell below contains the main title for the Log10(N) vs Time chart. Log10NMainTitle can contain from 0 to 255 characters.

Log10NLineStyle [A42]
NLineStyle: The cell below contains an index for the LineStyle of the Log10(N) vs Time curve. Log10NNLineStyle may be either 1=Continuous, 2=Dash, 3=Dot, 4=DashDot or 5=DashDotDot.

Log10NLineWeight [A44]
Log10NLineWeight: The cell below contains an index for the LineWeight of the Log10(N) vs Time curve. Log10NLineWeight may be either 1=Thin, 2=Medium or 3=Heavy.

Log10NLineColor [A46]
Log10NLineColor: The cell below contains an index for the LineColor of the Log10(N) vs Time curve. Log10NLineColor may be either 1=Black, 2=White (Not allowed), 3=Red, 4=Bright Green, 5=Blue, 6=Yellow, 7=Pink, 8=Turquoise, 9=Dark Red, 10=Green, 11=Dark Blue, 12=Dark Yellow, 13=Violet, 14=Teal, 15=25% Gray or 16=50% Gray.

In worksheet column B:
TimeInit [B1]
TimeInit: The cell below contains the Initial Time (in hours) at the start of the solution. TimeInit must be greater than or equal to 0.

dt [B4]
dt: The cell below contains the Time step or delta-t increment (in hours) between successive iteration points of the solution and therefore between successive points on the charts. dt must be greater than 0. and less than or equal to 0.5.

TimeFinal [B7]
TimeFinal: The cell below contains the Final Time (in hours) at which a solution will be calculated. TimeFinal must be greater than or equal to 0. and greater than or equal to TimeInit.

NInit [B10]
NInit: The cell below contains an initial guess used to compute the growth function N. NInit must be greater than 0.

apar1 [B13]
apar1: The cell below contains the independent coefficient in the temperature dependence of the a parameter a(Temp) = apar1 * EXP(apar2*Temp). apar1 must be greater than 0.

apar2 [B16]
apar2: The cell below contains the rate coefficient in the temperature dependence of the a parameter a(Temp) = apar1 * EXP(apar2*Temp). apar2 must be greater than 0.

bpar1 [B19]
bpar1: The cell below contains the independent coefficient in the temperature dependence of the b parameter b(Temp) = 1 + bpar1 * EXP(-bpar2*Temp). bpar1 must be greater than 0.

bpar2 [B22]
bpar2: The cell below contains the rate coefficient in the temperature dependence of the b parameter b(Temp) = bpar1 * EXP(-bpar2*Temp). bpar2 must be greater than 0.

kpar1 [B25]
kpar1: The cell below contains the independent coefficient in the temperature dependence of the k parameter k(Temp) = kpar1 * EXP(kpar2*Temp). kpar1 must be greater than 0.

kpar2 [B28]
kpar2: The cell below contains the rate coefficient in the temperature dependence of the k parameter k(Temp) = kpar1 * EXP(kpar2*Temp). kpar2 must be greater than 0.

Napar1 [B31]
Napar1: The cell below contains the first coefficient in the temperature dependence of the Na parameter Na(Temp) = 1000+(1000000-1000)/(1+EXP(Napar1 * (Napar2-Temp))). Napar1 must be greater than 0.

Napar2 [B34]
Napar2: The cell below contains the second coefficient in the temperature dependence of the Na parameter Na(Temp) = 1000+(1000000-1000)/(1+EXP(Napar1 * (Napar2-Temp))). Napar2 must be greater than 0.

TempInit [B37]
TempInit: The cell below contains the Initial Temperature (in degrees C) of the Temperature profile at the start of the solution. It is the first of three parameters of the Temperature profile. TempInit must be greater than 0.

Tpar1 [B40]
Tpar1: The cell below contains the second of three parameters of the temperature profile. Tpar1 must be greater than 0.

Tpar2 [B43]
Tpar2: The cell below contains the third of three parameters of the temperature profile. Tpar2 must be greater than 0.

In worksheet column C:
nPts [C1]
nPts: The cell below contains the number of solution points (rows) to be generated, stored in columns D through G on Sheet1 and B through F on Sheet2 and plotted on all the charts.

EqnRoot [C4]
EqnRoot: The cell below contains the expression of the growth equation and displays the root of that equation at the current Time step. At each Time step the Solve macro uses Excel's iterative Goal Seek command to drive the EqnRoot value to zero by changing the value in the NCurr cell. Computing the NCurr values is the main goal of this workbook. How close EqnRoot comes to zero is controlled by the value assigned in the Clear macro to Application.MaxChange.

TimeCurr [C7]
TimeCurr: The cell below contains the Time (in hours) at the current solution point. TimeCurr is repeatedly updated by the Solve macro as it calculates and the value is copied to the appropriate row in the Time column at the end of each iteration.

TimePrev [C10]
TimePrev: The cell below contains the Time (in hours) at the previous solution point. TimePrev is repeatedly updated by the Solve macro as it calculates.

TempCurr [C13]
TempCurr: The cell below contains the current Temperature (in degrees C) computed from the temperature profile expression in that cell. TempCurr is repeatedly updated by the Solve macro as it calculates and the value is copied to the appropriate row in the Temp(Time) columns of Sheet1 and Sheet2 at the end of each iteration.

TempPrev [C16]
TempPrev: The cell below contains the previous Temperature (in degrees C) computed from the temperature profile expression in that cell. TempPrev is repeatedly updated by the Solve macro as it calculates.

NCurr [C19]
NCurr: The cell below contains the growth solution from the current Time step. Before iteration begins NCurr is set to NInit and thereafter NCurr is repeatedly updated by the Solve macro as it calculates and its value is copied to the appropriate row in the N(Time) column at the end of each iteration. Computing the NCurr values is the main goal of this workbook.

NPrev [C22]
NPrev: The cell below contains the growth solution from the previous Time step. Before iteration begins NPrev is set to NInit and thereafter it is repeatedly updated by the Solve macro as it calculates.

aSum [C25]
aSum: The cell below contains the sum of the previous and current values of the a parameter found in the aPrev and aCurr cells. aSum is repeatedly updated by the Solve macro as it calculates.

aCurr [C28]
aCurr: The cell below contains the current a parameter value computed from the expression in that cell. a is the scaling factor of the population size (in the traditional logistic model, a=1). aCurr is repeatedly updated by the Solve macro as it calculates.

aPrev [C31]
aPrev: The cell below contains the previous a parameter value computed from the expression in that cell. aPrev is repeatedly updated by the Solve macro as it calculates.

bSum [C34]
bSum: The cell below contains the sum of the previous and current values of the b parameter found in the bPrev and bCurr cells. bSum is repeatedly updated by the Solve macro as it calculates.

bCurr [C37]
bCurr: The cell below contains the current b parameter value computed from the expression in that cell. b is the scaling factor of the available resources term (in the traditional logistic model, b=1). bCurr is repeatedly updated by the Solve macro as it calculates.

bPrev [C40]
bPrev: The cell below contains the previous b parameter value computed from the expression in that cell. bPrev is repeatedly updated by the Solve macro as it calculates.

kSum [C43]
kSum: The cell below contains the sum of the previous and current values of the k parameter found in the kPrev and kCurr cells. kSum is repeatedly updated by the Solve macro as it calculates.

kCurr [C46]
kCurr: The cell below contains the current k parameter value computed from the expression in that cell. k is a proportionality constant similar to the one in the traditional model. kCurr is repeatedly updated by the Solve macro as it calculates.

kPrev [C49]
kPrev: The cell below contains the previous k parameter value computed from the expression in that cell. kPrev is repeatedly updated by the Solve macro as it calculates.

NaSum [C52]
NaSum: The cell below contains the sum of the previous and current values of the Na parameter found in the NaPrev and NaCurr cells. NaSum is repeatedly updated by the Solve macro as it calculates.

NaCurr [C55]
NaCurr: The cell below contains the current Na parameter value computed from the expression in that cell. Na is the asymptotic number of cells that the habitat can support (in the traditional logistic model, it is sometimes dubbed Nmax). NaCurr is repeatedly updated by the Solve macro as it calculates.

NaPrev [C58]
NaPrev: The cell below contains the previous Na parameter value computed from the expression in that cell. NaPrev is repeatedly updated by the Solve macro as it calculates.

In worksheet column D:
Time [D1]
Time: The cells below contain TimeCurr, the current Time (in hours), for each step in the solution process and are used as the x-coordinate values in the Temp vs Time, N vs Time and Log10(N) vs Time charts.

In worksheet column E:
Temp(Time) [E1]
Temp(Time): The cells below contain TempCurr, the current Temperature (in degrees C), for each Time step in the solution process and are used as the y-coordinate values in the Temp vs Time chart.

In worksheet column F:
N(Time) [F1]
N(Time): The cells below contain NCurr, the current N growth value, for each Time step in the solution process and are used as the y-coordinate values in the N vs Time chart. Computing the N growth values is the main goal of this workbook.

In worksheet column G:
Log10(N(Time)) [G1]
Log10(N(Time)): The cells below contain Log10(NCurr), the base-10 logarithm of the current N growth value, for each Time step in the solution process and are used as the y-coordinate values in the Log10(N) vs Time chart.

On Sheet2:
In worksheet column A:
Sheet2 Chart Options: [A1]
Sheet2 Chart Options: The following cells allow the user to set the values of certain attributes of the charts on this worksheet. In some cases restrictions are placed on the values that may be entered.

aAxisMin [A2]
aAxisMin: The cell below contains the minimum value for the a axis on the a vs Temp chart. aAxisMin must be greater than or equal to 0.

aAxisMax [A4]
aAxisMax: The cell below contains the maximum value for the a axis on the a vs Temp chart. aAxisMax must be greater than 0. and greater than aAxisMin.

aAxisTitle [A6]
aAxisTitle: The cell below contains the title for the a axis on the a vs Temp chart. aAxisTitle can contain from 0 to 255 characters.

aMainTitle [A8]
aMainTitle: The cell below contains the main title for the a vs Temp chart. aMainTitle can contain from 0 to 255 characters.

aLineStyle [A10]
aLineStyle: The cell below contains an index for the LineStyle of the a vs Temp curve. aLineStyle may be either 1=Continuous, 2=Dash, 3=Dot, 4=DashDot or 5=DashDotDot.

aLineWeight [A12]
aLineWeight: The cell below contains an index for the LineWeight of the a vs Temp curve. aLineWeight may be either 1=Thin, 2=Medium or 3=Heavy.

aLineColor [A14]
aLineColor: The cell below contains an index for the LineColor of the a vs Temp curve. aLineColor may be either 1=Black, 2=White (Not allowed), 3=Red, 4=Bright Green, 5=Blue, 6=Yellow, 7=Pink, 8=Turquoise, 9=Dark Red, 10=Green, 11=Dark Blue, 12=Dark Yellow, 13=Violet, 14=Teal, 15=25% Gray or 16=50% Gray.

bAxisMin [A16]
bAxisMin: The cell below contains the minimum value for the b axis on the b vs Temp chart. bAxisMin must be greater than or equal to 0.

bAxisMax [A18]
bAxisMax: The cell below contains the maximum value for the b axis on the b vs Temp chart. bAxisMax must be greater than 0. and greater than bAxisMin.

bAxisTitle [A20]
bAxisTitle: The cell below contains the title for the b axis on the b vs Temp chart. bAxisTitle can contain from 0 to 255 characters.

bMainTitle [A22]
bMainTitle: The cell below contains the main title for the b vs Temp chart. bMainTitle can contain from 0 to 255 characters.

bLineStyle [A24]
bLineStyle: The cell below contains an index for the LineStyle of the b vs Temp curve. bLineStyle may be either 1=Continuous, 2=Dash, 3=Dot, 4=DashDot or 5=DashDotDot.

bLineWeight [A26]
bLineWeight: The cell below contains an index for the LineWeight of the b vs Temp curve. bLineWeight may be either 1=Thin, 2=Medium or 3=Heavy.

bLineColor [A28]
bLineColor: The cell below contains an index for the LineColor of the b vs Temp curve. bLineColor may be either 1=Black, 2=White (Not allowed), 3=Red, 4=Bright Green, 5=Blue, 6=Yellow, 7=Pink, 8=Turquoise, 9=Dark Red, 10=Green, 11=Dark Blue, 12=Dark Yellow, 13=Violet, 14=Teal, 15=25% Gray or 16=50% Gray.

kAxisMin [A30]
kAxisMin: The cell below contains the minimum value for the k axis on the k vs Temp chart. kAxisMin must be greater than or equal to 0.

kAxisMax [A32]
kAxisMax: The cell below contains the maximum value for the k axis on the k vs Temp chart. kAxisMax must be greater than 0. and greater than kAxisMin.

kAxisTitle [A34]
kAxisTitle: The cell below contains the title for the k axis on the k vs Temp chart. kAxisTitle can contain from 0 to 255 characters.

kMainTitle [A36]
kMainTitle: The cell below contains the main title for the k vs Temp chart. kMainTitle can contain from 0 to 255 characters.

kLineStyle [A38]
kLineStyle: The cell below contains an index for the LineStyle of the k vs Temp curve. kLineStyle may be either 1=Continuous, 2=Dash, 3=Dot, 4=DashDot or 5=DashDotDot.

kLineWeight [A40]
kLineWeight: The cell below contains an index for the LineWeight of the k vs Temp curve. kLineWeight may be either 1=Thin, 2=Medium or 3=Heavy.

kLineColor [A42]
kLineColor: The cell below contains an index for the LineColor of the k vs Temp curve. kLineColor may be either 1=Black, 2=White (Not allowed), 3=Red, 4=Bright Green, 5=Blue, 6=Yellow, 7=Pink, 8=Turquoise, 9=Dark Red, 10=Green, 11=Dark Blue, 12=Dark Yellow, 13=Violet, 14=Teal, 15=25% Gray or 16=50% Gray.

NaAxisMin [A44]
NaAxisMin: The cell below contains the minimum value for the Na axis on the Na vs Temp chart. NaAxisMin must be greater than or equal to 0.

NaAxisMax [A46]
NaAxisMax: The cell below contains the maximum value for the Na axis on the Na vs Temp chart. NaAxisMax must be greater than 0. and greater than NaAxisMin.

NaAxisTitle [A48]
NaAxisTitle: The cell below contains the title for the Na axis on the Na vs Temp chart. NaAxisTitle can contain from 0 to 255 characters.

NaMainTitle [A50]
NaMainTitle: The cell below contains the main title for the Na vs Temp chart. NaMainTitle can contain from 0 to 255 characters.

NaLineStyle [A52]
NaLineStyle: The cell below contains an index for the LineStyle of the Na vs Temp curve. NaLineStyle may be either 1=Continuous, 2=Dash, 3=Dot, 4=DashDot or 5=DashDotDot.

NaLineWeight [A54]
NaLineWeight: The cell below contains an index for the LineWeight of the Na vs Temp curve. NaLineWeight may be either 1=Thin, 2=Medium or 3=Heavy.

NaLineColor [A56]
NaLineColor: The cell below contains an index for the LineColor of the Na vs Temp curve. NaLineColor may be either 1=Black, 2=White (Not allowed), 3=Red, 4=Bright Green, 5=Blue, 6=Yellow, 7=Pink, 8=Turquoise, 9=Dark Red, 10=Green, 11=Dark Blue, 12=Dark Yellow, 13=Violet, 14=Teal, 15=25% Gray or 16=50% Gray.

In worksheet column B:
Temp(Time) [B1]
Temp(Time): The cells below contain TempCurr, the current Temperature (in degrees C), for each Time step in the solution process and are used as the x-coordinate values in the a, b, k, and Na vs Temp charts.

In worksheet column C:
a(Temp) [C1]
a(Temp): The cells below contain the y-coordinates of a, the population size scaling factor, for the a vs Temp chart.

In worksheet column D:
b(Temp) [D1]
b(Temp): The cells below contain the y-coordinates of b, the available resources scaling factor, for the b vs Temp chart.

In worksheet column E:
k(Temp) [E1]
k(Temp): The cells below contain the y-coordinates of k, the proportionality constant, for the k vs Temp chart.

In worksheet column F:
Na(Temp) [F1]
Na(Temp): The cells below contain the y-coordinates of Na, the asymptotic number of cells that the habitat can support, for the Na vs Temp chart.

Generation of Non-Isothermal Growth Curves Using a Generalized Logistic (Verhulst) Model in Excel
Content last updated: February 4, 2008