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Math 121 Calculus for Biology
Spring Semester, 2012
Lab Help
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24-Apr-12
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San Diego State University
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The first problem investigates the growth o fish using von Bertalanffy equation and allometric modeling. It uses the chain rule to find the maximum rate of weight increase, which is important information for the farm fishing industry. The second problem examines three models for the
population growth of gregarious birds, including the Discrete Logistic
Growth model, the Discrete Logistic Growth model with emigration, and a
Cubic model that demonstrates the Allee effect.
Question 1: This problem is very similar to the material in the lecture notes, where we fit data to find the length of a fish as a function of its age using von Bertalanffy's equation. Also, we create an allometric model for the weight of a fish as a function of its length. The questions that you answer are very similar to the ones in Problem 13 in the homework problems. The fit to the von Bertalanffy equation is done very much the same way as you do the first question in this lab, while the fitting to the allometric model follows the techniques learned in Math 121 using Excel's trendline. This is readily accessed by right clicking on the data. You may need to increase the number of significant digits given to you by trendline. To obtain 5 significant figures, we right click on the formula, selecting Format Trendline Label, choose Number and Scientific, and finally select 5 Decimal Places. As in the first problem, the use of Maple might be valuable for helping with the differentiations (though you can practice your skills by doing them by hand). For this problem, the first and second derivatives are zero at t = 0, because the curve is very flat there. To get the point of inflection and the value of the function and the derivative at the point of inflection you need the following commands (assuming you see the point of inflection is between t = 1 and t = 20:
> p := t -> model goes here;
> dp := diff(p(t),t); # This gives the derivative
> sdp := diff(dp,t); # This gives the second derivative
> tp := fsolve(sdp=0, t = 1..20); # This gives the point of inflection
> p(tp); evalf(%); # This gives the population at the point of inflection
> subs(t = tp, dp); # This gives the derivative at the point of inflection
Question 2: This study begins with a set of data for the
population growth of some birds, a time series. This is the usual manner of presenting data. However, the discrete dynamical models that we study use an updating function. In this problem you should work with one spreadsheet that creates the updating function, then later use another spreadsheet (or different location on the same spreadsheet) to analyze the time series.
To begin we concentrate on fitting the updating function to the data. Follow the directions carefully to obtain the right alignment of the population data (only) to use Excel's trendline polynomial fit and to get the right quadratic function here. Note that the function you are producing here is the updating function (Logistic Growth or quadratic function) for finding the next population, and you are NOT using the time data for this part of the problem. This portion of the problem has you graphing Pn+1 vs Pn.
On a separate spreadsheet (or different part of the one where you created the Updating function), make two columns with the original time series. Next to these two columns create a column for the model, which uses the best fitting updating function you found above. Create named variable, P0, that you use to begin your simulation of the logistic growth model (using the updating function). Starting the model with P0, use the updating function to simulate the population growth in the cell below. Find the sum of square errors between the data and this model simulation. With Excel's Solver, you adjust only the initial population to find the best fit of the model to the time series data.
In the next part of the question, you use your techniques from class (possibly with the help of Maple) to find the equilibria of the model and determine the local behavior of the model near those equilibria (Stable or Unstable, Monotonic or Oscillatory). This part returns you to understanding the importance of the graph of the updating function.
After working through the Logistic Growth model, you examine two alternative models. First you consider the system not being closed and consider the possibility of constant emigration on the dynamical model. The second alternative is fitting a cubic equation for a closed model, so passing through the origin. Both of these models have one additional parameter, so should fit the data better than the logistic growth model. You use Excel to find the polynomial fits, then use Excel's Solver to fit the best initial condition. You will also use the techniques learned in class for finding equilibria and studying the local behavior near those equilibria.
The techniques employed to solve this question are similar to the ones that you have used before. The updating function simply uses the polynomial fit under Trendline in Excel (order 2 or 3). Be sure to format your model formula to have enough significant figures (preferably using the scientific notation option). The fitting of the time series is done by defining P0 on the spreadsheet, then entering P0 as the first element of the model simulation. The discrete dynamical model is simulated by filling down the model column using the updating function fit by Trendline. Another column in the spreadsheet is used to compute the square error between the time series data and the model. Excel's Solver is used to find the least sum of square errors (minimum) by changing only P0.