Due Monday, April 8

Total Points: 10 (with some possible extra credit)

Instructions: Submit this homework as a document. It can be entirely typed on a computer and uploaded on Blackboard, or completed as a combination of handwritten and printed (computer generated plots must be printed). Answer questions in complete sentences and explain how you obtained your answers. Only provide answers and include plots, do NOT send in code! Collaboration and discussion - openly and on the discussion forum - is highly encouraged, but do not provide and publicize actual answers.

I. Fitting Competition

This exercise is built directly off of the lab exercise from Thursday, March 31. The link to the lab is here. The lab contains links to the datasets and code.

Recall - the goal - essentially, was to recreate the following analysis by Gause:

Figure 1: Gause’s plots of paramecium growth and competition.

The first thing you need to do is load the two datasets single.csv and micture.csv into R, using using the FILE/IMPORT DATADET point and click or using the read.csv() command

single <- read.csv("single.csv")
mixture <- read.csv("mixture.csv")

The single dataset has data on the growth of two species of paramecium: P. caudatum and P. aurelia separately. The mixture dataset has data from an experiment where they’re both present.

The second thing you’ll need to do is open and run all the code in the fittingfunctions.R file.

1. Plot the data

Make plots of both datasets. One plot for the single data, and one for the mixture data, with two lines on each plot.

The best way is using R following the steps in the lab, but you are welcome to do this in Excel or by hand or any other way you like.

2. Fit the logistic data

Use the fitLogistic() function to fit a logistic curve to each of the two single growth curves. Report the carrying capacity and instrinsic growth curve for each of these.

Make a new plot of the data including a fitted line, using the linesLogistic() function.

3. Fit the competition data

The simulateCompetition() function numerically generates the 2 species competition process we discussed in class, i.e.:

\[{dN_1 \over dt} = r_1 N \left(1-{N_1 \over K_1} - \alpha {N_2\over K_1}\right)\] \[{dN_2 \over dt} = r_2 N \left(1-{N_2 \over K_2} - \beta { N_1\over K_2}\right)\]

And the getR2() function quantifies the quality of that fit, with 1 being a perfect fit.

3a) Following the instructions in the lab, see how good of a fit you can get of the competition model. Report estimates of \(K_1\), \(K_2\), \(r_1\), \(r_2\), \(\alpha\) and \(\beta\), and the final \(R^2\) value.

3b) Plot the fitted curves.

4. Analyze the competition

The isoclines of a two-species competition model are given by:

\[N^*_2 = K_2 - \beta N_1\] \[N^*_1 = K_1 - \alpha N_2\]

Plot these isoclines using the estimate of \(\alpha\) and \(\beta\) obtained from the fitted competiton model. You can do this by hand, in Excel, or in R. It is relatively easy to draw straight lines in R, using the abline() function - which takes an intercept and slope as arguments. An example is below, drawing a red line with intercept 80 and slope -1.5, and a blue line with intercept 40 and slope -0.5:

plot(0,0, xlim = c(0,100), ylim = c(0,100), xlab = "N1", ylab = "N2")
abline(80, -1.5, col = "red")
abline(40, -.5, col = "blue")

Based on analyzing isoclines, do you think that these two species of paramecium can coexist? If not - why not? If yes - at what levels?