© 2018 Aaron A. King.

Ecological competition

In this exercise you will explore a model of ecological competition between two species. Let \(N_1(t)\) be the abundance (number of individuals) of species 1 and \(N_2(t)\), the abundance of species 2, both at time \(t\). The following equations describe the dynamics of the two populations under competition. \[\begin{gathered} N_1(t+1) = \frac{r_1\,N_1(t)}{1+a_{11}\,N_1(t)+a_{12}\,N_2(t)}\\ N_2(t+1) = \frac{r_2\,N_2(t)}{1+a_{21}\,N_1(t)+a_{22}\,N_2(t)} \end{gathered}\tag{1}\] In the above, all the parameters (i.e., \(r_1\), \(r_2\), \(a_{11}\), \(a_{12}\), \(a_{21}\), \(a_{22}\)) can be assumed to be non-negative. In addition, you can assume that \(r_1>1\) and \(r_2>1\).

Specific tasks:

  1. Explain the meaning of each of the parameters in the model. Pay attention to the units of each parameter.
  2. Show that you can choose the units of \(N_1\) and \(N_2\) so that \(a_{11}=a_{22}=1\). To do this, let \(n_1(t) = a_{11}\,N_1(t)\) and \(n_2(t) = a_{22}\,N_2(t)\). Show that \(n_1\) and \(n_2\) are dimensionless. Substitute these expressions into Eqs. 1 and simplify. Show that the resulting equations are equivalent to what you would obtain by setting \(a_{11}=a_{22}=1\) in Eqs. 1.
  3. What happens when \(r_1<1\)? What happens when \(r_2<1\)?
  4. Show analytically that the system has up to four equilibria. Find mathematical expressions for these.
  5. Write R codes to numerically compute the dynamics of the two populations given values of the parameters and the initial conditions. Produce both time series (\(N_1\) and \(N_2\) vs \(t\)) and state-space (\(N_2\) vs \(N_1\)) plots.
  6. Using your numerical solver, find parameters and initial conditions such that each of the following outcomes occurs. In each case, provide graphical evidence of the outcome.
    1. Species 1 outcompetes species 2, i.e., species 2 is driven to extinction.
    2. Species 2 outcompetes species 1.
    3. The two species coexist indefinitely.
    4. The outcome of competition depends on initial conditions, i.e., whichever species is present at larger densities at \(t=0\) outcompetes the other.
  7. Explain why these equations do or do not make sense as a model of competition. Keep in mind that it is possible for them to make sense in some ways and not in others.

Be sure to explain your reasoning carefully at every stage. Your goal should be that a first-year biology student who has not taken this course should be able to follow your reasoning.

Important Note: You will turn in each draft of your report via the course Canvas site. Upload both the Rmarkdown document and a PDF version of your report. Choose the filenames according to the following formula: Challenge_X-Y.Z where X is the challenge problem number, Y is the draft number (1, 2, or 3), and Z is the appropriate extension (.Rmd or .pdf).