Please place your name(s) in the upper-right corner.

  1. Yesterday Professor Aaron discussed the intricacies of writing one’s own numiercal solver (in his words, “simulation”). There was an emphasis and discussion around \(\Delta t\). What were the two most substantive things you learned about \(\Delta t\) yesterday?





  2. For the following Lotka-Volterra equations for mutualism, please Please plot two cases of nullclines below: where mutualism is weak and where it is very strong. Include trajectory arrows, slopes, equilibira, and axis intercepts (in the positive plane): \[ \begin{align} \frac{\mathrm{d}N_1}{\mathrm{d}t} = r_1N_1\left(\frac{K_1 - N_1 + \alpha _{12}N_2}{K_1}\right)\\ \frac{\mathrm{d}N_2}{\mathrm{d}t} = r_2N_2\left(\frac{K_2 - N_2 + \alpha _{21}N_1}{K_2}\right) \end{align} \]















  1. Consider the slopes of each nullcline from the problem above. Think about the following: If we fixed the slope of the nullcline for \(N_2\) at 1 and the nullcline for \(N_1\) is \(>1\), will they cross? If you increase or decrease \(K_1\), will whether or not they cross be affected by \(K_1\)? If the slope of the nullcline for \(N_2\) is 2, then what would be the slope of the nullcline for \(N_1\) for the nullclines to cross (i.e., for the species to stably coexist)? What would \(\alpha _{12}\) be? What about if the slope of the nullcline for \(N_2\) is 3? What would \(\alpha _{12}\) be? Write the following: Using your reasoning, please write a general mathematical statement (a single inequality) that gives the conditions under which the nullclines cross; i.e., the coexistence criterion.






















  1. What is meant by facultative and obligate mutualism both ecologically and mathematically?






  2. Please draw two phase planes with nullclines for the cases where: (1) one species is a facultative mutualist and the other obligate, and (2) both species are obligate mutualists.