Please place your name(s) in the upper-right corner.
On Monday you learned how to write a nested loop to vary
parameters across simulations. I’d like for you to write R code for `a
nested loop to determine the volume (in, say, units of cm) of a cuboid,
\(V = lwh\), varying the length 0.1–1,
width 10–20, and height 1–10.
Consider the case of commensalism where one species benefits and
the other isn’t effected: \[
\begin{align}
\frac{\mathrm{d}N_1}{\mathrm{d}t} = r_1N_1\left(\frac{K_1 - N_1 + \alpha
_{12}N_2}{K_1}\right)\\
\frac{\mathrm{d}N_2}{\mathrm{d}t} = r_2N_2\left(\frac{K_2 - N_2 +
0N_1}{K_2}\right).
\end{align}
\]
Please find the nullclines and plot them. What are the types of
dynamical behaviors we can expect from commensalism?
Your reading on competition showed that there were 4 different
types of dynamical behaviors: stable coexistence, species 1 outcompetes
species 2 (species 2 goes extinct), species 2 outcompetes species 1
(species 1 goes extinct), and unstable equilibrium where the species
that ultimately outcompetes the other is dependent on the initial
densities. Draw phase planes with the four types of dynamical behaviors
of competition. Label the \(x\) and
\(y\) intercepts of each nullcline and
try to see if you can come up with criteria (i.e., mathematical
inequalities) to determine the dynamical behaviors based on the
parameter values.
If you have time, do the same thing as question 2, but with amensalism where the reciprocal effects are negative and null.