Activity 4.4.4

In this activity, we will consider several ways in which two species might interact with one another. Throughout, we will consider two species $R$ and $S$ whose populations in year $k$ form a vector $\xvec_k=\twovec{R_k}{S_k}$ and which evolve according to the rule $$\xvec_{k+1}=A\xvec_k$$.

1. Suppose that $A = \left[\begin{array}{rr} 0.7 & 0 \\ 0 & 1.6 \\ \end{array}\right]$.

   Explain why the species do not interact with one another. Which of the six types of dynamical systems do we have? What happens to both species after a long time?

2. Suppose now that $A = \left[\begin{array}{rr} 0.7 & 0.3 \\ 0 & 1.6 \\ \end{array}\right]$.

   Explain why $S$ is a beneficial species for $R$. Which of the six types of dynamical systems do we have? What happens to both species after a long time?

3. If $A = \left[\begin{array}{rr} 0.7 & 0.5 \\ -0.4 & 1.6 \\ \end{array}\right]$, explain why this describes a predator-prey system. Which of the species is the predator and which is the prey? Which of the six types of dynamical systems do we have? What happens to both species after a long time?

4. Suppose that $A = \left[\begin{array}{rr} 0.5 & 0.2 \\ -0.4 & 1.1 \\ \end{array}\right]$. Compare this predator-prey system to the one in the previous part. Which of the six types of dynamical systems do we have? What happens to both species after a long time?