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

Suppose that \(A = \left[\begin{array}{rr} 0.7 \amp 0 \\ 0 \amp 1.6 \\ \end{array}\right] \text{.}\)
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?

Suppose now that \(A = \left[\begin{array}{rr} 0.7 \amp 0.3 \\ 0 \amp 1.6 \\ \end{array}\right] \text{.}\)
Explain why \(S\) is a beneficial species for \(R\text{.}\) Which of the six types of dynamical systems do we have? What happens to both species after a long time?

Suppose now that \(A = \left[\begin{array}{rr} 0.7 \amp 0.5 \\ 0.4 \amp 1.6 \\ \end{array}\right] \text{.}\)
Explain why this describes a predatorprey 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?

Suppose now that \(A = \left[\begin{array}{rr} 0.5 \amp 0.2 \\ 0.4 \amp 1.1 \\ \end{array}\right] \text{.}\)
Compare this predatorprey 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?