(e) If \(x=0\), what happens to \(y\left(t\right)\)? How is this indicated in the phase-plane? If \(y=0\), what happens to \(x\left(t\right)\)? How is this indicated in the phase-plane?
Answer:
As \(x=0\), \(\frac{dy}{dt}\) increases, which means that \(y\) increases.
As \(y=0\), \(\frac{dx}{dt}\) increases, which means that \(x\) increases.
(g) For each of the initial conditions given below, describe how the number of species of A and B change with time and what the situations will look like in the long run.
i) \(x(0)=2\ and\ y(0)=1.8\)
Answer: Population of A will increase over time and the population of B will decrease since it is a competitive environment. In the long run, the population of B would tend to zero.
ii) \(x(0)=2\ and\ y(0)=2.3\)
Answer: Population of A will decrease over time and the population of B will increase since it is a competitive environment. In the long run, the population of A would tend to zero.
iii) \(x(0)=2.2\ and\ y(0)=2\ \)
Answer: Population of B will decrease over time and the population of A will increase since it is a competitive environment. In the long run, the population of B would tend to zero.