find the two bifurcation points. If we find these values as \(p\) and \(q\text{,}\) where \(p < q\text{,}\) discuss the equilibria for \(b < p, b = p, p < b < q, b = q\text{,}\) and \(b > q.\) By “discuss” here, we mean how many equilibria are there for given values of \(b\text{.}\)
Sketch a phase line for \(a = 0\text{.}\) What are the equilibrium points here?
Sketch phase lines for \(a = 12, 10, 9, 8, 5, 0, -7, -27, -40, -55\text{,}\) and \(-72\text{.}\) It might be best to use technology, like Desmos, to help with this. What is the bifurcation point here?
Generate a bifurcation diagram: use \(a\) as the horizontal axis value, and \(y\) the vertical. For the given values of \(a\) in part (b) above, draw phase lines, with dots at the \(y\)-values that give equilibria over each different \(a\) value.
4.
Use the differential equation above and assume that \(y(t)\) is the number of velociraptors in my theme park (in tens) during month \(t\text{.}\) Also assume \(a\) is the number of velociraptors I harvest each month (so negative values indicate that I am purchasing more velociraptors to add to my herd).
Assume I start with \(40\) velociraptors, and do no harvesting. Do I expect my herd to grow, remain stable, or vanish over time?
Answer the same question above, but starting with \(0, 50, 75, 100\text{,}\) and\(120\) velociraptors.
Answer the same questions above, but now I harvest \(50\) velociraptors each month.
What is the maximum number of velociraptors I can harvest each month, if I want to continue harvesting that amount indefinitely? (Hint: this question is about the bifurcation point). How does the initial number of velociraptors I have enter into this?
