You are an ecologist. Years of monthly census data reveal rabbit and fox populations rising and falling in linked cycles. A colleague proposes the Lotka-Volterra model:
\(k = 0.07\) · \(a = 0.002\) · \(r = 0.03\) · \(b = 0.00004\)
Work on your whiteboard. Your teacher gives codes to unlock each phase.
Set \(F = 0\) in the rabbit equation. What type of ODE do you get? Separate variables and solve to find \(R(t)\), given \(R(0) = R_0\).
Set \(R = 0\) in the fox equation. Solve to find \(F(t)\), given \(F(0) = F_0\). What happens to the fox population as \(t \to \infty\)? Why does this make biological sense?
The interaction terms are \(-aRF\) and \(+bRF\). Explain in your own words why the interaction depends on the product \(RF\) rather than on \(R\) or \(F\) alone. Why is \(b \ll a\,\)?
When both species are present, can you separate variables and solve the rabbit equation independently? Explain carefully why or why not. What makes this system coupled?
At an equilibrium point both populations are constant. Translate this into mathematical conditions on the derivatives. Substitute \(k, a, r, b\) to write two explicit equations.
Factor each equation completely. Each equation produces two factors — find all combinations, i.e. all equilibrium points \((R, F)\).
For each point: state the coordinates, describe what it represents ecologically, and decide whether it is biologically meaningful. Which equilibrium is more interesting for the park, and why?
Call the ecologically meaningful equilibrium \(\bigl(R_{\text{eq}},\, F_{\text{eq}}\bigr)\). If both populations are exactly at these values right now, what will happen next month? Next year? Write one sentence a park ranger with no mathematics background would understand.
For \(\dfrac{dy}{dt} = f(t,y)\) with step size \(h\), the update is \(y_{n+1} = y_n + h\cdot f(t_n, y_n)\). Define \(f(R,F)\) and \(g(R,F)\) explicitly. Then write the two recursion formulas for \(R_{n+1}\) and \(F_{n+1}\) with \(h = 1\).
Use \(R_0 = R_{\text{eq}} + 100\), \(F_0 = F_{\text{eq}}\). Perform two Euler steps (\(h = 1\)). Complete the table on your whiteboard.
| \(n\) | \(R_n\) | \(F_n\) | \(\dfrac{dR}{dt}\) | \(\dfrac{dF}{dt}\) | \(R_{n+1}\) | \(F_{n+1}\) |
|---|---|---|---|---|---|---|
| 0 | ||||||
| 1 |
Set up two recursive sequences on your GDC using the formulas from 3b. Axes: \(R\) from 0 to 4500, \(F\) from 0 to 150. Plot one point every 20 steps, connect in order with direction arrows. Plot and label both equilibrium points.
Task 3c (already done): \(R_0 = R_{\text{eq}}+100\), \(F_0 = F_{\text{eq}}\) · 140 months · Case A: \(R_0 = R_{\text{eq}}\), \(F_0 = 2F_{\text{eq}}\) · 160 months · Case B: \(R_0 = 3000\), \(F_0 = 20\) · 180 months
Use a different colour for each trajectory. What do all three have in common? What differs?
(i) What shape do all three trajectories form? What does this say about long-term behaviour?
(ii) Which trajectory has the longest period? Shortest?
(iii) Populations reported as \((2\,000,\; 80)\) — will rabbits increase or decrease next month?
(iv) For Case B, trace one full cycle and explain each quarter ecologically.
(i) Why is a phase portrait more useful than separate graphs of \(R(t)\) and \(F(t)\)?
(ii) List three real-world factors the model ignores.
(iii) How does Euler's method introduce error? How could you reduce it?
(iv) What changes to \(\bigl(R_{\text{eq}},\, F_{\text{eq}}\bigr)\) if \(k\) decreases due to hunting?
(v) Apply one Euler step starting exactly at \(\bigl(R_{\text{eq}},\, F_{\text{eq}}\bigr)\). What do you get? Estimate the cycle period from your portrait.