The Fox & the Rabbit

At an equilibrium point both populations are constant — neither grows nor shrinks. Translate this into mathematical conditions on the derivatives. Substitute the values of \(k, a, r, b\) to write two explicit equations.

Factor each of your two equations completely. Each equation produces two factors; setting each factor to zero gives a possible value of \(R\) or \(F\). 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 the rabbit population is exactly \(R_{\text{eq}}\) and the fox population exactly \(F_{\text{eq}}\) right now, what will happen to both populations next month? Next year? Write one sentence a park ranger with no mathematics background would understand.

For a single ODE \(\dfrac{dy}{dt} = f(t,y)\) with step size \(h\), the Euler update is \(y_{n+1} = y_n + h\cdot f(t_n, y_n)\). Our system has two rates of change. Define \(f(R,F)\) and \(g(R,F)\) explicitly using the given constants.

We now have two coupled updates per step. Write the complete recursion formulas for \(R_{n+1}\) and \(F_{n+1}\) in terms of \(R_n\), \(F_n\), \(h\), \(f\) and \(g\). Use step size \(h = 1\) month.

Use initial condition \(R_0 = R_{\text{eq}} + 100\), \(F_0 = F_{\text{eq}}\). Perform two steps of Euler's method by hand (\(h = 1\)). Show every calculation clearly in this table on your whiteboard.

Set up the two recursive sequences on your GDC using the formulas from Task 3b. Generate 140 points \((R_n, F_n)\). Verify that your first two steps match the hand calculations in Task 3c.

On your whiteboard, draw a phase portrait: axes \(R\) from 0 to 4500, \(F\) from 0 to 150. Plot one point every 20 steps, connect them in order, and add arrows to show the direction of time. Plot and label both equilibrium points from Phase 2.

On the same axes, generate and plot trajectories for the following cases. Use a different colour for each, add direction arrows, and label each starting point.

Case A — Double the foxes: start at \(\bigl(R_{\text{eq}},\; 2F_{\text{eq}}\bigr)\), run 160 months.

Case B — Many rabbits, few foxes: start at \((3000,\; 20)\), run 180 months.

What do all three trajectories have in common? What differs between them?

(i) What shape do all three trajectories form? What does this tell you about the long-term behaviour of the ecosystem?

(ii) Which trajectory has the longest period — the time for one complete loop? Which has the shortest?

(iii) A ranger reports current populations of \((2\,000\text{ rabbits},\; 80\text{ foxes})\). Use your phase portrait to predict: will rabbits increase or decrease over the next few months? Will foxes?

For Case B (start: \(3\,000\) rabbits, \(20\) foxes), trace one complete cycle and explain each stage:

(i) With abundant rabbits and very few foxes, what happens to the fox population first? Why?

(ii) As foxes increase, what happens to the rabbit population? Why?

(iii) As rabbits decline, what then happens to the foxes? Why?

Explain each quarter of the loop in ecological terms.

Why is a phase portrait more useful than two separate graphs \(R(t)\) and \(F(t)\) plotted against time? What information does the phase portrait reveal that the time-graphs hide? Give a specific example using your trajectories.

The Lotka-Volterra model predicts perfect closed loops that repeat forever. Real census data shows loops that almost repeat but drift slightly over time.

(i) List at least three real-world factors that the model ignores.

(ii) How does Euler's method introduce its own error, separate from the model's simplifications? How could you reduce it?

(iii) What would happen to the equilibrium point \(\bigl(R_{\text{eq}},\, F_{\text{eq}}\bigr)\) if the park introduced hunting — reducing \(k\) (the rabbit birth rate)?

Apply one step of Euler's method with \(h = 1\) starting exactly at the non-trivial equilibrium \(\bigl(R_{\text{eq}},\, F_{\text{eq}}\bigr)\). What do you get? What does this confirm about the equilibrium point? Can you estimate the period of the cycle — the number of months for one full loop — from your phase portrait?