The situation
Begin with 90 rabbits and 12 foxes. Rabbits reproduce, foxes die without enough food, and encounters between them reduce rabbits while supporting fox growth. Use the same basic model as Module 4.2, with different parameters:
\[ \frac{dR}{dt}=aR-bRF, \qquad \frac{dF}{dt}=cRF-dF. \]
| Quantity | Base value | Meaning |
|---|---|---|
| \(R_0\) | 90 rabbits | initial prey population |
| \(F_0\) | 12 foxes | initial predator population |
| \(a\) | 0.8 per month | rabbit birth rate |
| \(b\) | 0.02 per fox per month | effect of foxes on rabbit losses |
| \(c\) | 0.005 per rabbit per month | effect of rabbits on fox growth |
| \(d\) | 0.6 per month | fox death rate |
| \(\Delta t\) | 0.02 month | simulation time step |
Simulate from 0 to 24 months. Then run a second scenario with 18 initial foxes, changing no other parameter. This is a comparison of model scenarios, not a recommendation to release animals in a real habitat.
What your report should show
1. Model and assumptions
Briefly explain the four flows and why the encounter term \(RF\) has opposite effects on rabbits and foxes. State two important simplifying assumptions.
2. Verification
Calculate the first update for both species by hand using the base case. Display the first two simulated rows and compare them with your calculation.
3. Base-case behavior
Show one clearly labeled graph of rabbits and foxes over time. In a few sentences, describe how the populations change and whether their peaks occur together. Report the largest rabbit population within the 24-month window and its time.
4. One experiment
Before rerunning the model with 18 initial foxes, predict whether the largest rabbit population in the 24-month window will be higher or lower than in the base case.
Run the second scenario with the same time step and parameters. Make a comparison graph with the two rabbit trajectories. Report each scenario’s largest rabbit population in the 24-month window and explain what changed. Because populations oscillate, also look at the curves rather than judging the entire outcome from one maximum.
You may copy and modify your simulation loop. A function is not required.
5. Conclusion
Answer the wildlife manager’s question in a short paragraph. State what the model predicts over two years and name one reason a real wildlife decision would need more evidence.
Evaluation
This project is graded holistically on a 10-point scale. I will look for a correct, readable model; a hand check that matches the code; clear graphs and comparisons; and an interpretation that respects the model’s limits.
Before submitting
- Render from a fresh R session and inspect the HTML output.
- Label axes with units and distinguish the graphed lines.
- Make sure you compare the same 24-month window in both scenarios.
- Submit both the
.qmdand.htmlfiles to Moodle.