If I'm on your qualifying exam committee, below are some topics we might cover. They are just kickoff points. This is not a comprehensive list.
1. Define density-dependent growth. How can it be identified from data?
2. Write an equation for a population with density-dependent growth. Analyze its stability analytically. Sketch a time series or describe how one could be derived.
3. What is chaos and where can we find it?
4. Define discrete-time and continuous-time models, autonomous and non-autonomous rates.
5. Why is diversity highest in the tropics?
6. What is demographic stochasticity and when is it important?
7. What makes communities stable?
1. What is a p-value? How should we interpret it? How is it commonly misinterpreted?
2. Define Bayes' theorem. Provide a quantitative example of Bayesian reasoning.
3. When should we trust a model? How should we decide between models?
4. What are credible intervals and confidence intervals, and how are they calculated?
5. How do we adjust for multiple hypothesis testing?
6. Can causality be inferred from observational data, and if so, how?