Recently Jeremy made the point that we can’t expect ecology grad students to learn everything useful under the sun and asked in a poll what people would prioritize and toss. More math skills was a common answer of what should be prioritized.

As somebody who has my undergraduate (bachelor’s) degree in mathematics I often get asked by earnest graduate students what math courses they should take if they want to add to their math skills. My usual answer is nothing – the way math departments teach math is very inefficient for ecologists, you should teach yourself. But its not a great answer.

In a typical math department in the US, the following sequence is the norm as one seeks to add math skills (each line is a 1 semester course taken roughly in the sequence shown)

- Calculus 1 – Infinite series, limits and derivatives
- Calculus 2 – Integrals
- Calculus 3 – Multivariate calculus (partial derivatives, multivariate integrals, Green’s theorem, etc)
- Linear algebra – solving systems of linear equations, determinants, eigenvectors
- Differential equations – solving systems of linear differential equations, solving engineering equations (y”+cy=0)
- Dynamical systems – y
_{t+1}=f(y_{t}) variations including chaos - Probability theory (usually using measure theory)
- Stochastic processes
- Operations research (starting with linear programming)

That’s 7 courses over and above 1st year calculus to get to all the material that I think a well-trained mathematical ecologist needs! There are some obvious problems with this. First few ecologists are willing to take that many classes. But even if they were, this is an extraordinary waste of time since over half of what is taught in those classes is pretty much useless in ecology even if you’re pursuing deep into theory. For example – path and surface integrals and Green’s theorem is completely irrelevant. Solving systems of linear equations is useless. Thereby making determinants more or less useless. Differential equations as taught – useless (to ecologists very useful to physicists and engineers). Measure-based probability theory – useless. Linear programming – almost useless.

Here’s my list of topics that a very well-trained mathematical ecologist would need (beyond a 1st year calculus sequence):

- Multivariate calculus simplified (partial derivatives, volume integrals)
- Matrix algebra and eigenvectors
- Dynamical systems (equilibrium analysis, cycling and chaos)
- Basic probability theory and stochastic processes (especially Markov chains with brief coverage of branching processes and master equations)
- Optimization theory focusing on simple calculus based optimization and Lagrange multipliers (and numerical optimization) with brief coverage of dynamic programming and game theory

Now how should that be covered? I can see a lot of ways. I could see all of that material covered in a 3 semester sequence #1/#2, #3, #4/#5 if you want to teach it as a formal set of math courses. And here is an interesting question. We ecologists often refuse to let the stats department teach stats to our students (undergrad or grad) because we consider it an important enough topic we want our spin on it. Why don’t have the same feelings about math? Yet as my two lists show math departments are clearly focused on somebody other than ecologists (mostly I think they’re focused on other mathematicians in upper level courses). So should ecology department start listing a few semesters of ecology-oriented math on their courses?

But I could see less rigorous, more integrative ways to teach the material as well. For example, I think in a year long community ecology class you could slip in all the concepts. Dynamical systems (and partial derivatives) with logistic/ricker models and then Lotka-Volterra. Eigenvectors and Markov Chain’s with Horn’s succession models or on age-stage structure, then eigenvectors returning as a Jacobian on predtor-prey. Master equations on Neutral Theory. Optimizaiton on optimal foraging and game theory Yes the coverage would be much less deep than a 3 semester sequence of math only courses, but it would, I think, be highly successful.

I say “I think” because, I don’t know anywhere that teaches the math this way. I teach a one semester community ecology grad class and try to get a subset of the concepts across, but certainly don’t come anywhere close covering everything that I wish were covered (i.e. my list above). And I know a lot of places have a one-semester modelling course for grad students. But teaching their own math courses, or teaching a math-intensive ecology sequence I haven’t come across.

What do you think? Have I listed too much math? or left your favorite topic out? How should this be taught? How many of our students (undergrads, just all grads, only a subset of interested grads) should this be taught to?.