What math should ecologists teach

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)

  1. Calculus 1 – Infinite series, limits and derivatives
  2. Calculus 2 – Integrals
  3. Calculus 3 – Multivariate calculus (partial derivatives, multivariate integrals, Green’s theorem, etc)
  4. Linear algebra – solving systems of linear equations, determinants, eigenvectors
  5. Differential equations – solving systems of linear differential equations, solving engineering equations (y”+cy=0)
  6. Dynamical systems – yt+1=f(yt) variations including chaos
  7. Probability theory (usually using measure theory)
  8. Stochastic processes
  9. 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):

  1. Multivariate calculus simplified (partial derivatives, volume integrals)
  2. Matrix algebra and eigenvectors
  3. Dynamical systems (equilibrium analysis, cycling and chaos)
  4. Basic probability theory and stochastic processes (especially Markov chains with brief coverage of branching processes and master equations)
  5. 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?.

44 thoughts on “What math should ecologists teach

  1. I am an ecology grad student currently taking Linear Algebra. I find linear algebra tremendously useful for thinking about modeling, understanding multiple regression from first principles, and its utility for analytically solving simulation models to understand their core structure. While mathematica or other tools could perhaps do this for me, I still find the course far from “useless”.

    • Thanks Colin – I agree there is a whole different set of criteria of math to teach to understand statistics. I still dare say you could get a small fraction of what they cover in linear algebra to understand regression. I don’t think Cramer’s rule is ever going to do you much good for example.

  2. It’s interesting that you’ve almost described the math program for my micro-economics undergrad, although they focus less on dynamics and much more on optimization (I think macro programs typically involve a bit more dynamics and stochastic processes, although not nearly as much dynamics as a population/community ecologist would use).

    The math was generally integrated into the classes I took; You’d learn optimization theory and Lagrange multipliers as you were covering the theory of demand and supply curves. I think that was the most useful way of learning it; You’d pick up the math needed at the same time as you were learning why you needed it, as you were talking about with your year long community ecology class.

    • I do think the overlap in math requirements with economics is high (an with physics and engineering low). And it is useful but not surprising to hear that they teach it better in econ (something for ecologists to model ourselves after).

  3. Hi Brian,
    Your sequence seems rather logical. I am strongly biased though, as we’ve recently argued for that exact sort of training ( https://peerj.com/articles/285/ ). I would just add that
    1) The math programme for economists looks often like your wishlist (provided it looks alike in the US and in Europe; some biology sections get similar math to economics – at least mine did)
    So if there’s an economics department, courses there might be an option.
    2) Some knowledge you discard can speed up learning of more advanced topics: dynamical systems are easier when you know differential equations beforehand, and systems of equations through matrices can help to understand eigenvectors and eigenvalues.
    3) The measure-theoretic presentation of probability looks the most “dangerous” to me given it transforms rather applied math into seriously abstract ones. That could put off ecologists (except the most math-oriented) from a topic central in both statistics and theory. Perhaps physicists or chemists get other probability classes with a more applied bent? (I would assume measure theory is mostly for math majors?)
    Otherwise, the suggestions of Ellison and Dennis could be practical http://www.esajournals.org/doi/abs/10.1890/080209 [Panel 2]

    • Thanks for the links! Completely agree on the similarity to economics. I didn’t get into a lot of detail on the differential equations, but I’m not sure a lot of time spent learning approximation methods for static solutions to 2nd order differential equations and inhomgenous equations etc. is of much use in ecology. I personally think diving right into a dynamical systems view is much more intuitive for ecologists. Yes measure theory is only useful to math majors. Some campuses the math department teaches an “applied probability” course that doesn’t use measure theory, but a lot of campuses I’ve been, the only probability course is measure theoretic.

      • Wow! There are campuses who don’t offer an undergraduate calculus based probability class? I’d probably argue that if a theoretical ecologist could only take one math class after calculus that would be the one. It is the foundation of statistics, stochastic population models, game theory etc. It’s hard to get anywhere without understanding what a pdf or random variable is. Dynamical systems (sometimes called nonlinear dynamics) would be a close second. With one course in dynamical systems and calculus based probability 80% of ecological theory opens up to you (sometimes with a little bit of work, but a good understanding of those two courses goes a very long way!)

  4. Thank you for articulating so well what I’ve been saying since coming to the USA in 2003. In my Canadian UG I took 2 courses in calculus and 2 in Linear Algebra (including eigenvalues and eigenvectors). The rest of your list looks like the courses I wish I’d taken. When I first started looking into how math was taught here I was baffled as to why Eigenanalysis was buried so deep.

    The idea of integrating more math into the courses where it’s needed is worth a longer look. I already incorporate a bit of Eigenanalysis into my population dynamics class because it’s so necessary for structured population models. I could do a better job with dynamics and Eigenanalysis.

    I wonder if Economics/engineering/physics can do a better job of integration because the students pursuing those fields are already “math willing”. Seems like many that start Biology/Ecology do so in part because they are science willing but math phobic.

    • Thanks Drew. Is 2 semesters of linear algebra standard for science majors in Canada? I can’t remember what McGill required when I was there. Anyway, I don’t’ doubt the US is near the bottom of undergraduate math preparation. And I am very curious to hear how undergrad prep varies in different countries.

      There is no doubt a wide gap in “math willingness”. I don’t have great answers. Seems to me the choice is either to fork and provide a math-intensive branch for math willing people. Or to dilute it a bit and teach it in an integrative context mixed in with the ecology so the motivations are obvious. I actually think most graduate students in ecology (undergrads are completely different story) but graduate students are math willing – just math scared. Going slow enough and keeping it well motivated in the ecology can work (and be valued by the students).

      • I agree with the UG vs. GS distinction; I was thinking in terms of UG students. The trouble with providing a math intensive branch for undergrads is that we don’t have enough faculty to specialize like that, at least at UNL. Or enough students.

        The requirement for Honours Zoology at U of Alberta was 2 semesters of math and a semester of statistics or computing; I did 4 of math and 2 of statistics. I guess I sensed it was important but didn’t really have any guidance in what to take.

        I think it has to be built into the entire curriculum, as you say. If not, then most students have forgotten even the little bit they knew when they reach a senior class like mine that suddenly throws derivatives and eigenvalues at them.

      • I’d forgotten about that post. Hard to tell how self-selected your sample is, but one theme that emerges for me is basically annoyance that “you make us take a class where there is no effort to link it to ecology (1st year calculus) and then you mostly never use math again” which suggest an integrative approach in undergraduate ecology could go down well.

        I’m not entirely sure about this though. My experience teaching a required 300 level ecology class (and TAing another in grad school) is that you have to really hammer home the math or else students think they can get away with just ignoring it, and that when you do spend lots of time on it, many students struggle and find it uninspiring (again at the undergrad level).

        My university has a large group of faculty studying science education at the university level but there is not a lot of focus on upper level ecology pedagogy (just as there is not a lot of history or philosophy of science specific to ecology), unlike say physics pedagogy which is studied at all grade levels. Apparently we’re not an appealing field to outsiders! But wouldn’t it be great if these folks were empirically testing this stuff?

  5. I’m a first-year graduate student with an interest in mathematical ecology, and the last option you describe (learning the necessary math along the way in biology-oriented courses, rather than taking classes in the math department) matches pretty well what my training was like in undergrad. As an undergrad at Princeton I didn’t take any pure math classes beyond first-year calculus, but I managed to pick up most of the math concepts you list by taking Theoretical Ecology in the EEB department and a class called Introduction to Biological Dynamics in the Molecular Biology department – if you’re wondering what it would actually look like if someone taught a course designed to be “math for biologists,” I think that one’s a good example, although it was not specifically geared toward ecologists (http://tglab.princeton.edu/mol510/). That approach was certainly more efficient than taking all of the relevant courses through the math department, and I’m much more motivated to learn the math if I can see its direct utility for studying ecology – though sometimes I wonder if I should have actually taken linear algebra so my knowledge of it didn’t feel so fragmented.

  6. Hi Brian,

    Great post, I really love your list (yay for master equation and more stuff stochastic processes). I also agree on your points about a lot of this being material we could integrate into the curriculum — indeed I wasn’t aware that these mathematical topics tended to be more ‘out-sourced’ than statistics. I’d particularly like to see statistics integrated into a course on stochastic processes; I’ve never understood why these things are taught as if they were separate worlds. Ideally using examples that can be done analytically at first but also numeric ones.

    The only part on your list that I’m not convinced by really is all the calculus; I bet you could drop drop them entirely and teach what you need as you go; it seems that students have often forgot that by the time they get to the relevant courses and need it re-taught anyhow. (I don’t think knowing a bunch of trig-integral identities brings someone any closer to understanding the definition of ‘expectation’, while learning discrete probability sums first probably gives you a better understanding of what integration is about anyway).

    I actually I think I’ve seen a fair share of the material taught exactly as you describe. Not sure if you’re familiar with Alan Hastings’ book, ‘population biology: concepts and models’, but he has long used it to teach dynamical systems (focusing just as you suggest on equilibrium analysis, oscillations, and chaos) & matrix algebra to first year grad students who have long forgotten their calculus. I also really like Nisbet & Gurney’s “modeling fluctuating populations” (didn’t like Russ Lande’s ‘Stochastic population dynamics’ as much); also love Bartlett’s tiny classic: ‘stochastic population models.’ Gillespie’s little pop gen course is also a pretty nice example of integrating teaching of stochastic mathematics in a clear ecological/evolutionary context; & Ewens book (math pop gen) goes a bit deeper but I never had a course out of it. There’s also a new calculus book by some eminent theoretical ecologists that might be worth a look http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP002970.html

    Anyway, great post and I really like your proposed curriculum.

    • Glad you like the list. I in general agree about keeping calculus limited. I chopped the 3rd semester to about 1/3 its usual size you could probably go further. If one is willing to go down that road and reclaim 1st year calc from mathematicians (which I think is a great idea if resources allow), then I agree. Epsilon-delta proofs. Much of the 2nd semester on tricks for integrals. Some of the more theoretical stuff like the mean value theorem and detailed proofs of the fundamental theorem of calculus (integral=antiderivative). These could all go away in applied course. I liked the table of contents in the book you linked to.

      PS we share a similar taste in modelling books!

  7. Nowadays purely mathematical models are just that – theoretical exercises. Most won’t pass peer-review. Peer-reviewers would rightly demand links to actual data and biologically realistic constraints on model parameters.
    So while I overall agree with your sequence of courses, adding to it data science, time series analysis and Bayesian statistics is essential. Future math ecologists got to be skillful at linking their models with actual data.
    Some would argue that such courses are not really math courses, which is a fare point, but they are close enough to be considered as an integral part of math ecologist “math curiculum.”

    • “Most won’t pass peer-review.”

      Um, you’re aware that there are entire journals devoted to theoretical modeling, right? Theory has a legitimate life of its own, independent of data. See Caswell 1988 Ecologial Modelling. Seriously, you really should click through and read it, it’s a classic, and I think you would find it very thought-provoking. It’s addressed to readers who hold precisely the views you seem to.

      As for the “rightful” demand that models have “realistic” parameters, sorry, but R. A. Fisher would like a word with you: 🙂


      With respect, I think you have an overly narrow view of the ways in which mathematical models can be used to help us learn about the world. The use you seem to have in mind–fitting models to data–is undoubtedly important. Indeed, I’ve extolled its importance myself here, here, and here. But it is far from the only way in which mathematical models are helpful. Here are a couple of posts discussing this point:


      • >>”you’re aware that there are entire journals devoted to theoretical modeling, right?”
        My PhD advisor (Yang Kuang, the author of classic Delay Differential Equations: Applications in Population Dynamics) and my postdoc advisor (Simon Levin) are either editors and/or published in the very journals you refer to (e.g. JTB, MBE, BMB, AmNat and Ecol Mod). Actually, BMB is the first journal I published in, which at the time was very heavy on theory.

        >>”I think you have an overly narrow view of the ways in which mathematical models can be used to help us learn about the world.”
        Why pigeonholing here?
        Unfortunately, mathematically correct but biologically flawed models still get published in the field. Such models can be very thoroughly analyzed (equilibria, stability, limit cycles, bifurcation diagrams, dissipativity). Even when the analysis is impeccable, what is the value of the resulting elaborate mathematical structure if its biological foundation is flawed?

        Furthermore, cheap computational power allows one to easily run models that are poorly constrained biologically. It is easy to generate seemingly plausible output.
        Such abuses of mathematical modeling in ecology only contribute to the irreproducibility crisis in science.

        Fortunately, journals are starting to catch up on this and demand biological validity and firmer links to actual ecological data (at least, that’s the impression I get from peer-reviewing for the journals mentioned above).

        Hence, in my view, a well rounded education in mathematical ecology should include the foundational courses Brian was referring to but also enrich students with abilities to extract, work with and analyze vast data that are increasingly available.

      • I’m not pigeonholing you, I’m taking what you wrote at face value. If I’ve misconstrued your brief comments, then by all means elaborate. But you simply said, flat out, that any model that’s not “linked to actual data” with “biologically realistic constraints on model parameters” is a “pure mathematical exercise” that “rightly” “won’t pass peer review”. Those are very strong claims with which I completely disagree, for reasons I explained. If those aren’t the claims you intended to make, I hope you’ll clarify.

        It would be helpful if you could provide examples of what you mean by the sort of purely theoretical models that you think are biologically flawed. Give citations, name names. For instance, is Fisher’s analysis of why there are only two sexes an example of the sort of purely theoretical model you’re thinking of? The May-Wigner theorem relating stability and complexity? The “0-1” diet rule from optimal foraging? The ideal free distribution? Levins’ original metapopulation model? The Lotka-Volterra equations? The Rosenzweig-MacArthur model? Neutral theory? Stoichiometric consumer-resource models that assume plants or algae with fixed rather than flexible stoichiometry for the sake of simplicity (e.g., my own work with Dave Vasseur)? Ratio-dependent predator-prey models? The work of Bob Holt and others on “community modules”? Log-linear autogressive models as a phenomenological description of time series data? “General Ecosystem Models” of the sort Drew Purves is trying to develop and parameterize? Chessonian coexistence theory, for its unrealistic assumption of stationarity? Something else? My point in giving this tediously long list is to emphasize that there are all sorts of theories and models out there that are “unrealistic” in all sorts of *quite distinct* ways but that nevertheless have proven useful for all sorts of *quite distinct* reasons. Further, none of the models I listed has been fit to data as far as I know, with a few exceptions (e.g., neutral theory, ratio dependent predator-prey models) in which those fitting exercises often are quite uninformative for various reasons. So I’m sorry, but if you criticize biologically unrealistic models that can’t be fit to data without elaborating or giving any examples, then I can only assume you meant to criticize every single model I listed and many others I didn’t. Seriously, I’m not trying to be deliberately dense here–I honestly have no idea what sort of work you mean to criticize.

        Let me also say that I’m sure there are flawed theoretical models that serve no useful purpose, theoretical or otherwise. I am not arguing that there is no bad theory out there. But it’s up to you to clarify which sorts of work you intend to criticize.

      • I do think Irakli staked out an extreme position and in some ways manifestly false (pure models do pass peer review). But as a prescription, models ultimately do need some smash against reality (even the “toy” or strategic models like May advocated). Lotka-Volterra would be nowhere near as influential as they are if it hadn’t been for Gause. If they never smash against reality, then I would have to agree they’re not advancing science. Now whether that has to happen in the scope of a single paper or single researcher or over the community and body of research is a whole other question. And whether it OUGHT to happen in a single paper is a whole other question again.

      • @ Brian:

        Ok, as long as we define “some smash against reality” sufficiently broadly. For instance, the May-Wigner theorem undermines some vague pre-theoretical intuitions about stability and complexity. Sharpening and improving our pre-theoretical intuitions definitely advances empirical science, even though it doesn’t involve any comparisons with data. Another example are the extremely simple toy models I discuss in my IDH paper. No comparisons with data involved–the function of the models is to identify faulty intuitions and faulty logic, thereby allowing us to rule out some hypotheses a priori. So “some smash against reality” needs to be defined as “any way in which models help us better understand reality, including many ways having nothing to do with comparisons to data or ‘realistic’ assumptions.” See that old post on Wimsatt’s work for an attempt at a comprehensive list of the ways in which models can help us learn about reality.

        And even here, I think you’re still missing the fact that pure theory has a legitimate life of its own. For instance, Caswell 1988 gives the example of Joel Cohen’s cascade model as an answer to a purely theoretical question: What’s the simplest possible model capable of reproducing certain features of food web structure? That question is of theoretical interest, even if the answer has nothing to do with the processes that actually generate food web structure. Now, if someone wants to say that they personally don’t care about theoretical questions, that’s totally fine. Just as someone might not care about atomic physics or sociology or Finnish literature. But in my admittedly-anecdotal experience, when an ecologist dismisses something as “pure theory”, they don’t usually mean “I personally don’t care about that, but others do and that’s totally fine.” What they usually mean is something more like, “That topic has no value, period.”

        Now, I actually do think entire fields of study can go off the rails and become, if not totally valueless, then of little value. Deconstruction in literary theory might be an example (see, e.g., http://www.newcriterion.com/articles.cfm/Quisling-criticism–the-case-of-Paul-de-Man-5464). But in general, I think that’s a hard case to make, and we shouldn’t be too quick to make it.

      • I think reality intrudes more than you give it credit! In the case of May’s stability work, he says effectively “this works shows there is no generic complexity–>stability relationship, if they exist in biology it is because of special properties of biological systems”. Which I think was a very fruitful framing, and indeed all sorts of work measuring special properties of ecological systems that increase stability ensued. Probably most strongly Tilman’s work that empirically found stability increasing with richness and some nice papers showing why that is. I’ve got a paper myself using the breeding bird survey and measuring the balance of population level forces, complementarity, etc.

        Similarly – would Cohen’s cascade model ever have existed (or been valued) if it didn’t explain empirically measured properties found to be common across empirical networks? To me that is just showing model can come after data as well as before.

        Parameterizing a model may not always be necessary, but I can’t think of many models I am interested that don’t connect to empirical data at least qualitativley through testing qualitative predictions, post hoc explanations, challenging ideas induced from data, etc.

      • @Brian:

        “I can’t think of many models I am interested that don’t connect to empirical data at least qualitativley through testing qualitative predictions, post hoc explanations, challenging ideas induced from data, etc.”

        As long as you’re defining “connecting to empirical data” in that very broad way, I’m fine with that. Indeed I think it’s just a different way of phrasing my point.

      • Jeremy here took the stance of a defender of classical models against those unwashed masses that are unaware “that there are entire journals devoted to theoretical modeling,” don’t know Caswell’s work, and have “overly narrow view” of models.
        It’s a sympathetic stance to take for sure, but a misplaced one. Needless to say I know Hal (Caswell) and have a lot of respect for his work. My formal graduate education consists almost entirely of mathematics and theoretical ecology. So you are really preaching to the choir here with your pro-model and pro-theory arguments. When data are not available, then a model and even a pure “thought experiment” is the way to go. For example, I explicitly used stoichiometric “thought experiments” in my 2002 Trends Ecol Evol and 2014 eLife papers.

        However, did you notice that my initial post started with “Nowadays…”? Indeed, nowadays data are increasingly more abundant and easily accessible. So it becomes increasingly difficult to justify publishing purely theoretical models, especially when data for validating the models are available. As a referee for AmNat, BMB, EcoMod, JBD, JTB, MBE and TPB, I see more and more that editors expect firm links to actual ecological data.
        For example, an AmNat editor recently rejected a paper I was refereeing largely because it was a purely theoretical model and no attempt was made to link it to actual ecological data.
        All this brings me back to the original point I was making: it’s imperative to include data science and statistics into the standard curriculum of mathematical ecology.

      • @ Irakli:

        Thanks for the clarification, now I understand where you’re coming from.

        As a description of where the field is going–models linked to data are increasingly valued, models not linked to data are decreasingly valued–I think you’re probably right. Presumably that’s one reason why theoreticians increasingly seem to be abandoning general ecology journals (with the exception of Am Nat) for theory journals.

        And yes, one reason for this trend is probably the increasing availability of data, though I suspect it’s also down to increasing availability of powerful model-fitting software, and perhaps for other reasons too. And yes, if that’s the way the field is going, then that’s a strong argument for changing how we train our students.

        I would merely add that I think there’s a downside to these mostly-welcome trends. They may be causing ecologists to forget the value of pure theory in its own right, and to lose track of the full range of ways in which mathematical models help us understand the world. Insofar as ecologists feel that theory is only something you do when you don’t have data and WinBUGS, well, I think that’s a mistake.

      • This discussion raises some very important points. I think we can all agree with Irakil that we’ve seen modeling papers that add arbitrary complexity without much biological motivation, while also acknowledging that papers advancing pure theory without explicit empirical data is still alive and well today, right?

        While I think it has been great to see data and model fitting playing a more active role in theory, I think it also raises much more potential for concern. Is there a risk that just demanding some “link to data”, such as fitting a model to the data, could be more misleading still? I’m not sure that routine training in statistical methodology is sufficient to avoid this trap, while purely theoretical approaches can often shed more light on these issues (http://doi.org/10.1111/2041-210X.12285 is one good recent example)

  8. Nice list, Brian! I’d add some game theory, spatial models and non-equilibrium dynamics to the list for the future mathematical ecologist.

    But there will never be too many mathematical ecologists in training in any one place, so how would you justify these topics for so few? Maybe intensive summer schools like our ELME summer programs at KBS when we got junior theoreticians from around the world together for 3-week advanced courses on adaptive dynamics (2011) and metacommunities (2013).

    What about the future empirical ecologist — what math training is essential for them (by far the majority)?

    • All good points and questions (and good reminder about ELME).

      To your last question, I personally think the list is the same just a different level of detail/depth. All ecologists need to be conversant with eignevectors, equilibrium, optimization even if they can’t sit down and calculate the derivatives to make it happen on a new equation.

  9. I’m surprised no one has mentioned how the physicists handle this. “Less rigorous, more intergrative” is exactly the stereotype of physicist math from a mathematician’s perspective. For full disclosure, I’m a physics PhD who now works in epidemiology.

    For physics, most of the math (other than calculus) is introduced as it is needed, with the spin that it is needed. This works because the math is valued by the students and the teachers, and so realistic time is allotted for it. Sometimes it interrupts the physics lesson proper for maybe 20 minutes (the first time we see how to solve something like dx/dt = -x). Sometimes it takes a month or more (partial differential equations for electricity and magnetism).

    For undergrad, the only math department courses required by every US program cover calculus (which makes sense for physics). Some departments also require linear algebra and diff eq, but many don’t. There is often a 1 semester mathematical physics course (“math for physicists”, both at the undergrad and grad level), but these often don’t work very well because they are too divorced from application to be interesting but too shallow to be useful as math classes. When the material comes up in context during other courses, it often has to be retaught.

    You all know the cultural issues in ecology that’ll make this harder than it is in physics, but I think that field provides an excellent model for how to teach math when the math is not the point.

    • Interesting (and parallels what I saw in my college roommates who majored in physics). Across yours and other comments, one might induce the general principle that fields which are serious about math (physics, economics) do teach the math themselves (don’t just tell their students to go take more math courses) but teach it in an integrative fashion.

      • The other thing that would be good about about an integrated approach outside of math would be an Ecology program could integrate numerical and simulation techniques with analytical lessons as a cohesive unit, which I think would really accelerate understanding of complex ideas like population vs. individual dynamics. Especially if it was worked into class time instead of just through assignments.

  10. Also, the best book I’ve ever read on stochastic processes is Phil Nelson’s Biological Physics textbook. More lucid and interesting than anything I’ve seen from math, physics, applied math… and I use probability and stochastic processes almost every day. I don’t have it in front of me, but I bet it’s pretty good on dynamical systems too.

  11. Great blog as always and many great comments. Strongly support more math for all scientists.

    But before you go kicking to much “useless” math out of the curriculum, don’t forget:

    1. math is a tool for thinking about relationships. Its not just for number crunching. More ability to think about relationships is always better.

    2. You’re not just training biologists. You’re educating scientists. Its an interdisciplinary world. Breadth matters.

    I hated math. But its all turned out to be very useful, both in science and in life.


  12. Really nice post. As a CS undergrad with a penchant for math, I took about two-thirds your list of math classes (and some in high school: #1,2,6) along with discrete mathematics. Loved it all and wish I could’ve taken more. But what I really like is your distilling of the essentials into the 3-course sequence. I think understanding the concepts behind those math disciplines is the important part for ecology theory, and I have never felt very satisfied by learning math within a biology context. (Or computer science context, for that matter.) For example, I don’t like learning just a touch of differential equations in an ecology modeling course, because I don’t know what I’m missing. And I rather feel that the math is oversimplified when it’s taught “on the side”.

    BTW, I’m interested that statistics isn’t included in your list as “math”. At my (small-ish) undergrad, stats was included in the math department. At my (large) grad school, stats was its own department. How many additional semester courses would be necessary to teach a good foundation of ecology-useful stats? 1? 2? 3?

    • There is probably a whole other post on stats. I would argue that the level stats are practiced in research ecology requires at least two semesters. I teach a 2nd semester stats class and always tell students they’ll need a 3rd semester when they figure out exactly what they want to do, but most of them don’t take a 3rd semester – they self teach at that point.

  13. Thanks Brian!

    I was glad to see that this list matched well with the list I was given when I started my MS.

    I was never a math oriented person, but I get frustrated when I don’t understand modelling papers. Hence, I decided to increase my math literacy during my MS and PhD. I have found that auditing math courses is a good way to get a good understanding while glossing over the details that are clearly irrelevant to ecology and evolution. The potential pitfall is that auditing requires self discipline to do the homework even though you aren’t required to. But, if you attempt the homework (but don’t spend a ton of time on it as you would if you were taking the course for credit) you can get a lot out of auditing a course.

    Your post makes me wonder if it’s time for an Ecological Mathematics text book. Anyone interested in writing one? Or, maybe the textbooks described in other comments are enough.

    • Its in the vein of teaching the math using ecological motivation but not integrative (i.e the math dominates the ecology), but the Otto & Day book “A Biologist’s guide to mathematical ecology” is excellent (and covers material not that different from my list. Stephen Ellner’s new book “Dynamic models in biology” is not specific to ecology but also very good.

  14. I took most of the courses you listed and a few more in stats, but I think my lower level discrete math course ended up being the most fun and practical. I’m not sure how typical my course was, but it turned out to be very applicable to the work I ended up doing in ecology – basic probability, recurrence relations, and graph theory in particular stand out.

  15. The messages contained within this article are very important. Unfortunately, in the UK students can drop mathematics at the age of 16 and still expect to do a biology degree. I worry that the main problem is that biologists only realise they need the maths when it is already getting too late. This survey tells us what we probably already suspect to be true -most research biologists wish they knew more mathematics: https://peerj.com/articles/285/

    If only I could convince my undergraduates that this was the case, and that even if they don’t carry on doing ecological/biological research then having these skills will set them apart.

    Aside from your excellent list of skills, I’d add that proper training in programming is also very important and I’d say R and something akin to C are good requirements. As a bare minimum I’d say all ecologists need to know linear algebra and at least some programming.

  16. Hey,

    honestly your first list is right on the money. I studied engineering for two years, and am now pursuing Restoration Ecology (so even less math ?) but I intend on taking the extra time to get math credit. I see dynamical systems and differential relations being vital to all ecological studies. There is just no compensation or me. It’s too important.

    I think eigenvalues are necessary for solving discrete dynamical systems, and so understanding determinants would be too. I think calculus is too important to be left out. It will have a large role for me to play in optimization, which is very important.

    I think it depends on what a person see’s themselves doing. I intend on pursuing a graduate in ecological fields, so statistics and probability would probably be better for me. I will probably not be doing any modelling, but providing the data for such, so learning dynamical systems and chaos theory would probably be irrelevant, and I should spend time learning biochem. and ecology.

    Leaving these choices to a young undergraduate would be wrong though, many people probably wont even know what they can do with what math at 17. So I think an ecology undergraduate should have to learn calculus, linear algebra, differential equations, probability, dynamical systems, control theory, logic and proofs (though not usually a conventional topic)… I don’t know what stochastic processes are.

    Heck, I intend on learning all these things in my own time. I also intend on studying topology, fractal, abstract algebra.. I see it all being useful for myself and any scientist, but reason will have to be made if we still intend on practicing ecology.

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