by Vishwesha Guttal

 Period-Doubling bifurcation diagram, discovered by student groups in my class (from Aug 2013).

I have been teaching a course on theoretical ecology for several years now. Given that this is a graduate-level (also open to undergrads as an elective) course, I have enormous freedom in how I design and teach it. This allows me to do two things: (1) I introduce a number of contemporary topics that are typically not taught in an introductory theoretical ecology course; some examples include emphasis on random walks and stochastic population models, understanding evolutionary dynamics via Price equation, etc. (2) I teach only half of the course material, and allow the other half to be discovered and learnt by students on their own via various group activities. Here I describe some examples of this active learning based teaching I have tried to adopt in this course.

The course I teach is a semester long, with 1 lecture (1.5 hours) and a lab (~3 to 4 hours) per week, and runs for about 13-14 weeks. The size of the class is usually 24-40,  most of whom are undergraduate math/physics/biology majors , a few are engineering students and a very few are ecology PhD students. So, there is a wide diversity in the class in terms of mathematical skills. However, basic calculus and basic programming are pre-requisites for the course.

Each week, I first teach a concept in a lecture mode. In the lab that follows, students work in groups to solve an in-class-assignment during which they discover something new, and often counter-intuitive, to what I taught in the previous class. The most challenging part for me is to come up with an assignment that is both doable and intellectually stimulating for students. I want them to feel ‘oh wow, I didn’t expect that from the concept I knew of or learnt previously’ at the end of every lab. To illustrate this, let me take you through the first two weeks of my class in some detail.

Week 1: Building the simplest population dynamics model

In the first lecture of the course, I start with a discussion on the basic philosophy of mathematical modelling, interesting ecological questions that mathematical models can help us answer, the difference between mathematical and statistical modelling, etc. I then narrow down to the problem of change, or dynamics, in ecology and evolution. By the end of the one and a half hour lecture, I show students how to build the simplest population dynamical model, based on counting number of births and deaths per individual per generation. This will be familiar to ecologists as discrete-time density-independent growth (or decline) of populations, although I don’t use this jargon at this stage. We learn that in this model, populations either grow, or decay,  exponentially, leaving little room for any other possibilities.

Next comes our first laboratory session where I begin by revisiting the model and its results. We then discuss what insights and new questions arise from this simple modelling exercise. Clearly not all populations go extinct. Those that do survive, don’t keep on growing exponentially. Therefore, one important question that we all agree upon is to understand what regulates growing populations in the real world. I then ask students to work in groups to submit an in-class assignment, answering the following questions:

(a) List factors that regulate population growth.

(b) Can you modify the simple model we derived to account for limited resources?

As I said earlier, I have not introduced the term of density (in)dependence, etc at this stage. Therefore, they are not constrained to think within the limits of known concepts, and I often hear interesting suggestions many on expected lines as well, ranging from predation, pathogens, changing birth and death due to environmental degradation, competition for resources, etc as factors that limit populations. After some further discussion, I suggest that we consider resources as the limiting factor. Given that models can be written in so many ways, I do constrain them a little bit by saying do not explicitly model resource dynamics or competition with other species.

After two hours of discussions student groups come up with a diverse and interesting set of solutions, with some being very simple and others rather complicated, in mathematical terms; the details of solutions student groups come up with can be a separate article in itself. Here is something interesting: When I present all the models suggested by different groups to the entire class, and ask them which one shall we proceed with for further analyses – I am always amazed how often the entire class agrees that the simplest model is the best model to analyse.  It seems as if the principle of parsimony in model building is a natural instinct! In the case of this particular exercise, this simplest model invariably happens to be the (discrete-time) logistic model.

Week 2: Analysing the model

In the second lecture, I teach the idea of stable fixed points, where populations do not change and return back upon perturbations. I also teach mathematics of stability analyses, and show stable fixed points as a function of the growth parameter for the discrete-time logistic model.

The lab of the second week is the most interesting one. I ask them to systematically simulate the logistic model, and see if the analytical results are right. Before they start, I reiterate the notion of fixed-points. I also discuss the notion of predictability – asking them what they think causes unpredictability in the real ecological world. The most frequent answer is that stochasticity breaks down predictability in biological systems. For the simulations, to check the notions of stability and predictability, they simulate population trajectories starting from different initial conditions. If they all reach same population sizes, the populations are both stable and predictable.

As they start simulations, the first interesting insight is the discovery of cycles; so the notion of stable fixed points for populations is insufficient. Moreover, cycles can arise endogenously in populations. The first time students see the bifurcation diagram of a stable population give away to a cycling population is often mesmerising to many students and to me as a teacher as well.  The second exciting part is the discovery that that even two very close trajectories can dramatically diverge for large growth parameter values (chaotic regime) .That predictability breaks down even if the model has no stochasticity,  is another fascinating take-home to everyone.

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My motivation to teach this course has been to facilitate the process of students discovering a variety of theoretical ecology concepts on their own via group discussions. Therefore, my intention is not to show a large number of models and their mathematical analyses. Although theory and mathematical models can be analysed objectively, I try to focus a lot on the process of  building the models. This is a subjective exercise, given that each scientist may have slightly different focus on what is an important question, what aspects of biological reality must be captured in the model and what can be ignored, and ultimately, the purpose of modelling itself. The emphasis on the process of model building means that the quantum of materials I teach is perhaps less in comparison to a regular theoretical ecology course.

An interesting challenge I face every year is that the pace, and hence content, of the classes differs quite a bit based on students’ background and the questions they ask. Some years are dominated by ecology students with little math background, whereas other years it’s mostly mathematics and physics majors with little prior ecology knowledge. So I need to redesign both lectures and laboratory work to keep students interested and engaged. The above lecture and lab outlines for the first two weeks are therefore only illustrative.

I am certain that many have tried active learning methods in theoretical ecology/evolution courses. I would like to hear how you go about them!

Author biography: Vishwesha Guttal is an assistant professor at the Centre for Ecological Sciences, Indian Institute of Science, Bengaluru, India. His lab works on critical transitions in ecosystems, collective animal behaviour and self-organised pattern formation in ecology. He tweets @vishuguttal