# On the value of simple limiting cases: Lotka-Volterra models and trolley problems

I’ve talked in the past about the many different ways in which models can be false, and how false models are actually useful because they’re false, not despite being false. Of course, that doesn’t mean that all ways of being false are useful, or that certain ways of being false are always useful. See, for instance, this old post on when it is or is not helpful to try to build a “null” or “neutral” model of some phenomenon.

In this post I’ll focus on a couple of ways in which deliberately simplified models can be useful, even if they’re too simple to even approximately describe any real-world situation. The Lotka-Volterra models of competition and predator-prey dynamics are examples of the sort of highly-simplied models I’m thinking of. Simple models can be useful for clarifying concepts and sharpening intuitions, and as baseline (aka limiting) cases (and perhaps in other ways too, but for this post I’ll just focus on these two uses). And since ecology is far from the only field in which simple models are used in these two ways, I’ll illustrate these two uses of simple models by talking about simple models from a different field. So let’s talk about trolley problems in moral philosophy!

Trolley problems

A trolley problem is a deliberately simplified moral dilemma involving a runaway trolley. In its original version, there’s a runaway trolley speeding down a track. Ahead of it are five people tied to the track, who will be killed if the trolley runs over them. You’re standing some distance away, next to a lever that, if pulled, will divert the trolley onto a side track. The problem is, there’s one person tied to the side track, who will be killed if the trolley runs over him. There’s no time for you to call for help, or untie the people, or anything like that. Your only choice is to pull the lever or not. What should you do?

Clarifying concepts and sharpening intuitions

A trolley problem describes a very simple, artificial, unrealistic situation. Which is the point. Removing extraneous and distracting details helps us think clearly. Here, the original trolley problem clarifies what the moral philosophy of utilitarianism is and why it’s appealing. The utilitarian view is that the morality of actions depends on their consequences. To clarify this idea, it helps to have an artificial setting in which the consequences of one’s actions are unusually (artificially, unrealistically) direct and serious. Having five people live and one die is better than the reverse, so according to utilitarianism you’re morally obliged to pull the lever. It’s hard to imagine a more clear-cut illustration of what utilitarianism means and why one might want to be a utilitarian.

The original trolley problem also clarifies the consequences of a non-utilitarian position. For instance, you could take the view that human lives are incommensurable, and further that what’s morally right is independent of the consequences of our actions. If that’s your view, then you have to be prepared to live with choices that might have terrible consequences (think of the motto “let justice be done though the world perish“). Probably many people, without thinking about it much, would say that you should always “do the right thing” even if that would hurt other people. Which sounds perfectly reasonable, even noble—until you’re confronted with a trolley problem and are forced to stop and think. Possibly, your intuitions about what’s right might change once you’re forced to stop and think about them. (As an aside, I don’t mean to suggest that the utilitarian position is right, or wrong. I’m just highlighting how simple models can put pressure on our intuitions, thereby forcing us to think harder about what we believe and why).

Simple models play the same role in science. For instance, the Lotka-Volterra competition model often is used to introduce the very concept of “interspecific competition” for good reason. It’s the simplest possible model of that phenomenon, and so makes clear what exactly is meant by the term “interspecific competition”. It’s also the simplest possible model of “niche differences” and so clarifies what that concept means (anything that weakens interspecific competition relative to intraspecific competition). It also helps sharpen—and correct—one’s intuitions about the consequences of competition. For instance, if your intuition is that the best competitor should always exclude the others, the Lotka-Volterra model can correct that intuition, by revealing when that intuition is correct and when it isn’t.

It isn’t just students whose unaided intuitions sometimes need to be exposed to critical scrutiny. In my posts and paper attacking zombie ideas about the intermediate disturbance hypothesis, I’m actually not attacking the idea that disturbance or environmental fluctuations can promote coexistence and help maintain diversity. Rather, what I’m doing is attacking certain widespread, incorrect ideas about how disturbances and environmental fluctuations can promote coexistence and maintain diversity. Much as how, in moral philosophy, someone might use the trolley problem to clarify why pulling the lever is the right thing to do, rather than arguing that the lever shouldn’t be pulled at all.

Baseline cases

Another use of simple models is as a baseline case (also known as a limiting case). By first understanding the simple model, and then modifying it in some way, we reveal the consequences of the modification. In this way, the simple case serves as a necessary starting point for the analysis of more complex cases.

In moral philosophy, there are many different versions of the trolley problem. Comparing and contrasting our moral intuitions about what’s the right thing to do in different situations helps clarify what exactly our intuitions are. It also clarifies why we have the moral intuitions they do, and exposes those intuitions to scrutiny that they might not survive.

For instance, consider a modified version of the trolley problem. As before, there’s a runaway trolley speeding down a track, to which five people are tied. But now, you’re on a bridge above the track. The only way to stop the trolley is to drop a sufficiently-heavy weight in front of it—and as it happens, standing next to you on the bridge is a very fat man. Should you push him onto the track to stop the trolley?

Many people who would pull the lever without hesitation in the original trolley problem balk at the idea of pushing the fat man onto the tracks in the second version. Which reveals something about our moral intuitions, since the simple utilitarian calculation in both cases is the same. Your choice in both cases is between one death, and five. We can learn something about what’s right, and why it’s right, by trying to figure out why many people have contrasting intuitions about the two trolley problems.

It’s much the same in ecology. For instance, I teach predator-prey dynamics in my population ecology class by using the original Lotka-Volterra predator prey model as a baseline case. Then I introduce various modifications to the model—making prey growth negatively density dependent, giving the predator a type II functional response rather than a linear functional response, etc. Contrasting the dynamics of the modified models with the dynamics of the original model reveals the consequences of the modifications. I actually build a “periodic table” of different assumptions about predator and prey biology, so that students can see the systematic patterns in model behavior that emerge.

The rationale for simple baseline cases, to which one then adds complications, is basically the same as for any manipulative experiment. Want to know what effect factor X has on the behavior of your study system? Remove or otherwise manipulate factor X and see what happens! Want to know if the effect of factor X is contingent on factors Y and Z? Manipulate all three factors in a factorial fashion and see what happens! Etc.

Are simple models actually useful in these ways?

Some would argue that simple models are not useful in the two ways I’ve suggested. In moral philosophy, trolley problems and other strange thought experiments (“What if we’re all just brains in vats?”) are sometimes derided and parodied as the apotheosis of seminar room cleverness. Rather than revealing our moral intuitions and helping us to think clearly about them, or serving as a starting point for reasoning about everyday moral dilemmas, trolley problems just reveal what we’d do in bizarre situations that have no connection whatsoever to the real world. On this view, our contrasting reactions to different trolley problems just show that people get confused when faced with bizarre situations. Crooked Timber has a fun little post that talks about this criticism of “trolley problem philosophy” (accompanied with whimsical cartoons illustrating various trolley problems).

The analogous criticism of simple models comes up in ecology too, and also gets aimed at deliberately-simplified experimental systems like microcosms and mesocosms. It’s claimed that simple models, and simplified experimental systems, aren’t simplified versions of the real world. Instead, they are completely artificial constructs that are completely incomparable with the real world. Studying simple models, and simplified experimental systems, is at best irrelevant, and at worst warps rather than improves our understanding of how nature works.

I can appreciate this criticism of trolley problems, though I mostly disagree with it. As John Holbo at Crooked Timber says (see link above), the silliness and artificiality of trolley problems can itself be seen as a feature rather than a bug. As a philosophy professor leading a seminar, you can introduce a trolley problem as a way of “pumping” students for their moral intuitions. And then you talk about how the unrealistic nature of the trolley problem is likely to distort our responses. For instance, in trolley problems the consequences of our actions are unusually direct and serious. In the real world, the consequences of our actions often are diffuse, far removed from us, and difficult to trace. What does that do to our ability to do the right thing? Which is itself a good lead-in to talking about real-world moral dilemmas, since many real-world moral dilemmas are difficult either because they don’t arise in our everyday experience, or because we don’t realize they even exist (e.g., because we don’t recognize the diffuse consequences of our actions). “Everyday” moral dilemmas—i.e. moral dilemmas that are easily-recognizable as such, and that crop up more often in the real world than do runaway trolleys—aren’t something most people experience every day.

In ecology, I don’t think the analogous criticism of simple models, or deliberately-simplified experimental systems, really has much traction. Simplified models in ecology are more obviously useful than trolley problems are in moral philosophy. It’s so obvious that one can add in realistic complexities to a simple model (or to a simplified experimental system), and so obvious that this helps one understand the effects of those complexities, that it’s hard to see how anyone could deny the value of doing this. The value of trolley problems in moral philosophy is plausible but hypothetical. But in ecology people have published scads of papers in which they actually have used simple models and simplified experimental systems to learn things about the real world (e.g., this old post). As long as you’re willing concede the scientific value of some sort of simplification and manipulation of the real world (and if you don’t, you’re denying the value of all manipulative experiments, wherever and however conducted), you’ve given up any in-principle objection to simple models and simplified experimental systems. Plus, our future world is likely to be a simplified and maybe even bizarre version of our current world. For instance, in the near future there are going to be fewer species in the world, and the remaining ones are going to be distributed quite differently than in our current world, leading to ecologies that are far outside of our everyday experience in some respects. We’re likely to run into difficulty handling “everyday” moral dilemmas if all we know how to think about are the sorts of decisions we really do make every day. For much the same reason, we’re likely to have difficulty thinking about possible future ecologies if all we know how to think about is currently-existing ecology. I suspect that ecologists who dismiss simple models and microcosms as valueless distortions of the real world are just reacting based on intuitions that they haven’t really thought through.

Which is another reason to value simple models, and simplified experimental systems, in ecology. Thinking about why they’re valuable forces you to scrutinize your intuitions about how to do science. I was interested to read John Holbo’s comment that his philosophy students recognize how odd it is that they mostly want to pull the lever, but don’t want to push the fat man. As he says, “They are utilitarians with reservations, but can’t articulate their reservations in a plausible way.” This is very much my own experience with people who question the value of simple models and microcosms in ecology–they struggle to articulate their intuitive reservations in a plausible way that stands up to conscious scrutiny.

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I'm an ecologist at the University of Calgary. I study population and community dynamics, using mathematical models and experiments.

## 6 thoughts on “On the value of simple limiting cases: Lotka-Volterra models and trolley problems”

1. Awesome post, I completely agree with your views on these sort of models as a way to sharpen our intuitions and as limiting-cases that we should learn to understand. It is too easy to add features to a model, and much more insightful to find the essence.

I refer to these sort of models as heuristics, but I think it is only one type of three. There is also abstractions that are generalized and robust models, any reasonable changes to them result in a qualitatively (and often quantitatively, if measured properly) same model. An example of this would be reasonable models of computation (Turing Machines, lambda calculus, etc). I personally think these sort of models are sorely lacking in biology and ecology, and focusing on them would provide (in the long term) great benefit in understanding biological questions. Finally, there are insilications, these are the hyper-realistic models that one often finds in physics (especially among theorists of specific experiments; think of the computational models used to calculate the trajectory of the Mars rover rocket). Here, all the parameters of the model can be measured empirically, and the final prediction can be compared directly and quantitatively against experiment. I think this is what the public thinks most theorists work on, but in my experience it is rarely part of our work. Such models also seem to be lacking in non-molecular biology, but it looks like people are trying to bridge that gap. In fact, I think there is more focus on developing these sort of models than abstractions, even though I don’t think biology will be able to realize (at least with how things look right now) insilications that apply beyond a single experiment (or small family of closely related experiments; compare this to classical mechanics in physics that is used for everything from building bridges to the aforementioned Mars rocket (I don’t think they need to take corrections from general relativity)).

Is your periodic table of LV models available somewhere? I really enjoy looking at taxonomies of models, and would be very happy to read about yours. For instance, I like this taxonomy for models of cooperation (although I think it needs to treat spatial models a little bit more seriously):

West, S.A., Griffin, A.S., & Gardner, A. (2007). Evolutionary explanations for cooperation. Current Biology, 17(16): R661-R672.

2. You might enjoy reading R. M. Hare. He uses a statistical approach to ethics (without naming it that way). His point is that we generalize from core cases to build up rules for behavior, but always place an asterisk in them for special cases. The core of his point is that simple models drive our ethical thinking by letting us generate a relatively simple set of rules. We know that ‘edge cases’ can occur, but we treat them in a fundamentally different way. We cope by using a two track form of reasoning. Most cases can be handled by simple deontological reasoning, but some must be treated using case by case utilitarian reasoning.

I mention this, because trolley problems and all kinds of reasoning from the extreme can seem to force us into bizarre conclusions even as they reveal something fundamentally correct about the world. He represents a kind of cautious skepticism about the value of what others have called ‘intuition pumps.’ These fabricated examples do make clear some genuine insights into our beliefs, but they are unstable ground. It is far from clear that the extremes reveal anything other than what is true at the extreme.

A familiar example is lying. Kant seems to be right that beneficent lies cannot be a core part of our behavior. Other people need that information even if you aren’t always happy to share it. But, why does Grandma need to know you hate this Christmas sweater? Isn’t it right to just lie, say thank you, and make a beloved family member happy? It doesn’t have any serious consequences for moral behavior or human well being. It just lets her be happy while you tuck the reindeer-themed sweater into the back of your closet.

My point is, to paraphrase Collingwood, that our theories are answers to questions and not all conceivable questions actually arise. Trolley problems of all sorts are needed to flesh out the full weight of our concepts. It often happens that our ideas only apply to some limited set of problems. Nonetheless, we shouldn’t make the mistake of assuming that possible questions actually map onto the world as it is. Or, that what is true under the most extreme possible situation means all that much for the ordinary events.

• Thanks very much for this. I’ve heard of Hare but haven’t actually read him. Your description of his approach to trolley problems and extreme cases more generally certainly sounds similar to my own thinking.

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