by Andrew Saintsing, Graduate student at the Poly-PEDAL Lab ,Berkeley
I’m far from an expert on ecology, but I looked forward to reading John P. DeLong’s Predator Ecology (published by Oxford University Press) when the ICB journal social media coordinator offered me a copy in exchange for a review. Having spent the past five years in graduate school measuring the respiratory rates of running cockroaches, I’d say I qualify as a physiologist, but I really appreciate ecology. I think it’s fascinating how the two fields intersect. As far as I see it, both concern themselves with the way energy and materials flow through systems. They just focus on different levels: physiologists look within an organism, and ecologists look among groups of organisms.
An informative and enjoyable read, Predator Ecology provides a deep dive into the functional response, a mathematical relationship between the abundance of prey and the foraging rate of a predator. I feel fairly confident that I learned about the functional response in my undergraduate ecology course, but I can’t say that I remembered much of anything about it when I first picked up DeLong’s book. Luckily, that didn’t matter. DeLong quickly and efficiently introduces the conceptual underpinnings of the functional response to ensure every reader, regardless of incoming knowledge, he gets them up to speed. He then spends the rest of the book exploring the parameters of the model, complicating its assumptions, and raising questions that have yet to be answered.
After finishing the book in a couple of days, I think you’ll feel ready to start doing your own research on predator-prey interactions. Even if you don’t consider yourself an ecologist, though, Predator Ecology is worth perusing. In his efforts to find real-world meaning in the mathematical parameters of the functional response, DeLong models how to think through integrating theory, math, and empirical data. Accomplishing that synergy is a goal common to all biological disciplines, so any biologist should benefit from seeing someone working through it.
I spoke with DeLong over Zoom once I’d finished the book and asked him some of my lingering questions. I’ve transcribed the interview and shared a portion of it below. It’s been edited for length and clarity and annotated where appropriate.
Why did you write this book?
Because I couldn’t pull all of this stuff together in my own head. The literature was very confusing. There were so many perspectives and ways of writing about this topic, and there was no synthesis. I was trying very desperately to pull it together for myself, so that I could then say something intelligent to my students. I literally had to write it down. Once I started writing it down, I realized there’s a need.
The book is very focused on the functional response. Would everybody who’s looking at predator ecology start with the functional response?
I’m not sure that they would, but I think sooner or later they would get there. They might be studying wolf movements, and it might take them a while before they connect the dots between spatial ecology and foraging ecology. But if you were interested in rates of foraging and you know what determines how many moose a wolf would eat, you would start pretty close to the functional response. In most ecology textbooks, it’s really very brief. It just shows the three types and then moves on. So, many people who would have taken a basic ecology course would say, Oh, yeah. I know what the functional response is. But then that’s probably all they would have read about it in an ecology class.
[Note: Some background information might be useful for this question and answer. The Michaelis-Menten equation is used to model enzyme kinematics. An enzyme catalyzes a reaction that turns a substrate into a product. For a given enzyme, the rate of product formation depends on the concentration of the substrate, but that rate approaches a maximum as the concentration of the substrate goes to infinity. The curve of the Michaelis-Menten equation is called a rectangular hyperbola. The graph of a Type II functional response is also a rectangular hyperbola — as prey abundance or density goes to infinity, the number of prey a predator can catch and consume reaches a maximum. However, the Type II functional response is typically written using Holling’s disc equation, which has a different set of parameters.]
One thing that I thought was really interesting: you’re going through the functional response, and you say an alternative equation could be the Michaelis-Menten. My undergraduate curriculum was very chemistry- and medical school-focused. It was funny to see Michaelis-Menten described as an alternative because it’s such a big deal in biochemistry. That got me down this whole rabbit hole of all the things that look like Michaelis-Menten and the functional response. Apparently, it’s a lot of things.
I think the comparison between those two forms of that curve is really enlightening. People who grow up thinking about Michaelis-Menten look at the parameters, and they mean something to them. But the parameters are different. They have different units, so they don’t mean anything to somebody who’s more familiar with Holling’s disc equation. Those parameters might signify something biological. It emphasizes the need and our systemic failure to be able to understand that math is a representation of real biological processes, and that every symbol, every constant should be, in your brain, translatable to the process that you’re interested in. My personal experience is that we learn math without understanding that symbols are really symbols, and that they represent something, and that it is actually really important for us to understand what it is that they represent. When you come at the same curve and you write it in two different ways, it means totally different things to different people. It’s difficult for me to understand what the Michaelis-Menten version of the symbols mean for a predator. A half-saturation constant — what is that for my predator? And what does it really mean if I’m thinking about my system? But handling time I can understand.
The subversive educational mission in the book is to make people see the biology in the symbols. We don’t learn math that way, and then we think math is dumb and not useful, and we forget it as quickly as we can. It’s really useful if you just can map it back on to the things you are interested in.
How did you get so interested in math?
When I was a little kid, math was my favorite subject. I always did great in math, but there was no point to math, so after calculus I didn’t take anymore because nobody ever clarified to me that there was any value in it. Then, after a long circuitous trip through various things including geology and hydrology, eventually I found biology. I was not a biology major or a math major. I was an environmental science major. When I went back to grad school, I was not doing any math at all. I was doing more traditional wildlife biology, bird migration stuff. I took a modeling course, and I thought it was cool, but it wasn’t the gamechanger. Eventually though, when I switched — I did a master’s, and then I did a PhD — in my PhD, I started to see the value a bit more. Connecting meaningful empirical observations to math was the thing. It was attractive as a whole. I only have two pure theory papers. I’ve avoided pure theory papers. I have some pure empirical papers. But I’ve been trying to connect them ever since.
I wouldn’t consider myself to have a very strong mathematical background. I haven’t gotten through all of the calculus levels, but I think that relatively simple math can get you almost everywhere you want to go in biology.DeLong
Do you ever find a mathematical expression that you just can’t see how it maps on to actual biological phenomena and you feel like it needs to be modified to actually reflect what’s going on?
Oh, yeah. All the time. It may be that the original derivations were sensible, but then by the time you’re done with it, you may not be able to connect the dots to a process anymore. Or there are all sorts of little off-the-shelf phenomenological functions that people drop in here and there that serve a purpose, but if you look at the units, you can’t make sense of what they are. It’s pervasive, and it makes it difficult for people to embrace that side of mathematical biology. If you’re an empiricist and you’re used to measuring something, you’re like, “I know exactly what that number represents. I can see it.” Then you look at the model, and you’re like, “I have no idea what that represents.” Maybe they didn’t explain it, or maybe the model doesn’t have any clear mapping. It’s been too simplified, or the derivation didn’t start at the right place.
How do biologists have a better relationship with math, where we can get more consistently meaningful mathematical models out of empirical data?
[Note: Once again, some background information may be necessary. In his answer, DeLong mentions the “three-quarter power.” This refers to Kleiber’s Law. In the 1930s, Max Kleiber measured the metabolic rates of mammals of various sizes. He plotted out metabolic rates as a function of body mass and found that metabolic rates scaled to body mass raised to the three-quarters power. Physiologists and ecologists have grappled with that number ever since, with some refuting it and some trying to explain it. West, Brown, and Enquist suggested that quarter-power scaling reflected the central role that branching networks (e.g., vertebrate circulatory systems) with “fractal-like” geometry play in dictating the rate of metabolic processes.]
It’s a big question, and it’s a hard problem, but it’s going back to the basics. Starting from someplace that you know has to be true. I don’t know how you would do this for your work, but I think that metabolic rate is a problem that needs this same sort of treatment. There’s a lot going on in metabolic rate, and so everybody argues about it from different points that go on in a long chain of events. But in the end, there is a closed form starting point. There has to be some kind of boundary around the exchange of carbon or oxygen that’s inviolable. Then you can start to break it apart where every step has to be true.
If you don’t do that, you jump to places where you’re like, “OK, well, here’s a curve.” You’re not going to be able to understand it. Trying to understand what three-quarter power means: it’s an exponent. What the hell are the units of three-quarters? That gets really weird really fast. Then you’re like, “Well, fractals.” I love thinking about all that stuff, but it is difficult to translate. I can’t look out at nature and say, “Oh, yeah. Three quarters.” I can’t point to anything that tells me what that parameter is in real life. It’s this emergent property. We started with a statistical observation and then tried to go backwards. I think that’s why it’s such a problem. We didn’t start with something that had to be true.
There’s a chapter about Type III functional responses. The way people talk about them is one of those things that is just so confusing. There’s a lot of work on them, but it hasn’t generated any real understanding. I felt frustrated with where we’re at. This phenomenon is invoked in so many papers by so many people, but it’s not explainable, and the parameters have no biological meaning. This is screaming for some new treatment, so I was playing around before I stumbled upon my version of an interpretable Type III functional response. Many of the things that I’m trying to get at in the book came together with that. Going back to the basics and trying to understand that every symbol in a function should have biological meaning that you can understand. The traditional Type III functional response model has weird implications. There’s a lot of opportunity to go back. A lot of things that we learn about in textbooks and that people talk about need to be revisited.
Pick up a copy of Predator Ecology to learn more about John P. DeLong’s thoughts on the functional response. After you’ve read the book, consider exploring our journal Integrative and Comparative Biology for related papers that could complement DeLong’s writing. For instance, in one chapter, DeLong discusses how the traits of predators and prey influence parameters in the functional response equation. He mentions speed of movement, capacity for acceleration, and perception. A 2015 paper from Talia Moore and Andrew Biewener titled Outrun or Outmaneuver: Predator-Prey Interactions as a Model System for Integrating Biomechanical Studies in a Broader Ecological and Evolutionary Context discusses how biomechanical experiments might actually be used to understand real-world interactions. Meanwhile, Shelley Adamo’s paper The Integrated Defense System: Optimizing Defense against Predators, Pathogens, and Poisons, published this year, explores how difficult it can be to zero in on traits related to predator-prey interactions because of how integrated animal systems can be. But those are just two examples. Regardless of what excites you about predator-prey interactions, you’ll likely find something relevant in Integrative and Comparative Biology.
Free read by DeLong et al
A Unifying Framework for Understanding Biological Structures and Functions Across Levels of Biological Organization
M A Herman, B R Aiello, J D DeLong, H Garcia-Ruiz, A L González, W Hwang, C McBeth, E A Stojković, M A Trakselis, N Yakoby
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connect with Andrew Saintsing via Twitter @AndrewSaintsing