Do you know what a type I functional response is? Are you sure? (UPDATED)

A “type I” functional response is a linear functional response. An individual predator’s feeding rate at any moment in time equals aN, where a is the “attack rate” (i.e. the constant per-predator, per-prey feeding rate), and N is the current prey density. If you plot aN vs. N, you get a straight line with zero intercept and positive slope a, hence the term “linear” functional response.

At least, that’s the way I and many others teach it. But apparently a fair number of people teach “type I” functional responses quite differently, if a Google image search on “type I functional response” is to be believed. Apparently, many people teach that a type I functional response is nonlinear! That is, they teach that a type I plot of individual predator feeding rate vs. prey density looks like this:

Nonlinear type I functional response, from an entomology course website

I have to say, that looks really weird to me! To my eye, that’s basically a type II functional response, except it’s piecewise linear rather than smoothly nonlinear. If you “sanded down” the “kink” in the figure above, you’d basically have a type II functional response. As every undergraduate knows, a type II functional response is what you get when you assume that predators aren’t capable of consuming prey infinitely fast, but rather have some finite “handling time” per prey captured, and can’t search for prey while also handling captured prey. In the limit of infinite prey density, predators with a finite handling time T can capture prey at a rate of 1/T (i.e. they spend all their time handling, because the instant they stop handling one prey item they capture another, since prey density is so high). A type I functional response, as I teach it, is the limiting case of a type II as you allow the handling time to go to zero.

So here’s my question: why do so many people teach that a type I functional response has an asymptote? Is it for historical reasons? Because for the life of me, I can’t trace the historical origin of this notion. In their original predator-prey models, Lotka and Volterra assumed a linear functional response, not a piecewise linear function with an asymptote. So did Nicholson and Bailey in their original 1935 host-parasitoid model. And Holling (1959a Can. Entom. 91:293-320), the source for the classification of functional responses into three basic “types”, says the following in the passage at the end of his paper where he defines those types (boldface emphasis added):

The functional responses could conceivably have three basic forms. The mathematically simplest would be shown by a predator whose pattern of searching was random and whose rate of searching remained constant at all prey densities. The number of prey killed per predator would be directly proportional to prey density, so that the rising phase would be a straight line…A more complex form of functional response has been demonstrated in laboratory experiments by De Bach and Smith (1941), Ullyett (1949a) and Burnett (1951, 1956) for a number of insect parasites. In each case the number of prey attacked per predator increased very rapidly with initial increase in prey density, and thereafter increased more slowly approaching a certain fixed level.

And in Holling (1959b Can. Entom. 91:385-398; the “disc equation” paper), he comments on the linear functional responses assumed by Lotka, Volterra, and Nicholson & Bailey. He notes that linear functional responses are unrealistic at high prey densities (which they are). But unless I missed it (did I?), nowhere in these papers does he suggest redefining linear functional responses so that they aren’t linear any more. In other words, Holling himself seems to have been perfectly clear from the get-go on the difference between what came to be called type I and type II functional responses, with the former being linear, period. Did he change his usage in later papers I haven’t read? C’mon commentariat, help me out–I’m clearly embarrassingly ignorant about the history of the functional response literature and I need you to help me become less ignorant! (UPDATE: Yes, I did miss it, see the comments. The source for the piecewise linear definition seems to be Fig. 8 in Holling 1959).

To be clear, my claim here is not that functional responses must be defined or taught as Holling defined them. I’m not attempting proof by authority here. And I would never claim that we must always define and teach concepts according to how they were originally defined historically. I’m just trying–and failing!–to trace the history of a really basic concept I thought I and everyone else agreed on. It’s a really weird feeling. How can it possibly be that ecologists don’t all agree on the definition of what I thought was a fairly precise concept (in contrast to, say, “niche”)? Especially given that that concept is so basic it’s taught in every undergraduate ecology course!

From my digging, it looks like the two different ways of defining “type I” functional responses map roughly onto something like a conceptual/empirical divide. I teach them as purely linear because, conceptually, that’s the simplest limiting case. You can only understand why more realistic nonlinear functional responses have the shape they do, if you first understand the admittedly-unrealistic limiting case of linear functional responses. This approach of starting simple and then adding complications one by one isn’t just pedagogically useful–it’s scientifically useful. It’s a way to build up to an understanding of complicated, realistic biological situations. Indeed, this was Holling’s (1959a,b) own approach to understanding the shape of predator functional responses; he argues at length for this approach. And in the modern theoretical literature, every predator-prey or food web model I’ve ever seen that includes predators with “linear” functional responses assumes just that: linear functional responses. Not some piecewise linear function that asymptotes at high prey density. And in at least some of those papers, those linear functional responses are called “type I”.

In contrast, it looks to me like people who define type I functional responses as piecewise linear are tailoring them to a specific empirical case: certain species of filter feeders with very short handling times, that forage at a reduced rate when their guts are full so as to only take in food as fast as it can be digested. For instance, Jeschke et al. 2004 take this point of view in their review of the empirical literature on functional responses. And the undergrad textbook from which I learned (2nd edition of Begon, Harper, Townsend), takes the same view, albeit without really explaining in any detail where piecewise linear type I functional responses might come from, mechanistically, or why they are all that different from type II.*

Having thought about it a bit, I still prefer my way of approaching this. I think that, for both pedagogical and research purposes, you should start with the simplest limiting case, however unrealistic, and then add in biological complexities one by one. That’s what we do in other areas of ecology, like population growth, where we start with the simplest limiting case (exponential or geometric growth) and then add in density-dependence. I don’t like having a classification scheme for functional responses that doesn’t even include the simplest limiting case among the possibilities. And I don’t really see the point of a classification scheme that distinguishes two different classes of functional responses that both reach an asymptote at high prey densities, based on whether or not they approach that asymptote “sharply” or “gradually”. That seems too much like hair splitting to me. Especially since type II functional responses can vary a lot in how “gradually” they approach their asymptote, just depending on the attack rate parameter:

Three different type II functional responses with different attack rates but the same handling time, and thus the same asymptote. Higher attack rates lead to functional responses that closely approach the asymptote before decelerating sharply.

But that’s just one man’s opinion. What do you think? How should “type I” functional responses be defined? As linear, or piecewise linear? Or maybe we should get away from the whole “type I, II, III” scheme entirely, in favor of an approach that embraces the full range of biological possibilities without trying to jam all functional responses into a few named categories? That’d arguably be in the spirit of Holling’s original thinking about the whole concept of a “functional response”. And it’s what we do in many other areas of ecology. For instance, there are lots of different ways in which a population’s per-capita growth rate might depend on its own density, but nobody ever talks about “type I” vs. “type II” vs. “type III” density dependence. So maybe it’s time to just say goodbye to the whole tradition of named types of functional responses. Anyway, surely something has to change. Because it can hardly be optimal, scientifically or pedagogically, to have a whole named category of functional response being defined totally differently by different people.

*As an aside, it’s not universally agreed that filter feeders often are best described as having piecewise linear functional responses, rather than type II or some other shape.