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:
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:
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.
actually we have a paper in Ecology Letters that s going to be published online early within a couple of days where we suggest to skip the FR categories for an allometrically defined continuum:
Body masses, functional responses and predator-prey stability
Kalinkat, G; Schneider, FD; Digel, C; Guill, C; Rall, BC; Brose, U
I really appreciate this possibility to promote our work so unabashedly 😉
The other point I’ll make, which I probably should’ve made in the post, is that I don’t really understand how the “slow your foraging rate when your gut is full” argument gives you a piecewise linear functional response. After all, if you put a filter feeder with an empty gut into an environment with a very high prey density, it will initially forage at a rate proportional to prey density, with the same constant of proportionality as when prey density is lower. Only after some period of time, when the gut fills, does the forager then drop its feeding rate. So even if the foraging biology of filter feeders works exactly as assumed by those who define type I functional responses as piecewise linear, it seems incorrect to me to plot that functional response solely as a function of prey density. In doing so, I think you’re implicitly, and incorrectly, assuming that gut fullness necessarily is proportional to current prey density (up to the point where the gut is 100% full). Am I wrong?
Can anyone cite a reference where someone actually writes down and solves an explicit mathematical model of the “gut fullness” argument and derives the functional response shape? I have a vague recollection that Peter Abrams did some work on food web models with gut size-limited herbivores in the 1980s; will have to try to look up that work at some point…
Again it’s Jeschke et al:
Predator functional responses: discriminating between handling and digesting prey. (2002, Ecol Mon)
who have worked on this.
How much the gut fullness affects your results is critically related to the experimental duration that varies widely within the literature on functional responses. If you do an experiment that is short enough than a true linear response can actually be found. The ecological relevance of this, however, might be limited. I’m just about to figure that out.
Personally, I wouldn’t bother too much about linear (“true type I”) and piecewise linear (“flawed textbook type I”) as I assume both to be rather unrealistic. Nevertheless, one should be aware that the concept and the use of the functional response is so widespread (because with many organisms it is so easy to set up respective experiments) that almost consequentially the definitions that can be found vary also widely and the approach and motiviation might differ depending on the questions being asked. E.g. does the population ecologist want to parameterise a population model vs does the behavioral ecologist want to understand the foraging process?!
Thanks for the link to Jeschke et al. As I suspected, their “steady state satiation” model, which incorporates both handling time and digestion rate, doesn’t actually generate a piecewise linear functional response, not even in the limit when handling time is zero. It produces type II functional responses, but with different mechanistic bases depending on whether the predator is handling time limited or “digestion limited”. Of course theirs isn’t the only possible model of a digestion-limited consumer. But it’s hard for me to imagine any plausible way of modifying the model so as to get a piecewise linear “type I” curve.
Given that Jeschke et al. 2002 describe their own model of digestion-limited predators as leading to *type II* functional responses, it seems kind of odd to me that in their 2004 review they’d present putatively digestion-limited filter feeders as having piecewise linear functional responses. But I didn’t read their review super-carefully, maybe they talk about this at some point…
Until recently, I considered the piece-wise linear to be a rather ugly implementation of “it’s proportional, but it can’t increase forever”. But then I discovered the paper of Stig Sjöberg (1980) that introduced queueing theory to foraging ecology. It derives both Holling’s Disk Equation and the piece-wise linear as special cases of the same model in which food items are lined up in the gut to be digested one at a time, first-in, first-out. Predators are assumed to search at a maximal rate, unless their gut is full, in which case they cease searching. In the case of a gut capacity of only a single food item, Holling’s Disk Equation is recovered, which wouldn’t surprise Buzz Holling, who included a resting phase for satiated predators as part of the handling time (Holling 1959). In the limit of infinite gut capacity (measured in food items) with the processing times of food items proportional to their size, we get the piece-wise linear.
As to your first question: you are right that it takes some time before the distribution of consumer states equilibrates with the surroundings. Imagine a group of lions released from a cage without prey onto the Serengeti with gazelles. Their average capture rate will initially be higher because they are all in searching mode and none are in handling mode! One may neglect this if the lions are very quick to find and handle prey or if you’re interested in the capture rates in the long run and gazelle densities don’t change. In other words, a separation of time scales may lead to a quasi-steady state assumption, as implicit in Holling’s (1959) derivation of the Disk Equation (and explicit in Sjöberg’s). Whether such an assumption is warranted depends on the system and the question at hand.
By the way, how nice to find a place on the ‘net where fellow ecologists freely speak their minds unpeer-reviewed!
C.S. Holling (1959). Some Characteristics of Simple Types of Predation and Parasitism. The Canadian Entomologist, Vol. XCI, No. 7, pp. 385-398
Stig Sjöberg (1980). Zooplankton feeding and queueing theory. Ecological Modelling 10, pp. 215-225
Thank you for this Daan, I wasn’t aware of Sjoberg (1980), that’s a very useful reference.
I’ve wondered about this, too. I usually use whichever version the book I’m teaching from says, but then tell the students that it is also drawn the other way. But I’ve never been able to figure out why it sometimes gets drawn with the piecewise approach.
Oh thank goodness–I’m *so* glad to know I’m not the only one who wonders about this! I’ve been dreading a bunch of people all telling me that I’m totally ignorant, and that everybody knows the piecewise approach was derived in some famous paper of Holling’s or something.
I also assumed type-I = linear, but had heard rumors of this other interpretation. Seems that the Holling 1959 paper you cite is the source of the confusion. To highlight another part of the same passage describing the type-I: “**the rising phase** would be a straight line”, which strongly hints at a “non-rising phase”. If there’s any doubt, the upper left panel of figure 8 illustrating it is (approximately) piecewise linear. Ugh.
To clear up this confusion, I hereby propose to call the “linear up to a maximum” functional response “type-1.5″, or in roman numerals, type-IS. Who’s with me?
The type-1.5 functional response does come in handy in modeling, in that it captures the effect of saturation without the destabilizing effects of a type-II, while keeping the algebra easier. Learned this from Armstrong 1994 (Limnology & Oceanography 39: 597-608), an excellent paper that few non-aquatic ecologists seem to know about for some reason…
it might come handy but is there a biological justification for it? Does Holling give one? I can’t get my hands on the paper right now.
By accounting for saturation, the type-1.5 is at least more realistic than the linear type-1, which is a very popular choice in modeling.
“By accounting for saturation, the type-1.5 is at least more realistic than the linear type-1”
Except that, as noted in another comment, if you actually model sources of saturation mechanistically (in particular digestion limitation), you’re apparently hard-pressed to actually derive a piecewise linear function. So a piecewise linear function “accounts” for saturation only a purely phenomenological way. And there’s a place for that, especially if it makes the math convenient. But I don’t know that it’s the kind of thing that needs it’s own name. So while “type 1.5” functional response is a cute idea, I’m not sure I’d want to see it taken up. 🙂
I bet you’re right Chris. So it probably traces back to Holling after all.
I know that Armstrong 1994 paper, but apparently not very well, I don’t remember it using a piecewise linear functional response. Or maybe I’m thinking of a different paper of Armstrong’s?
Total aside: Robert Armstrong really is one of the great underappreciated ecologists of his generation. He’s not just the guy who discovered what we now call relative nonlinearity. Armstrong 1979 Ecology is a neglected classic. Basically everything that’s in much more well known later papers from Holt et al. 1994 and Leibold 1996 is in that paper. And the analytical method seems really clever to me and I find it very easy to understand. He has some very nice old diversity-stability stuff too. People joke that God would never get tenure because he only had one publication, but I think that’s more or less what happened to Robert Armstrong. Relatively few papers–never mind that they’re all brilliant.
Totally agree. The good news is he’s still kicking out the jams in biological oceanography. See http://somas.stonybrook.edu/people/armstrong.html for recent activity.
As a grad student, I went down to Princeton to visit him because I wanted to test diamond food web models like Armstrong 1979 for my dissertation. Saw him give a talk to the research group he was a part of. Basically, he was working on marine food web models that could be plugged into big global climate change models. He was talking about a model he’d developed and at one point said the zooplankton had a type III functional response. Somebody in the audience stopped him to ask him what that was–and it took him like 15 minutes to explain it to everyone’s satisfaction! This was totally down to the audience not being ecological, they were totally unfamiliar with what a functional response is or why it might take on any given shape. After everyone else left, I told him, “Don’t worry Robert, I was with you all the way, I totally understood the functional response part”!
Are you sure it keeps the algebra easier? A piecewise-linear response would likely make calculus harder in any case. I reckon the issue of analytical tractability is behind this phenomenon: “… 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.” If you’re teaching functional responses in a theoretical course with mathematical modelling, you use a linear response because working with piecewise functions is a pain. Maybe if you’re teaching the “concept”, you’re more likely to use the piecewise linear because it seems more reasonable (reality is bounded).
Here’s a thought: if you look at the standard picture for Hollings’ responses: http://upload.wikimedia.org/wikipedia/commons/a/af/FunctionalResponsesGraph.svg, the linear function has little choice but to intersect the graph’s top line. That kinda makes it look piecewise linear. Maybe the piecewise-linear response is the result of an optimal illusion?
As I said in the post, even if reality is bounded (which it is), to understand reality you often have to start by first considering unrealistic special cases. Exponential growth forever is unrealistic–but that’s how you start teaching population growth, even to empirically-oriented students. Because in order to understand the effects of realistic complications like density dependence, you have to understand what happens when you remove those complications.
Interesting suggestion that the piecewise linear idea emerges from a misreading of standard graphs. I recall a similar suggestion in a totally different context. Baseball statistician and historian Bill James once suggested that the whole idea of a sacrifice bunt (a common but controversial tactic by the hitting team) arose from a historical misunderstanding. James suggested that the idea used to be to try to bunt for a base hit, but if you got thrown out at least you’d have advanced the other runners on base. And somewhere along the way people misunderstood and got the wrong idea: that the batter was supposed to nobly give himself up and make out intentionally for the sake of advancing the runners.
I have no idea if James’ suggestion is true. But I’m afraid I doubt that yours is, though of course I could be wrong. 🙂
@Jeremy I get your point referring to the necessity to start teaching with simple models (often incorporating strange and unrealistic assumptions). However, one last reference that strongly recommend to read about *empirically derived* (i.e. realistic) functional responses is Sarnelle & Wilson (2008): Type III functional response in Daphnia. Ecology, 89, 1723 – 1732.
Makes the point that If you look carefully enough (by appropriate experimental design and statistics), you can find a type III almost everywhere. In conclusion, even a type II could be an unrealistic special case.
This is actually how I, many years back, learned what a type I functional response is. It was (is?) presented graphically this way in the early 90ies German editions of W Lamperts’ and U Sommers’ book Limnoecology (Limnoökologie), motivated by the same argumentation about filter feeders you gave. You understand my confusion when I finally read Hollings original papers :). But I don’t think that this book is the source of it. No idea where it comes from.
One piece of the history might be Hassell’s 1978 monograph, “The Dynamics of Arthropod Predator-Prey Systems.” I haven’t thought about the math in that monograph for a while; however, there is an appendix that might relate type I and type II functional responses. Hassell mentions that the type I response is a special case of the type II response. Hassell’s type I figure uses a solid line to represent the increasing part of the response, and a dashed line for the plateau. (It is almost as if the plateau is an optional part of the figure or a parenthetical aside.)
Hmm, interesting, I wasn’t aware of that. A solid line for a linear functional response with a dashed horizontal line for the asymptote is kind of an interesting way of trying to convey graphically what everyone knows–a linear functional response is surely unrealistic at sufficiently high prey densities.
Having said that, I could see where it could lead to confusion. If you say, correctly, that a linear functional response is a limiting case of type II, then a piecewise linear function is an incorrect illustration of that statement, no matter which lines are solid and which dashed.
In Holling 1959a, Fig. 8 shows a type I with an asymptote as said above. I suspect the confusion comes from the fact that Holling was not not really interested in what happens at high prey densities. In that Figure and the associated discussion, he discusses prey population regulation. A type II response generates a per capita prey death rate decelerating with prey density, for constant predator numbers, while a type I keeps the per capita death rate steady, at low densities at least – hence the difference between the two, one is more prone to generate limitation/regulation of the prey (if predator numbers are increasing with prey density as well, you get prey regulation at low prey density, even with a type I).
As a sidenote, several authors found good reasons for a fully constant intake rate that does not change with prey density, which somehow escapes the above classification. Sabelis et al., for mites, call it the “pancake” model. With a bit of stochasticity on top, this might be an interesting alternative model.
That’s the type-0!
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