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OUP CORRECTED PROOF – FINAL, 25/6/2021, SPi
Predator Ecology
Predator Ecology
Evolutionary Ecology of the
Functional Response
John P. DeLong
School of Biological Sciences, University of Nebraska-Lincoln
and Cedar Point Biological Station, USA
1
3
Great Clarendon Street, Oxford, OX2 6DP,
United Kingdom
Oxford University Press is a department of the University of Oxford.
It furthers the University’s objective of excellence in research, scholarship,
and education by publishing worldwide. Oxford is a registered trade mark of
Oxford University Press in the UK and in certain other countries
© John P. DeLong 2021
The moral rights of the author have been asserted
First Edition published in 2021
Impression: 1
All rights reserved. No part of this publication may be reproduced, stored in
a retrieval system, or transmitted, in any form or by any means, without the
prior permission in writing of Oxford University Press, or as expressly permitted
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above should be sent to the Rights Department, Oxford University Press, at the
address above
You must not circulate this work in any other form
and you must impose this same condition on any acquirer
Published in the United States of America by Oxford University Press
198 Madison Avenue, New York, NY 10016, United States of America
British Library Cataloguing in Publication Data
Data available
Library of Congress Control Number: 2021937953
ISBN 978–0–19–289550–9 (hbk.)
ISBN 978–0–19–289551–6 (pbk.)
DOI: 10.1093/oso/9780192895509.001.0001
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CPI Group (UK) Ltd, Croydon, CR0 4YY
Links to third party websites are provided by Oxford in good faith and
for information only. Oxford disclaims any responsibility for the materials
contained in any third party website referenced in this work.
Contents
Prologue vii
1. Introduction 1
1.1 Functional responses and food webs 3
2. The Basics and Origin of Functional Response Models 9
2.1 Types of functional responses 9
2.2 Predator dependence of the functional response 18
2.3 Relationship to alternative formulations in aquatic literature 20
2.4 The Rogers Random Predator equation 21
3. What Causes Variation in Functional Response Parameters? 27
3.1 Variation in functional response parameters 27
3.2 Breaking down the space clearance rate 29
3.3 Factors affecting space clearance rate 32
3.4 Breaking down the handling time 40
3.5 Limits on the parameter space 42
3.6 Other predators 44
4. Population Dynamics and the Functional Response 47
4.1 The functional response as a trophic link 47
4.2 Adding some complexity 48
5. Multi-species Functional Responses 55
5.1 The need for MSFRs 55
5.2 Extending the functional response to multiple prey types 58
5.3 An example with damselfly naiads 61
6. Selection on Functional Response Parameters 65
6.1 Why functional response parameters might change through evolution 65
6.2 A dynamic tug-of-war 68
6.3 Temporal variation in the strength of selection 70
6.4 Traits linked to functional response parameters 74
6.5 Links among predator–prey model parameters 75
7. Optimal Foraging 79
7.1 Picking prey types to increase fitness 79
7.2 Deriving the standard prey model optimal foraging rule 80
7.3 OFT remains useful and needs further testing 84
vi Contents
8. Detecting Prey Preferences and Prey Switching 89

8.1 Prey selection in the presence of alternate prey 89
8.2 Detecting prey switching 90
8.3 Null expectations from the functional response 92
8.4 Null expectations for Manly’s α 97
9. Origin of the Type III Functional Response 101
9.1 What generates a type III functional response? 101
9.2 Concerns about the standard type III model 106
9.3 An alternative type III model 108
10. Statistical Issues in the Estimation of Functional Responses 115
10.1 Curve fitting 115
10.2 Noise and the nature of foraging trial data 118
10.3 Differences between parameters 125
10.4 Type II or type III? 127
11. Challenges for the Future of Functional Response Research 133
11.1 MSFRs 133
11.2 Sources of variation in parameters and constraints 135
11.3 Functional response models 138
11.4 Linking functional responses from foragers to communities 140
11.5 Accounting for time spent on other activities 143
Epilogue 145
References 147
Index 166
Prologue
The motivation for this book was three-fold. First, I personally wanted to
learn more about functional responses. I found, however, that information
about functional responses in the literature is piecemeal. In no place could
I find a synthesis about them, despite the existence of thousands of papers
describing or parameterizing functional responses for all manner of predators
and prey. Second, there was clear conflict in the literature about what models
to use to describe functional responses, the biological meaning of the model
parameters, and why functional responses vary among predator–prey pairs
and across environmental or trait-based gradients.
Third, and perhaps most importantly, the functional response became the
core concept for my field-based course called Predator Ecology. I wanted to
provide my students with an overview of the fundamentals and the biological
relevance of functional responses, so that in short order they could inter-
pret papers, conduct their own experiments, and grasp how natural selec-
tion might be shaping predator–prey interactions and therefore food webs.
I needed to start synthesizing for the course, resolve conflicts in terminology
and models, and help students connect the math to the biological reality of
nature. That was the birth of this book.
So for me and anyone else, this book covers the fundamentals and then
offers a deep dive into what functional responses really are, how to think
about them, why they are relevant to pretty much anything ecological, and
where studies on functional responses might go in the future. This book is
what I needed when I started teaching the Predator Ecology class. This book
is intended for advanced undergraduate students and graduate students, as
well as anyone interested in functional responses. The book moves between
simple introductions, derivations of the core models, reinterpretations and
clarifications of the parameters and the functions themselves, and novel
hypotheses about functional responses and their consequences. For anyone
mostly interested in the concepts and biological relevance, it may be useful to
skip over some of the derivations and focus on the biological meaning of func-
tional responses and their parameters. Then come back to the equations later.
To support hands-on learning as well as new research into functional
responses, the book is accompanied by a full set of code to reproduce all
data and analysis-based figures in the book. This code is written for Matlab
viii Prologue
© but could be translated to other scientific programming languages. The

code, and associated data as necessary, are hosted in a zipped folder at
www.oup.com/companion/DeLongPE. Any corrections or updates to the
code will be posted at this site.
This book would not have been possible without the patience and support
of all the people who have taken and TA’d my Predator Ecology class both on
City Campus at The University of Nebraska-Lincoln and out at Cedar Point
Biological Station. Their involvement has helped to keep the momentum on
my predator ecology research going. I appreciate the helpful comments on
drafts of this manuscript from Stella Uiterwaal, Kyle Coblentz, and Mark
Novak. I could not have understood the root sum of squares expression (see
Chapter 3) without the help of Van Savage. I’d like to give a shout-out to
HawkWatch International (www.hawkwatch.org), where I got my start in
predator ecology shortly after college. Finally, none of what I do would make
sense without the love and support of Jess, Ben, and Pearl, who make my heart
soar like a hawk.
1
Introduction
Predators seem to be universally fascinating. Maybe that is because we

humans are predators, or because we can be prey for other predators
(Quammen, 2004). Maybe we are just morbidly curious about death.
Whatever the reason, I have noticed that nature shows tend to focus a lot
on predator–prey interactions: the drama of the predator’s hunt or the relief
of the prey’s escape. We are drawn to predation and intuitively understand
that it is a fundamental part of nature. Indeed, what a predator eats is often
among the first things that we learn about it, suggesting that who eats whom
is among the most central features of ecological systems, or at least central to
the way we imagine them (Sih and Christensen, 2001).
Predation is fundamental beyond the event of a predator capturing prey.
The rate of predation and the identity of the prey combine to direct the
flow of energy through ecological communities. As a result, predation plays
a key role in structuring food webs. Of course, there are other ways that
energy flows through communities that do not involve predation, such as
photosynthesis, herbivory, parasitism, decomposition, and the consumption
of detritus or nectar. These are all equally crucial, but consuming other
organisms is a widespread way of getting that energy, so understanding the
rate at which predators consume prey is a necessary part of understanding
ecological systems.
So what controls the rate at which predators consume prey? Many things.
Predator traits like claws, prey defenses like camouflage, habitat complexity,
hunger, and the presence or behavior of other predators all play their part.
Among these many influences, one of the biggest factors is the number of prey
available to be consumed. Generally, predators have a higher foraging rate
when there are more prey to be had—up to a point. The relationship between
foraging rate and prey abundance (or density) is known as the functional
response (Holling, 1959; Solomon, 1949) (Figure 1.1).
The functional response is a description of how many prey a predator would
be expected to eat given a particular amount of prey available to the predator,
wherever they are searching for food. That expected number depends on
the behavior and morphology of both predator and prey in the context of
Predator Ecology: Evolutionary Ecology of the Functional Response. John P. DeLong, Oxford University Press.
© John P. DeLong 2021. DOI: 10.1093/oso/9780192895509.003.0001
2 Predator Ecology
Steep/high functional response

Prey consumed or foraging rate
Shallow/low functional response
No prey, no predation
Prey abundance or density
Figure 1.1. Some generalized functional responses. Foraging rates must be zero when
prey are absent, so functional responses are anchored at the origin. The curves
increase as prey increases, but the shape of that increase, the presence of an
asymptote, the overall height, and possible bends in the curve all depend on the
specific conditions, morphologies, and behaviors of the predator and prey involved.
a particular habitat, so the functional response is really an emergent property

of the total foraging process (Juliano, 2001). The functional response is always
anchored at the origin—no prey, no predation (Figure 1.1). There also is
always some increase in foraging rate as prey increases. The shape of that
increase, and both an explanation for and description of that shape, depend
on many factors. These factors relate to how predator and prey move and
encounter each other, how predators choose what to attack, how good prey
are at escaping, and what other things predators have to do with their time
besides hunting.
Functional responses have a long history in the scientific literature (Jeschke
et al., 2002). On the empirical side, there are well over 2,000 functional
responses measured for hundreds of predator–prey pairs. These functional
responses have been digitized and archived in the recent FoRAGE database
(Functional Responses from Around the Globe in all Ecosystems) (Uiterwaal
et al., 2018). These measured functional responses cover numerous taxa from
Introduction 3
unicellular foragers of bacteria and phytoplankton (Roberts et al., 2010) to

wolves (Jost et al., 2005). A key goal of this literature is understanding what
determines the shape of the functional response for any given predator–prey
interaction.
On the mathematical side, many models have been proposed to describe
functional responses (Akcakaya et al., 1995; Beddington, 1975; Crowley
and Martin, 1989; DeAngelis et al., 1975; Denny, 2014; Hassell and
Varley, 1969; Holling, 1959; Skalski and Gilliam, 2001; Tyutyunov et al.,
2008; Vallina et al.,2014). With these models, functional responses have been
used to predict prey selection (Charnov, 1976; Chesson, 1983; Cock, 1978)
and invoked as building blocks of community and food web models (Boit
et al., 2012; Brose et al., 2006; Guzman and Srivastava, 2019; Petchey
et al., 2008; Rall et al., 2008; Rojo and Salazar, 2010). These different forms
of functional responses have been argued over and compared for their
implications and construction (Abrams, 1994; Abrams and Ginzburg, 2000;
Houck and Strauss, 1985; Skalski and Gilliam, 2001). Yet, we still do not have
resolution about what mathematical expressions in functional responses are
most useful, most biologically meaningful, or most general.
1.1 Functional responses and food webs
Even with the vast amount of work already conducted about functional
responses, I would argue that we have only just begun to understand the
causes and consequences of quantitative variation in functional responses.
This limited understanding is true in general, but it is even truer when it
comes to functional responses as they occur in the field. Ecological commu-
nities contain numerous species and uncountable numbers of individuals.
The number of predator–prey interactions in even a relatively simple food
web can run into the hundreds. In a typical cartoon food web (Figure 1.2),
just a handful of species leads to numerous pairwise foraging interactions
among species. Of course there are far more species in a real food web than
shown in the cartoon, and there may be separate functional responses for
each type of insect eaten by the shrews or for each type of grass eaten by
the caterpillars. We typically draw such feeding interactions with an arrow
pointing toward the forager. Thus, herbivores also have functional responses,
as do all the interactions leading up to the top predator, in this case a barn
owl (Tyto alba). For this reason, all of what follows in this book may apply
to organisms consuming plants in part or in whole (e.g., algivores) as well
as the carnivorous mammals that often come to mind when we think of
4 Predator Ecology
Figure 1.2. A simplified food web. Functional responses characterize the

prey-dependent foraging rate for all predators in the food web including the
herbivores, not just the top predators. In this way, functional responses set the rate of
energy flow in food webs from plants to apex predators.
predators. Predators may even forage on biological entities that cause disease,
such as viruses and other parasites (Anderson et al., 1978; Thieltges et al.,
2013; Welsh et al., 2020), expanding the relevance of functional response
beyond the typical food chains described in ecology textbooks. All of these
interactions can be described by functional responses, and chances are they
are mostly quantitatively different from each other (Jeschke et al., 2002). They
may even be quantitatively different among individual predators within a
population (Hartley et al., 2019; Schröder et al., 2016; Siddiqui et al.,2015).
Yet, to my knowledge, there are no cases where the functional responses of
Introduction 5
all the connected species in a food web have been characterized completely,
although researchers have done more to estimate strengths of interaction
among species in food webs in other ways (Gilbert et al., 2014; Wootton and
Emmerson, 2005). Thus, we may have long lists of prey types for many preda-
tors in food webs—who eats whom—but we generally do not know the rate
of predation on most if any of those prey types. Without this information, we
are typically restricted to making taxonomic or body size-based assumptions
about the shape and height of functional responses in food web models (Boit
et al., 2012; Brose et al., 2006; DeLong et al., 2015; Rojo and Salazar, 2010;
Schneider et al., 2012).
Because functional responses determine the flow of energy from lower to
higher trophic levels (Figure 1.2), they strongly influence rates of birth and
death and consequently the size of predator and prey populations. Variation
in the shape of a functional response therefore can have crucial effects on the
dynamics of populations. For example, a steep and high functional response
(Figure 1.1) means a lot of foraging and a chance that the predator can over-
exploit the prey, leading to instability or population cycles (see Chapter 4).
Thus, the shape of the functional response is important. Even slight changes
to functional responses can lead to big changes in the population dynamics
of trophically interacting species (Chapter 4) (Oaten and Murdoch, 1975;
Williams and Martinez, 2004). Overall, it is thought that functional responses
should be on the shallow side in nature, meaning that no particular predator–
prey interaction dominates all of the energy flow (McCann et al., 1998). Such
“weak” interactions minimize overconsumption and encourage persistence of
predator and prey populations. We will see in Chapter 6, however, that natural
selection is not likely to favor shallow functional responses for predators
because shallow functional responses limit energy uptake. Understanding
what pulls functional responses up or pushes them down is a central problem
in the evolutionary ecology of predator–prey interactions and food webs
as whole.
Because of their effect on population dynamics, functional responses play
a role in virtually all types of community dynamics (Houck and Strauss, 1985;
Rosenbaum and Rall, 2018), including trophic cascades (DeLong et al., 2015;
Levi and Wilmers, 2012; Ripple and Beschta, 2012; Schneider et al., 2012),
predator–prey cycles (Jost and Arditi, 2001; Korpimäki et al., 2004; Krebs
et al.,2001; Oli, 2003), and keystone predation (Paine, 1966). More than those
dynamics, predation plays a role in determining the magnitude of ecosystem
functions that many prey organisms conduct (Curtsdotter et al., 2019; Koltz
et al., 2018; Wilmers et al., 2012). Predation is also dependent on temperature
(Burnside et al., 2014; Dell et al., 2014), and so the impact of climate change
6 Predator Ecology
on ecological systems will in part stem from how changes in temperature and
precipitation alter the steepness and height of functional responses (Binzer
et al., 2012; Boukal et al., 2019; DeLong and Lyon, 2020). Indeed, it is clear
that temperature itself is a major driver of the shape of functional responses
(Daugaard et al., 2019; DeLong and Lyon, 2020; Englund et al., 2011; Islam
et al., 2020; Uiterwaal and DeLong, 2020; Uszko et al., 2017), indicating that
functional responses will play a key role in determining the effects of climate
change on ecological communities.
The functional response also plays a key role in species conservation and
management. For example, introduced predators are generally thought to
have steep and high functional responses in their invaded range (Alexander
et al., 2014). This high functional response may arise because of a mismatch
between the offenses of the novel predator and the defenses of the potentially
naïve prey, giving predators the edge in foraging interactions and a chance
to survive in their new communities. Without that boost in foraging, out-of-
place predators might not be able to persist and invade, since the functional
response controls the energy flow to the predator and thus its population
size. This hypothesized tendency may be one reason invasive predators,
when they do establish, can be particularly destructive to a wide range of
prey (Doherty et al., 2016). Similarly, biocontrol predators are released to
control the populations of problematic insects, including agricultural pests
and infectious disease vectors (Saha et al., 2012; Tenhumberg, 1995). An ideal
biocontrol predator is one with a high functional response on the target pest,
as a shallow functional response implies little mortality of the pest (Lam
et al., 2021; Monagan et al., 2017). Indeed, we can even use estimates of
the functional response to show that other pest control strategies such as
insecticides can alter the effectiveness of the predator (Butt et al., 2019).
Similarly, the functional response influences how well microbial predators
and zooplankton can control toxic red tides in marine systems (Jeong et al.,
2003; Kim and Jeong, 2004). Finally, humans compete with non-human
predators for prey such as game birds. Understanding the functional response
can be a crucial part of devising management strategies for native predators
that maintain the availability of harvested prey species (Smout et al., 2010). In
fisheries, the functional response is needed for understanding and managing
both harvested and unharvested fish populations, as the functional response
influences the ability of fish populations to grow (Hunsicker et al., 2011).
The generally positive slope of functional responses also suggests that any
behaviors a predator can use that can increase prey density can increase for-
aging rates and thus individual fitness and population growth. For example,
northern shrikes (Lanius borealis) have been observed singing in winter, with
Introduction 7
the effect of drawing in potential passerine prey, raising prey density and
thus foraging rates (Atkinson, 1997). Small forest falcons also may use calls
for luring prey with the same effect (Smith, 1969). In addition, searching
behaviors or nomadism would allow predators to increase foraging rates by
actively locating areas (local patches) that contain many prey (Korpimäki
and Norrdahl, 1991). Thus, movement behavior can allow predators to alle-
viate the constraint of the functional response, sometimes referred to as a
“numerical” response, but this will not be a focus of this book.
In short, the functional response describes an essential process in ecology,
with ramifications for nearly every aspect of natural systems and our ability
to manage them for ecosystem services, pest control, or conservation. The
remainder of this book therefore takes a deep dive into functional responses.
Chapter 2 covers the mathematical basis of functional responses, shows the
derivation and meaning of the standard models, and standardizes terminol-
ogy. Chapter 3 describes known variation in functional responses within
and across species. Chapter 4 shows how variation in functional responses
influences population dynamics. Chapter 5 expands the discussion to multi-
species functional responses, which are the functional responses of a predator
feeding on more than one prey type. Chapter 6 shows why and how natural
selection should act on traits that influence functional response parameters.
Chapter 7 describes optimal foraging theory in light of functional responses
and how we still do not understand foraging strategies (optimal or otherwise)
with respect to fitness. Chapter 8 introduces the idea of prey switching and
shows why functional responses are necessary for understanding or even
identifying prey preferences. Chapter 9 discusses the many possible origins of
sigmoidal functional responses. Chapter 10 discusses the nature of functional
response data and reviews statistical concerns in curve fitting. Chapter 11
suggests critical areas of research needed to understand more fully functional
responses and their consequences for ecological communities.
2
The Basics and Origin of Functional
Response Models
This chapter is the essential beginner’s guide to the functional response, its
derivation, the various forms, its connection to other models in the literature,
and what the parameters mean. It is the ground floor for the rest of the book.
Surprisingly, our understanding of the functional response as presented in
the literature is quite muddled, with confusion ranging from the terminology
used, to the various mathematical forms the functional response takes, to the
biological interpretation of functional response model parameters. I provide
a summary and forward-looking perspective on these issues.
2.1 Types of functional responses
Functional responses are curves relating a predator’s foraging rate on a specific

prey to the availability of that prey (Denny, 2014; Holling, 1959, 1965; Juliano,
2001; Solomon, 1949) (Figure 2.1). The foraging rate must be zero when there
is nothing to eat, so functional responses are anchored at the origin and have a
positive slope at low prey density. As prey density1 increases to higher density,
the functional response usually does one of three things: it may continue to
rise linearly (this is called a type I, linear functional response; Figure 2.1A); it
may approach an asymptote (type II, saturating functional response; Figure
2.1B); or it may increase very slowly at first, then increase more quickly, and
eventually approach an asymptote (type III, sigmoidal functional response;
Figure 2.1C). Although it may be that functional responses show more
variation than these three types (Abrams, 1982), and numerous alternative
formulations have been derived for various purposes (Gentleman et al., 2003;
Jeschke et al., 2002; Tyutyunov et al., 2008), this simple classification is still in
1 In this book, predator and prey numbers will be presented as either abundances (just a number in a
space) or as density (a number per space). Although sometimes they are presented in units of biomass, it
is important to remember that predator and prey are individual organisms, and the impact of predation
on population dynamics and evolution occurs through the gain or loss of individuals.
10 Predator Ecology
Type I Type II
200 50
Anchovies consumed d–1

(A) (B)
Rotifers consumed h–1

40
150
30
100
20
50
10
0 0
0 5,000 10,000 0 50 100 150 200 250
Anchovies m–3 Rotifers 40 mL–1
Type III Type IV
15 150
(C) (D)
Aphids consumed d–1
Ants consumed d–1

10 100
5 50
0 0
0 5 10 15 0 1 2 3
Aphids cm–2 Ants m–2 × 104
Figure 2.1 (A) A type I functional response for the comb jelly (Mnemiopsis leidyi)
consuming anchovies (Anchoa mitchilli), fit to equation (2.1). Data from Monteleone
and Duguay (1988). It is possible that this functional response is type I, but it also may
be just incompletely determined. (B) A type II functional response for the copepod
Mesocyclops pehpeiensis foraging on the rotifer Brachionus rubens, data from Sarma
et al. (2013) and fit to equation (2.5). (C) A type III functional response for the ladybird
beetle Eriopis connexa feeding on the aphid Macrosiphum euphorbiae, data from
Sarmento et al. (2007) and fit to equation (2.7). (D) A type IV functional response of the
spider (Zodarion rubidum) foraging on ants (Tetramorium caespitum) and fit to
equation (2.9). Data from Líznarová and Pekár (2013).
wide use and covers much of the ground necessary to understand functional
responses and what they represent (Real, 1977). Beyond the three main
forms, some suggest that there is also a type IV functional response (Figure
2.1D), with the foraging rate dropping again at very high prey density, but
this form has not been documented many times (Baek, 2010; Jeschke et al.,
2004). Although functional responses also exist for herbivores (Andersen and
Saether, 1992), this book will focus on the functional responses of predators,
that is, consumers that kill their prey, even if they only eat part of the prey.
The main reason for making this choice is that by not killing the plant, an
herbivore has a fundamentally different effect on the population dynamics
of its plant resources. Functional responses also exist for parasitoids that
The Basics and Origin of Functional Response Models 11
stun prey and lay eggs in their prey, but these will similarly be less of a
focus here.
Typically, we estimate the shape of a functional response experimentally.
The usual approach is to measure foraging rates with a series of foraging
trials in which an individual predator forages on some arbitrarily assigned
number of prey for some prespecified amount of time. The researcher then
counts the number of prey remaining and assigns the difference between the
prey offered and the prey remaining to predation, trying to account for the
possibility that some prey may have died during the experiment for reasons
other than predation. Then, the type of functional response can be determined
by fitting a model to the data with a variety of methods (see Chapter 10). Here,
a model is just an equation describing a process or pattern that we believe to
be relevant to our data—we will look at many of these shortly. For example, in
Figure 2.1 I fit models describing type I–IV functional responses to each data
set, and we can see that, with the right parameters, the equations describe the
shape of the data reasonably well. We will see later that there are other ways
to estimate a functional response model, but the vast majority of functional
responses have been estimated using a curve-fitting approach (Uiterwaal
et al., 2018). Given the importance of fitting functional response models
to the experimental data, and the importance of parameterized functional
responses for predicting population dynamics (see Chapter 4), it is important
to understand what the models are, where they come from, and how they
represent a simplification of the foraging process. The starting point here is
the type I functional response.
The type I functional response arises owing to (1) a foraging or defen-
sive behavior that does not change with prey density, (2) a predator–prey
encounter rate based on random contacts, and (3) the lack of a time cost for
the predators to deal with the prey they kill. The type I functional response is
analogous to chemical reactions, where the rate of a reaction is proportional
to the product of the concentration of the reactants (i.e., mass action). Not
all of the reactions that could occur among reactants will occur within a
given period, however, because they are distributed in space and it takes
time for molecules to come into contact. So this potential amount of reaction
is scaled by a reaction rate constant k. The rate of a chemical reaction,
A, between reactant R1 and reactant R2 is therefore A = k [R1 ] [R2 ], where
the brackets indicate concentration. For predator and prey, the mass action
term is the product of the predator density [C, consumer; think coyote] and
the prey density [R, resource; think roadrunner], or [C][R]. Mass action of
predator and prey represents literally all the possible encounters between each
predator in a population and each potential prey individual. As with reactants,
12 Predator Ecology
predator and prey are spread out in space, and it takes time for them to come
into contact. We therefore scale mass action of predator and prey by some rate
constant a, which here I will call the space clearance rate (see Box 2.1), and add
time spent by the predator searching for prey T s , to give the total amount of
foraging as F = aCRTs , where I have dropped the brackets for simplicity. In
the type I functional response, we assume that the only thing a predator does
with its time is search for food, so the total time available for a predator (T tot )
is equal to its search time (T s ), which means that we can update our equation
to F = aCRTtot . The functional response describes the per capita (i.e., per
individual) rate of foraging, f pc (in this book I will use a small f for foraging
rate and a capital F for number of prey individuals consumed). To get to the
foraging rate, we divide both sides of the equation F = aCRTtot by T tot , and
to get to per capita, we divide both sides of the equation by C. These two steps
yield the type I functional response:
fpc = aR (2.1)
Box 2.1. Why a should be called “space clearance rate.”
The functional response parameter a has many names. The terms attack efficiency,
attack rate or successful attack rate, attack constant, rate of successful search,
capture rate, maximum clearance rate, maximum per capita interaction strength,
instantaneous rate of discovery, rate of potential detection, and instantaneous search
rate have all been used as names for this parameter. These names, however, are not
good descriptions of the biological process captured by the parameter, as is clearly
[space]
evident from a unit analysis. The units of the parameter are . We can see
[pred][time]
this because the units of a line in the functional response space are the units of
[prey] [prey]
the rise ( ) over the units of the run ( ). These units clearly indicate that
[pred][time] [space]
[attacks]
the parameter is not a rate of attack, which would be . Nor is the parameter
[pred][time]
an efficiency, which are often unitless, as in the fraction of energy extracted from the
energy available in a fuel. Rather, by the units, the parameter a is the space (area or
volume) containing prey that is effectively cleared of prey by the predator per unit
of time. I formerly preferred the term area of capture for this parameter, as this is
close to the real meaning and has historical precedent in the area of discovery term
previously used for parasitoids (Hassell and Varley, 1969). In this book, however, I
advocate for renaming this parameter the space clearance rate, which is what the
units suggest is the biological process being captured.
It is sometimes thought that the type I functional response increases linearly

and eventually reaches an abrupt ceiling (occasionally called a “rectilinear”
shape), caused by the limits of gut capacity, but this ceiling is rarely observed,
difficult to distinguish from a gradual asymptote, and generally ignored in
practice (Denny, 2014; Fox, 2013; Jeschke et al., 2004). Jeschke et al. (2004)
suggested that in many cases without an observed ceiling (e.g., Monteleone
and Duguay, 1988), the functional response also could be viewed as incom-
pletely determined and therefore called a “linear” functional response. Only
when higher prey abundances are used would you then be able to determine
if the functional response increased linearly to a ceiling or began to taper off
to an asymptote.
The type II functional response is a modification of type I, where the
predator pays a time cost (called a handling time) when it captures prey. The
handling time includes the time needed for subduing, killing, consuming, and
digesting prey and then getting around to searching for more food again. This
time is subtracted from the searching time such that the more the predator
kills, the less time it searches. The derivation of the type II functional response
requires backing up a step from equation (2.1) to the number of prey captured
per capita (big F pc ) in search time T s :
Fpc = aRTs (2.2)
The traditional argument in the derivation of the type II functional response

is that predators must divide their time between searching for prey (T s ) and
handling prey (T h ), so the total time is Ttot = Ts + Th . The handling time cost
applies only for prey actually killed (F pc ), although it does in practice include
the time spent on unsuccessful captures per successful capture. Therefore,
the total time budget can be written as Ttot = Ts + Fpc h, where h is the
handling time per prey. Since F pc is already spelled out in equation (2.2), we
can substitute this into our time budget to get:
Ttot = Ts + aRTs h (2.3)
Then we divide the total amount of prey captured by the total time (equation
(2.2) for F pc divided by equation (2.3) for T tot ) and get:
Fpc aRTs
= fpc = (2.4)
Ttot Ts + aRTs h
14 Predator Ecology
Factoring out T s on the right-hand side gives the standard form of the type II
functional response:
aR
fpc = (2.5)
1 + ahR
The type II functional response is also called the Holling disc equation, after
C.S. Holling, who used students foraging for sandpaper discs to illustrate the
shape of the foraging curve. The type II functional response is the most widely
used functional response model.
The type II functional response is equivalent to the type I when h = 0.
Processing prey, however, can never truly take no time—there must always be
some time for digestion or managing prey—but that processing time might
not always cut into search time. As a result, type I functional responses
exist over at least some prey densities (Figure 2.1A), even though particular
examples of this type may be just incompletely determined. The derivation
of the type II functional response assumes that handling time and searching
time are mutually exclusive; that is, predators cannot continue to search for
additional prey while they are handling the prey they have already captured.
This turns out not to be completely true for all predators. For example, a filter-
feeding predator might continue right on filtering even as it clears prey from
the water and starts digesting it, so its handling time is zero and its functional
response is type I.
The best way to think of the handling time is the loss of searching time
owing to handling prey, as only when searching is interrupted does the
handling time cause the functional response to saturate, even if handling and
searching are not mutually exclusive. Thus, despite the name of the parameter,
predators do not have to be actively handling prey to incur handling time,
where active handling generally means that the predator is still manipulating,
biting, swallowing, or otherwise trying to ingest the prey. They only need
to experience a loss of searching time (Abrams, 2000). Different predators
may experience losses of searching time in different ways. For example, some
predators may be able to digest previous meals while searching for the next
one, in which case digestion per se would not be a component of handling
time. In other predators, such as some snakes, digestion may be so costly that
it could limit additional searching (Secor, 2008). As a result, behavioral identi-
fication of handling time as something you can see the predators doing often
represents only a portion of the handling time as defined in the derivation
of the functional response. Further, since predators are not 100% effective
in capturing prey, an estimated handling time from a functional response
experiment actually may include an amount of time wasted on unsuccessful

attacks per successful attack (Jeschke et al., 2002). Handling prey that are
not captured sounds like a violation of the assumptions; however, it can be
understood that the handling time for captured prey includes some average
number of failed attempts per captured prey.
If we do not factor out the T s from equation (2.4) as we did earlier, we
can write the type II functional response in a way that is more biologically
intuitive:
Ts Ts
fpc = aR = aR (2.6)
Ts + aRTs h Ttot
Equation (2.6) shows that the type II functional response is just the type I
Ts
model multiplied by the fraction of time actually spent foraging ( ). That
Ttot
is, if a predator spends 50% of its time searching for prey at some prey level, the
type II curve at that prey level will be half that predicted by the type I model.
The type II functional response approaches a maximum foraging rate of
1/h (asymptote in Figure 2.2A). This should make sense—a predator can only
consume one prey per hour if it takes an hour to handle the prey. The space
clearance rate can be seen on the functional response curve itself as the slope
of the curve as R → 0 (Figure 2.2A). This happens because as R gets very
small, the denominator approaches one, meaning that the curve collapses to
approximately type I very near the origin, where the equation is just that of a
line with a slope equal to the space clearance rate.
The type III functional response does not, to my knowledge, have a mecha-
nistic derivation. One simply puts an exponent (such as 𝜃; often called a Hill
exponent; Real, 1977) on the prey abundance, R𝜃 , transforming a rising and
saturating curve into one that has a sigmoidal shape (Figure 2.2B):
aRθ
fpc = (2.7)
1 + ahRθ
The type III functional response may be caused when predators avoid a
particular prey type when it is rare, but choose that prey more often when
it becomes more common. This prey-switching explanation is the most com-
monly invoked mechanism for generating a type III functional response, but it
is not the only possibility (see Chapters 8 and 9). It is worth pointing out that
the type III functional response is often drawn as bending up and reaching
the asymptote more slowly than a type II functional response (Denny, 2014).
This is not true when one simply adds a Hill exponent to a type II functional
16 Predator Ecology
Type II
25
(A)
Asymptote is 1/h
20
Slope is a
Foraging rate
15
10
0
0 10 20 30 40
Type III
25
(B)
20
Foraging rate
15
0.4
10
0.2
5
0
0 0.5
0
0 10 20 30 40
Prey density
Figure 2.2 How to see the parameters space clearance rate (a) and handling time (h) on
a functional response curve. (A) For the type II functional response, a is the slope of the
curve as it passes through the origin, and the functional response asymptotes at 1/h.
(B) For the type III functional response, the asymptote is the same as in the type II but
the space clearance rate is not easily visualized along the curve. The inset shows that
the curve rises more shallowly than a type II at first but then steepens to rise more
quickly. The type III curve also approaches the asymptote more quickly because the
effective a (aR𝜃) is less than a below R = 1 and greater than a above R = 1.
response while keeping the same space clearance rate, in which case the
type III curve crosses the type II curve and actually approaches the asymptote
more quickly (Figure 2.2B, inset). It should generally be the case, however,
that when estimating a functional response from data, the space clearance
rate will be smaller if a type III curve is used as a fitted model rather than a
type II curve (see Chapter 9).
As with the type II functional response, the saturation point on a type III
functional response is 1/h (Figure 2.2B). The location of the space clearance
rate on a type III functional response is much harder to visualize than on the
type II (see Chapter 9). It is helpful to realize that the Hill exponent is really
just a way to modify a such that predators do not clear much space when that
specific prey type is rare. That is, a can be a function of R (Juliano, 2001). The
typical way to do this is to rewrite a as an increasing function of prey density,
such as a = bRq , where q > 0 and the Hill exponent 𝜃 = q + 1, and substitute
this into equation (2.7):
(bRq ) R
fpc = (2.8)
1 + (bRq ) hR
Both the 𝜃 (equation (2.7)) and q (equation (2.8)) versions of the type
III functional response are used in the literature (Smout et al., 2010;
Vucic-Pestic et al., 2010).
The type IV functional response is the case when foraging rate declines at
high densities. That is, after prey levels have increased high enough for the
predator to approach the asymptotic foraging rate, further increases in prey
levels cause the foraging rate to decline again (Figure 2.1D). This effect could
be caused by predator confusion in the presence of a lot of prey or by the
risk to the predators caused by an increase in the number of dangerous prey
(Jeschke and Tollrian, 2005). This type of functional response appears to
be quite rare, and some have even suggested that it should be viewed as a
mathematical artifact (Morozov and Petrovskii, 2013). However, it has been
seen for the odonate predator Aeshna cyanea foraging on the cladoceran
Daphnia magna (Jeschke and Tollrian, 2005) and Zodarion spiders foraging
on a variety of ants, which may be related to the danger ants pose to their
predators in larger groups (Líznarová and Pekár, 2013). One way to model
this would be to propose that handling time increases as prey levels increase,
perhaps because the handling challenge is more difficult when faced with the
risk incurred from the presence of numerous uneaten prey. Thus, we could
simply add another exponent to prey levels in the denominator of the type II
functional response:
aR
fpc = (2.9)
1 + ahRg
As with the Hill exponent, here the g is just a phenomenological parameter

generating a particular effect rather than an easily interpretable biological
process.
18 Predator Ecology
2.2 Predator dependence of the functional response
All of the above functional response types are curves that link foraging rate
to one variable—prey density. Per capita foraging rates, however, usually also
decrease with the density of predators (Figure 2.3). This effect is often called
interference because each predator has a negative effect on the foraging rate of
all the other predators (DeLong and Vasseur, 2011; Hassell, 1971; Skalski and
Gilliam, 2001). The type II functional response can be modified in several
ways to accommodate interference competition (Novak and Stouffer, 2020;
Skalski and Gilliam, 2001; Tyutyunov et al., 2008). One common way is to
raise predator density to a power, called m for mutual interference, in much
the same way that 𝜃 was added to the functional response to create a type III
functional response (this is the Hassell–Varley form of interference) (Arditi
et al., 1991; Hassell, 1971):
aRCm
fpc = (2.10)
1 + ahRCm
50
40
Crickets eaten day–1
30
20
10
300
0 200
12
10
8 100
6
4 Crickets m–2
2 0
0
Spiders m–2
Figure 2.3 The foraging rate of the wolf spider Pardosa milvina foraging on the cricket
Acheta domesticus in response to both prey and predator density, replotted from the
data in Schmidt et al. (2014) and fit to equation (2.10). A saturating type II response can
be seen along the cricket axis where there is only one spider (dashed line), but the rate
of foraging declines at all prey densities as spider density increases, which is known as
predator interference.
The mutual interference exponent m is negative to indicate that adding more

predators reduces the foraging rate, but in principle the exponent could be
positive to indicate some mutually beneficial effect of more predators on
the average foraging rate, such as group foraging. As with the type III func-
tional response, what is happening here is that a is being made a decreasing
function of predator density, rather than an increasing function of prey
density as in the type III functional response:
(aCm ) R
fpc = (2.11)
1 + (aCm ) hR
The a in equation (2.10/2.11) is the same as the a in equation (2.5), because

equation (2.5) is just the special case of equation (2.10/2.11) when C = 1.
When m = −1, the functional response is dependent on the ratio of prey
to predator abundance, which is known as ratio-dependent (Akcakaya et al.,
1995). When m = 0, the functional response is prey-dependent, meaning only
dependent on prey levels and not dependent on predator levels. The ratio-
dependence idea stimulated some controversy about how important predator
density is to functional responses; for more on this, see Chapter 3.
Another common way of including interference in a type II functional
response is to add a “wasted time” term to the time budget (Beddington, 1975;
DeAngelis et al., 1975). This approach (the Beddington–DeAngelis form) has
a clearer derivation, where you simply add the amount of time wasted for
each predator–predator interaction. Specifically, we add the time wasted per
predator, w, times the number of predators minus one (so that there is no
interference when there is only one predator) to the time budget in equation
(2.3) and follow the same steps until you get:
aR
fpc = (2.12)
1 + w (C − 1) + ahR
These two models for predator interference (equations (2.11) and (2.12)) dif-
fer not just mathematically but in what the functions imply about the mech-
anism of interference. In the Hassell–Varley expression (equation (2.11)),
interference is caused by a reduction of the space clearance rate, which
might be caused by several behavioral changes in the predator that influence
their ability to move in space or acquire prey. In the Beddington–DeAngelis
expression (equation (2.12)), interference is caused by a loss of searching
time, meaning a change in the predator’s time budget. When fitted to data,
both models look very similar in practice, which may not be surprising
because both models describe the outcome of some cost to the predator
20 Predator Ecology
from interacting with other predators (Kratina et al., 2009; Skalski and
Gilliam, 2001). More detailed behavioral analyses are required to identify
the mechanisms generating predator interference and to guide the use of
specific interference models for any particular system. There is also no reason
to assume that all cases of interference are generated by the same mechanism,
so the two models presented here, or other models, might still be required to
properly describe the effect of predator abundance on foraging rates across
the different predators in food webs (Crowley and Martin, 1989; DeLong and
Vasseur, 2013; Skalski and Gilliam, 2001; Tyutyunov et al., 2008).
2.3 Relationship to alternative formulations

in aquatic literature
Many studies focused on aquatic invertebrates or microbes, as well as some

theoretical studies, use an alternative formulation of the functional response
known as a Michaelis–Menten equation:
Imax R
fpc = (2.13)
KR + R
where I max is the maximum ingestion rate and K R is the half-saturation

constant. This constant is the prey level at which the foraging rate reaches
half of its maximum value I max . What happens with this equation is that as R
gets very large, the relative effect of adding K R to R in the denominator gets
R
small. In other words, → 1 as R→ ∞. Thus, the asymptote is I max , which
KR +R
means I max = 1/h, just as before. You also can transform equation (2.5) into
equation (2.13) by multiplying the top and bottom by 1/ah (Fan and Petitt,
1994):
1
aR 1/ah R Imax R
h
fpc = = = (2.14)
1 + ahR 1/ah 1/ah + R KR + R
which also clarifies that the half-saturation constant KR = 1/ah . We will

not use this formulation in this book except once at the very end, because
the combined parameter K R reflects both searching and handling processes
and is therefore more difficult to understand and interpret biologically.
Nonetheless, the form continues to be used in the literature, perhaps
because of the convenience of having a maximum foraging rate parameter in
the model.
Another common rate in the aquatic literature is the “clearance rate,” which
is actually a little different than space clearance rate. This is calculated as the
foraging rate f pc divided by the prey levels. Returning to equation (2.6), we
can see that the clearance rate in the aquatic literature is equal to the product
of space clearance rate and the fraction of time spent searching:
Ts
aR
Ttot Ts
C= =a (2.15)
R Ttot
Thus, the difference between the space clearance rate that is a component of
the functional response (a) and the clearance rate in the aquatic literature is
that the former is a constant and the latter captures the effect of the time spent
handling (which increases with more prey) on the space clearance rate.
2.4 The Rogers Random Predator equation
An important alternative expression for the functional response arises from

accounting for the common experimental artifact of declining prey levels
during functional response experiments. A typical functional response exper-
imental set-up has an arena within which predators forage on some amount
of prey. As the predators forage, one could replace each consumed prey item
with a fresh prey item, thus keeping the prey levels constant. Doing this can
be quite disruptive to the predator or impractical, so in most cases the prey
simply decline in abundance through time as the predators consume them.
That means that with each prey captured, the expected foraging rate on the
new prey level declines, reducing the apparent foraging rate for the original
prey density. We can account for this declining prey density using what is
known as the Roger’s Random Predator equation (RRP; see Box 2.2 for the
derivation of equation (2.16); Rogers, 1972; Royama, 1971):
Re = Ro (1 − ea(hRe −t) ) (2.16)
In this equation, the foraging rates have been replaced by numbers of prey
offered (Ro ) and eaten (Re ), while a and h are still space clearance rate and
handling time, respectively, and the duration of the experiment is t. However,
because the RRP generally is used with the number of prey items offered,
not the number per unit area, the default units of space clearance rate in this
context are arenas (the physical space of the experimental area, regardless of
whether the arena is a surface or a volume) per predator per time.
22 Predator Ecology
Box 2.2. Derivation of the Rogers Random Predator equation.
We start with a differential equation that describes the rate of change in prey
abundance (R) through time (t) owing to predation:
dR aR
=−
dt 1 + ahR
This equation describes a rate of prey loss owing to predation described by a type II
functional response. If we integrate the equation over the course of an experiment
of duration t, we will get the cumulative loss of prey from the amount of prey offered
at the beginning, Ro , to the number of prey items left at the end of the experiment,
Rt . First we arrange the equation to aggregate R to the left-hand side and time to the
right-hand side:
(1 + ahR)
dR = −adt
R
and take the integrals from Ro to Rt and from 0 to t on the left-hand side and right-
hand side, respectively:
Rt R
t t R t
(1 + ahR) 1
∫ dR = ∫ dR + ∫ ahdR = ∫ − adt
Ro
R R
R R 0
o o
Integration yields:
ln Rt − ln Ro + ahRt − ahRo = −at
which we can rearrange as:
Rt
ln = −at − ahRt + ahRo
Ro
and then further to:
Rt
ln = a (−t − hRt + hRo )
Ro
Finally, exponentiating both sides followed by moving the Ro to the right-hand side
gives:
Rt = Ro ea(h(Ro −Rt )−t)

There are two tricks needed to get to the final equation. First, subtract each side
from both sides (i.e., switch sides and change sign):
−Ro ea(h(R0 −Rt )−t) = −Rt
Second, recognize that the number of prey actually eaten is the difference between
the number offered and the number that is left: Re = Ro − Rt . We therefore add Ro
to both sides:
Ro − Ro ea(h(Ro −Rt )−t) = Ro − Rt
Since the right-hand side is now the number of prey eaten, we just substitute and
factor out the Ro , bringing us to equation (2.16):
Re = Ro (1 − ea(hRe −t) )
Because Re occurs on both the left-hand side and right-hand side of

equation (2.16), it needs to be solved using numerical methods (i.e., searching
for combinations of parameters that satisfy the equation). More recently,
Bolker (2011) showed that the RRP can be simplified to isolate the Re on the
left-hand side and make fitting of the equation to functional response data
much easier:
W (ahR0 e−a(t−hR0 ) )
Re = Ro − (2.17)
ah
where W is the LambertW (or product log) function (see Box 2.3 for the
derivation of equation (2.17)). I will call equation (2.17) the Lambert Random
Predator equation. Remembering that a type III functional response is one
in which space clearance rate is a function of prey level, a = bRq , it might
seem that we can substitute this term into equation (2.17) to get the equivalent
type III formulation:
q )(t−hR )
W ((bRq ) hR0 e−(bR 0 )
Re = Ro − (2.18)
(bRq ) h
However, this is just an approximation and may produce slightly biased

estimates of the foraging rates (Bolker, 2011; Rosenbaum and Rall, 2018). For
more on type III functional responses, see Chapter 9.
24 Predator Ecology
Box 2.3. Derivation of the LambertW version of the Rogers

Random Predator equation.
Bolker (2011) starts with the original RRP (see text and Box 2.2) and does a bunch of
clever rearranging to get to a solution. Since only part of those steps is shown in the
original work, I will write out all of the steps here so I can remember them. Starting
with the original RRP:
Re = Ro (1 − ea(hRe −t) )
we can rearrange to get:
Re
1− = e−a(t−hRe )
Ro
Expand the exponential term:
Re
1− = e−at+ahRe = e−at eahRe
Ro
eahRo
The first trick is to multiply the right-hand side by and combine the first
eahRo
exponential term with the numerator and the second with the denominator:
Re eahRe
1− = (e−at eahRo ) ( ahR )
Ro e o
Now follow exponential rules and factor out:
Re
1− = (e−a(t−hRo ) ) (eah(Re −Ro ) )
Ro
Ro
The second trick is to multiply the exponent in the second exponential term by ,
Ro
which allows us to rearrange to:
Re R
ahR ( e −1)
1− = (e−a(t−hRo ) ) (e o R )
Ro
The third trick is to multiply both sides by ahRo and move the second exponential
term to the left-hand side:
Re R
[ahRo (1− e )]
[ahRo (1 − )] e Ro = ahRo (e−a(t−hRo ) )
Ro
Bolker did all these tricks to get to this special place. Once here, he offered a clever
solution to this arrangement by noticing that it takes the form of the product log,
which is that for some function y = xex , the solution is W(y) = x. In this example,
Re
x = ahRo (1 − ), and this means that the last equation can be written as:
Ro
Re
ahRo (1 − ) = W (ahRo (e−a(t−hRo ) ))
Ro
Now rewrite as:
ah (Ro − Re ) = W (ahRo (e−a(t−hRo ) ))
And with just a little more rearranging we get to equation (2.17):
W (ahRo (e−a(t−hRo ) ))
Re = Ro −
ah
3
What Causes Variation in Functional
Response Parameters?
The parameters of the functional response are not traits. They represent
processes such as hunting and digesting prey. Thus, all the traits that influ-
ence the way predators and prey encounter each other in space and the
morphologies and behaviors that influence capture, evasion, or digestion are
potential sources of variation in the functional response parameters. In this
chapter, I cover how we break the parameters down mathematically so that
the connection between the parameters and traits is more transparent.
3.1 Variation in functional response parameters
Functional response parameters can take on a wide range of values

(Figure 3.1). Across all kinds of predator and prey pairs represented in
the FoRAGE database (Functional Responses from Around the Globe
in all Ecosystems), variation in space clearance rate exceeds 15 orders of
magnitude, while variation in handling time exceeds 8 orders of magnitude.
This massive amount of variation in functional response parameters stems
from variation among individual predators as well as variation across pairs
of predator and prey species. Where does such variation come from?
Among the many possible sources of variation, body size of both predator
and prey seems to play a large role. There has been tremendous effort to
quantify how predator and prey body sizes influence functional response
parameters, both within and across species (Byström and García-Berthou,
1999; DeLong and Vasseur, 2012a; Gergs and Ratte, 2009; McCoy et al., 2011;
Miller et al., 1992; Rall et al., 2011, 2012; Rindorf and Gislason, 2005; Schröder
et al., 2016; Spitze, 1985; Uiterwaal et al., 2017; Uiterwaal and DeLong, 2018;
Vucic-Pestic et al., 2010; Weterings et al., 2015). Even when accounting for
body size, however, substantial variation in functional response parame-
ters remains. This additional variation can be linked to taxonomic identity
28 Predator Ecology
2D foragers
102
(A)
100
Handling time (days)
10–2
10–4
10–6
10–10 10–5 100 105
Space clearance rate (m2 pred–1 day–1)
3D foragers
105
(B)
Handling time (days)
100
10–5
10–10
10–10 10–5 100 105
Space clearance rate (m3 pred–1 day–1)
Figure 3.1 Pairs of space clearance rate and handling times for >2,000 predator–prey
pairs of all types of species from around the world. Data are from the global
compilation of functional responses called FoRAGE (Uiterwaal et al., 2018). Because the
units of space clearance rate depend on the actual type of space being used, the data
are plotted separately for two-dimensional interactions (A) and three-dimensional
interactions (B).
(Kalinoski and DeLong, 2016; Rall et al., 2012; Uiterwaal and DeLong, 2018,
2020), temperature (Bergman, 1987; Dell et al., 2014; Englund et al., 2011;
Rall et al., 2012; Uiterwaal and DeLong, 2020; Uszko et al., 2017), age, or
instar, which also may reflect variation in body size or energetic demands
(Gergs and Ratte, 2009; Hassanpour et al., 2011; Houde and Schekter, 1980;
McArdle and Lawton, 1979; McCoy et al., 2011; Spitze, 1985; Uiterwaal and
What Causes Variation in Functional Response Parameters? 29
DeLong, 2018; van den Bosch and Santer, 1993; Yaar and Özger, 2005),
sex (Walker and Rypstra, 2002), habitat complexity (Alexander et al., 2015;
Barrios-O’Neill et al., 2015; Cuthbert et al., 2019; Koski and Johnson, 2002;
Kreuzinger-Janik et al., 2019; Nilsen et al., 2009; Ware, 1972), aggressiveness
or choosiness about prey types (Michalko et al., 2017), individual variation in
dominance (Hartley et al., 2019), and nutritional state or energetic condition
(i.e., “hunger”) (Li et al., 2018; Lyon et al., 2018; McMahon and Rigler, 1965;
Nandini and Sarma, 1999; Schmidt et al., 2012).
Understanding how this variation arises and how specific traits or processes
mechanistically influence functional response parameters requires some dis-
section of the processes that generate predation events in the first place.
As with most model parameters in biology, the parameters of the functional
response are composed of other parameters. For example, the intrinsic rate
of population growth is a parameter r, but it is also the difference between
two other parameters—the rate of births b and the rate of deaths d. Thus,
to understand the space clearance rate and the handling time parameters,
we must take them apart and look at what other parameters combine to
produce them. Breaking down the parameters into their component parts will
help us understand the foraging mechanisms that lead to specific functional
response parameter values, why there is so much variation in parameters, and
how evolution that acts on traits linked to foraging may be mediated by the
consequences of the functional response (Denny, 2014; Jeschke et al., 2002;
Livdahl, 1979).
3.2 Breaking down the space clearance rate
The space clearance rate is the fundamental foraging parameter, as it

determines how fast predators can forage when they pay no time costs for
handling prey. Successful foraging requires the predator to encounter, attack,
and successfully kill prey (Endler, 1986; Gentleman et al., 2003; Roberts et al.,
2010; Sih and Christensen, 2001). We can use these three steps to break down
the space clearance rate into its component parameters. First, encounters
arise owing to the movement of predator and prey in space and are thus
proportional to the root sum of square of the velocity of the predator (V c )
and the prey (V r ): √V2c + V2r (Aljetlawi et al., 2004; Pawar et al., 2012) (for
derivation of the root sum of squares expression, see Box 3.1). The rate of
encounters is also proportional to the amount of prey in the environment,
so as either the predator or the prey increase in speed, or prey become
30 Predator Ecology
Box 3.1. Why the root sum of squares?
The root sum of squares expression (√V2c + V2r ) arises from recognizing that the
⇀
relative velocities (Vrelative ) of two objects determine the rate at which they collide.
Going back to a classic math problem as an example, the relative velocity of one
car passing another is the difference in their velocities. That is, the fast car is going
⇀ ⇀ ⇀
past the slow car at relative velocity Vrelative = Vfast − Vslow . If the fast car was not
moving, the slow car would be moving away at its own velocity, and if the slow
car stopped moving, the fast car would pass it at its own velocity. By analogy, gas
molecule-like predators (C) and prey (R) move past each other at a relative velocity
⇀ ⇀ ⇀
of Vrelative = VC − VR . The above root sum of squares expression comes from taking
the average of the relative velocities of all the predator and prey individuals (Pawar
et al., 2012). Because both predators and prey are on average not moving anywhere
(i.e., the displacement of all the different individuals in different directions sums to
zero), we can ignore their starting points and focus on the mean relative velocity.
And because the angle separating their velocities is as likely to be 90∘ as anything
else, we can use the Pythagorean Theorem to calculate the average relative velocity
as the hypotenuse of the triangle formed by the two velocities moving away from
each other from the same point and at 90∘ :
⇀ ⇀ 2 ⇀ 2
Vrelative = √VC + VR
which is written in equation (3.2) without the arrows for convenience. Note that
key features of different kinds of predators correctly map onto this expression.
The relative velocity of a sit-and-wait predator is that of the prey, and the relative
velocity of a predator foraging on sessile prey is that of the predator. That outcome
would not occur if we used the arithmetic mean of the two velocities.
more abundant, the encounters between predators and prey will increase.
Increasing predator abundance also increases encounters, but remember the
functional response is a per capita expression. Second, an actual encounter
arises only if predators are aware of the prey item, so we include a term
for the distance, d, at which a predator can detect prey. Detection will be a
length if the predator is searching in two dimensions and an area for predators
searching in three dimensions (Pawar et al., 2012). Importantly, this detection
distance is the distance that a predator can perceive a prey, not the maximum
distance that it can perceive anything. For example, a lion may be able to see
a zebra at a great distance, but it may not be able to distinguish it from the tall
grass until it is much closer. Third, after encountering the prey, the predator
must choose to attack the prey, and it does so with some probability, pa .
Finally, the predator has some chance of actually being successful given that
it has attacked the prey, ps . Putting this together, with the events transpiring
from right to left, the space clearance rate is:
a = ps pa d√V2c + V2r (3.1)
Equation (3.1) shows how the foraging process leads to the emergent param-
eter space clearance rate, but the equation also helps to clarify why the units
[space]
of space clearance rate come out as . The units of the velocities are
[pred][time]
[length]
(the distance traveled by an individual predator or prey in a unit of
[time]
[length]
time), the detection distance is either for a two-dimensional forager
[pred]
2
[length]
or for a three-dimensional forager, and the probabilities are unitless.
[pred]
Multiplying them together as in equation (3.1) makes the units of space
[area] [volume]
clearance rate for a two-dimensional forager or for a three-
[pred][time] [pred][time]
dimensional forager. We can then generalize the units of the numerator as
“space.” It is worth noting here that some sources suggest the space clearance
rate represents a preference for a particular prey type (Heong et al., 1991;
Smout et al., 2013). Preference for a prey type may influence space clearance
rate by influencing the probability of attack, but the space clearance rate itself
reflects preferences along with all of the other factors influencing predation
events, so space clearance rate should not be thought of as only reflecting
preference.
Equation (3.1) also can help clarify why space clearance rate, encounter
rate, attack rate, and foraging rate are not the same things, even though many
papers refer to space clearance rate as “attack rate” (Figure 3.2). Space clear-
ance rate is a property of the interaction between predator and prey that tells
us how effectively a predator clears searched space of their prey. Successful
foraging depends on encounters between predators and prey, which is the
product of prey density and the rate at which space is perused for prey, so
encounter rate E = d√V2c + V2r R. Some portion of those encountered prey are
attacked, giving the actual attack rate as A = pa d√V2c + V2r R. Some portion of
32 Predator Ecology
pspad Vc2 + Vc2 R
Space clearance rate
Encounter rate
Attack rate
Foraging rate
Figure 3.2 How the parameters involved in determining the space clearance rate are
repackaged into encounter rate, attack rate, and foraging rate.
these attacked prey are successfully predated, giving the total foraging rate for
a type I functional response (with no handling time) as:
fpc = ps pa d√V2c + V2r R = aR (3.2)
One of the useful things about equation (3.2) is that it is now easy to
imagine specific traits of predators or prey that could influence the foraging
steps and thus influence the parameter itself. Traits associated with speed of
movement, such as limb lengths, swimming ability, activity level, or body
size, can influence encounter rates. For example, smaller crabs (Panopeus
herbstii) foraging on mussels (Brachidontes exustus) that were more active
had a higher functional response than less active crabs, presumably through
an effect on searching velocity (Toscano and Griffen, 2014). Traits associated
with information gathering such as sight, smell, and the ability to sense
vibrations can all influence the detection distance. The probabilities of attack
can be linked to a predator’s interest in a type of prey, set possibly by potential
energetic or nutritional reward as food or the cost to the predator through
prey defenses such as toxins or spines. The probability of capture success
could likewise be influenced by traits that facilitate or disrupt capture, such
as acceleration in the predator or prey, grasping structures, or again, prey
defenses.
3.3 Factors affecting space clearance rate
All foraging activity occurs in physical space characterized by habitat and

dimensionality (e.g., a flat plane, the branching structure of a plant, the open
ocean, or the air). Functional responses therefore reflect the structure and
complexity of the foraging habitat that constrains or facilitates the foraging

process by determining where organisms can be or how they can move. If
habitat complexity alters movement patterns or the ability to detect or avoid
predators, then the space clearance rate should reflect this. For example,
kokanee salmon (Oncorhynchus nerka) showed a steeper functional response
foraging for Daphnia in high light than in low light, presumably owing to
greater prey detectability (d) in brighter conditions (Koski and Johnson,
2002), and lynx showed steeper functional responses in winter than in sum-
mer, potentially because of the difficulty of prey escape during winter (an
effect on ps ) (Nilsen et al., 2009).
Habitat complexity can reduce the encounters among predators and prey
and therefore can make a functional response shallower for some predators
(Alexander et al., 2015; Barrios-O’Neill et al., 2015; Cuthbert et al., 2019;
Ware, 1972). Moreover, increasing habitat complexity can generate the poten-
tial for refuges, altering not just the height of the functional response but
its shape. In the case of dragonfly larvae (Anax junius) foraging on tadpoles
(Rana pipiens), the functional response shifted to type III in the presence
of prey refugia, suggesting reduced encounters between predator and prey
particularly at low prey densities (Hossie and Murray, 2010). In other systems,
however, predators may show an increase in space clearance rate with habitat
complexity (Wasserman et al., 2016). In the case of the notonectid predator
Enithares sobria foraging on the water flea Daphnia longispina, the effect of
habitat complexity depended on temperature, suggesting that the physical
aspects of predator or prey movement interacted with the physiological effects
of temperature on predator, prey, or both (Figure 3.3; Wasserman et al., 2016).
Similarly, increasing the complexity of arena edges can reduce functional
responses by creating refugia (Hoddle, 2003), but increasing the area of
arenas can increase the functional response by concentrating prey along the
susceptible outer edge making prey more vulnerable (Uiterwaal et al., 2019).
In addition to habitat complexity, predators that forage in groups may have
higher detection, more encounters, or greater probability of success, increas-
ing the functional response relative to individual foraging (Fryxell et al.,
2007; Nilsen et al., 2009). Also, “hunger” levels, as determined by how much a
predator has recently fed, would logically influence space clearance rate, since
the decision to attack (pa ) or search velocity (V c ) should be motivated by how
much the predators need to eat (Charnov, 1976; Jeschke, 2007; Jeschke et al.,
2002). This expectation seems to be met in D. magna foraging on a range
of food types (McMahon and Rigler, 1965) and for Hogna baltimoriana wolf
spiders foraging for grasshoppers (Lyon et al., 2018).
34 Predator Ecology
100
90 No habitat
Low habitat
80 High habitat
Number of prey consumed
70
60
50
40
30
20
10
0
0 20 40 60 80 100 120 140
Number of prey oﬀered
Figure 3.3 Functional responses of the notonectid predator Enithares sobria foraging
on the cladoceran Daphnia longispina at 14∘ C. Data from Wasserman et al. (2016).
Functional responses were lower for both no and low levels of habitat complexity
(zero and two plant stalks in the arenas) than with high levels of habitat complexity
(four stalks).
As already indicated, one of the better-investigated traits linked to

functional response parameters is body size (Goss-Custard et al., 2006;
MacNulty et al., 2009; McCoy et al., 2011; Miller et al., 1992; Thompson,
1975). Because body size is often positively associated with the speed at which
both predators and prey move through the environment through a power-
law-like relationship (Calder, 1996; Peters, 1983), we might predict that
space clearance rate has a power-law-like relationship with either predator
and/or prey body size (Aljetlawi et al., 2004; McGill and Mittelbach, 2006)
(Figure 3.4). This kind of relationship has been found across species for
several types of organisms, including mammalian carnivores (DeLong and
Vasseur, 2012a), protists (DeLong and Vasseur, 2012b; Fenchel, 1980a, b),
ladybird beetles (Uiterwaal and DeLong, 2018), and across multiple taxa
together (Rall et al., 2012; Uiterwaal and DeLong, 2020). Although there is
an apparent “universal” scaling across all species (Figure 3.4A), there are
distinct scalings for some taxonomic groups (Figure 3.4B; Uiterwaal and
DeLong, 2020). The within-group scalings also may be shallower than the
overall scaling, such that the overall scaling intersects multiple groups and
obscures some potentially important variation. The body size-dependence of
All 3D foragers
1010
(A)
100
Space clearance rate (mL pred–1 day–1)
10–10
10–5 100 105
Protists, copepods, and ﬁshes

1010
(B)
Protists
Copepods
Fish
100
10–10
10–5 100 105

Predator body mass (mg)
Figure 3.4 The scaling of space clearance rate with body mass, based on data from the
FoRAGE database. (A) All the three-dimensional foragers in the database together scale
strongly with body mass, with an exponent of 0.82, not considering other factors.
(B) Breaking these foragers into three taxonomic groups that also tend to vary in body
size, it is clear that the overall scaling crosses some group-specific scaling patterns,
with the scaling exponents somewhat shallower for fishes (0.68) and copepods (0.56)
but in line with that for protists (0.89).
functional responses both within and across species may even be strong
enough to negate species-level differences, such as was shown for some
freshwater fishes foraging on brine shrimp (Miller et al., 1992). Interestingly,
because of the different power-law scalings of velocity for two-dimensional
and three-dimensional foragers, it also has been shown that the exponent
for the scaling of space clearance rate on body size is steeper for three-
dimensional foragers than two-dimensional foragers (Pawar et al., 2012).
36 Predator Ecology
Body size also influences space clearance rate within predator–prey pairs.
For example, the space clearance rate of individual least killifish (Heterandria
formosa) foraging on brine shrimp (Artemia salina) was positively correlated
with predator body size (Schröder et al., 2016). In other cases, variation is
associated with prey body size. Space clearance rates got smaller as prey body
size increased for both water bugs (Belostoma sp.) and dragonfly nymphs
(Pantala flavescens) foraging on red-eyed treefrog tadpoles (Agalychnis cal-
lidryas) (McCoy et al., 2011). This effect could arise through increased escape
̌
speeds for larger prey, affecting the probability of success (ps ) (Gvozdík and
Smolinský, 2015). Sometimes this body size effect is linked to age or instar of
the predator. For example, backswimmers (Notonecta maculata) are aquatic
insects that grow through multiple instars on their way to adulthood, and
they show clear variation in their functional response owing to age and body
size. Gergs and Ratte (2009) showed that as backswimmers grow through
these instars, their functional response for foraging on water fleas (D. magna)
changes (Figure 3.5). The largest backswimmer instars have a functional
response that increases with water flea size but declines again for the largest
water fleas (Figure 3.5A). This pattern indicates that the space clearance rate
peaks at intermediate prey sizes (Figure 3.5B). This peak, however, is lower
for the smaller backswimmers and higher for the larger backswimmers.
In the backswimmers, age and body size have strong effects on the func-
tional response, creating variation in how strongly specific backswimmers
interact with specific water fleas. Spitze (1985) found a similar pattern for
phantom midges (Chaoborus americanus) foraging on a different species of
water flea (Daphnia pulex). Why does such variation occur? Equation (3.1)
indicates it could arise through several avenues, but Gergs and Ratte (2009)
showed that encounter rates increased with water flea size (an effect on V c ),
larger backswimmers detected water fleas from farther away (an effect on d),
and the probability of success (ps ) peaked at intermediate backswimmer
size. In a different study, Streams (1994) found that encounter rates and the
probability of attack (pa ) both increased with body size of backswimmers
(Notonecta undulata), but the proportion of successful attacks (ps ) decreased
with predator body size. Thus, several aspects of the foraging process related
to body size, movement, and traits interact to generate variation in the
functional response.
Equation (3.1) shows that it is the product of multiple events that generates
space clearance rate. Thus, if some factors increase while others decrease with
some trait, it is not difficult for space clearance rate to peak at intermediate
values of that trait, as in the case with backswimmer body size. Given the
multiple ways in which predator and prey body size influence the components
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Title: Naturalismus, Idealismus, Expressionismus
Author: Max Deri
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Language: German
Original publication: Leipzig: Verlag von E. Seemann, 1921
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NATURALISMUS, IDEALISMUS, EXPRESSIONISMUS ***
Anmerkungen zur Transkription
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NATURALISMUS
IDEALISMUS
EXPRESSIONISMUS
VON
MAX DERI
FÜNFZEHNTES BIS SIEBZEHNTES TAUSEND
VERLAG VON E. A. SEEMANN IN LEIPZIG 1921

I. THEORIE.
Anpassung der Gedanken an die Tatsachen nennt man

Beobachtung, gegenseitige Anpassung der Gedanken aneinander
Theorie.
Dieser Grundsatz, den Ernst Mach formuliert hat, leitet die
folgenden Feststellungen. Er sagt von vornherein, daß alle
metaphysischen Gesichtspunkte aus der Untersuchung
ausgeschaltet bleiben sollen. Das Bewußtsein des Menschen wird
als Ergebnis eines jahrtausendelangen Anpassungsprozesses der
lebendigen Materie an das Milieu dieses Daseins aufgefaßt. Woher
dieses Universum, dieser „Kosmos“ stammt, wird nicht gefragt.
Ebensowenig, wieso die lebendige Substanz, aus der sich der
Mensch aufbaut, die Fähigkeit der Reaktion auf die Eindrücke der
Außenwelt besitzt. Diese beiden Voraussetzungen: das Dasein einer
„Welt“ äußerer Vorhandenheiten, und das In-ihr-Sein lebendig-
reaktionsfähiger Materie wird als Gegebenes hingenommen. Und
nur danach wird getrachtet, die gegenseitigen Abhängigkeiten
sowohl der äußeren Objekte, wie der inneren seelischen
Reaktionen, wie auch der beiden Gruppen voneinander möglichst
genau und unvoreingenommen festzustellen und zu beschreiben.
Beobachtung soll also der Vorgang heißen: die Tatsachen so
lange möglichst genau in Begriffen nachzubilden, bis diese den
Erfahrungen möglichst eindeutig und widerspruchslos zugeordnet
sind. Und T h e o r i e soll heißen: die auf Grund der anpassenden
Nachbildung der Tatsachen erlangten Begriffe so lange immer
wieder gegenseitig anzupassen, bis sie auch untereinander, im
gesamten Begriffskomplex, keinerlei Widersprüche mehr enthalten.
Dabei ist es nach neuerer Anschauung — die man besonders
eindringlich und überzeugend in den Schriften von Moritz Schlick
vertreten findet — keineswegs möglich, Eine bestimmte Theorie,
also Ein untereinander bis zur Widerspruchslosigkeit angepaßtes
Begriffsystem als das einzig und allein „richtige“ zu erweisen.
Sondern die moderne relativistische Anschauung, die jegliches
„Absolute“ als eine Fiktion erkannt zu haben glaubt, steht auf dem
Standpunkt, daß j e d e Theorie „richtig“ ist, d i e d e n Ta t s a c h e n d e r
äußeren Erfahrung ein geschlossenes Begriffsystem eindeutig
z u o r d n e t . Die „Eindeutigkeit“ wird gefordert: als Resultat der
Anpassung der Gedanken an die äußeren Tatsachen, in der Form,
daß jeder Tatsache auch nur Ein Begriffsgebilde zugeordnet
erscheint. Und die „Geschlossenheit“ in der Form, daß innerhalb des
Begriffsystemes kein Widerspruch der einzelnen gedanklichen
Gebilde untereinander bestehen bleibt. — So gelten heute etwa
b e i d e kosmischen Theorien, sowohl die ptolemäische, wie die
kopernikanische, beide als „richtig“, beide als „möglich“. Die Theorie
des Ptolemäus ordnet ihre Gedanken den Tatsachen der
Bewegungen der Himmelskörper in Der Art zu, daß sie annimmt, die
E r d e „stehe fest“, und die Sonne sowie der ganze Sternenhimmel
drehten sich um diese Erde als um ihren Mittelpunkt. Sie kommt
dabei dann weiterhin, durch Anpassung der Begriffsbilder
untereinander, zu einem eindeutigen System von
Bewegungsbahnen: zu Kreisen, Ellipsen, Epizyklen und anderen
komplizierten Kurven, in denen sich die einzelnen Himmelskörper
um den Zentralpunkt der Erde bewegen. — Die Anschauung des
Kopernikus dagegen nimmt die Anpassung der Vorstellungen an die
Tatsachen in Der Weise vor, daß sie die S o n n e als Zentralpunkt
auffaßt, und die Erde und die Planeten sich um diesen Mittelpunkt
bewegen läßt. Auch diese Anschauung kommt dann durch
Anpassung der Tatsachen-Abbildungen untereinander zu einem in
sich widerspruchslosen, geschlossenen Begriffsystem; hier aber
ergeben sich als Bewegungsbahnen der Gestirne keine
komplizierten Kurven mehr, sondern ausschließlich Kreise und
Ellipsen. — B e i d e Anschauungen also sind, nach moderner
Auffassung, möglich und „richtig“. Die Entscheidung aber, welcher
von diesen zweien man schließlich den Vorzug gibt, wird aus dem
Gesichtspunkte heraus getroffen, daß man — die restlose
Tatsachen-Entsprechung und die restlose innere Eindeutigkeit beider
Systeme vorausgesetzt — jedes vorzieht, das ein e i n f a c h e r e s , ein
weniger kompliziertes System von Begriffen ergibt. In diesem Sinne
gibt also letzthin das geistige „Ökonomie-Prinzip“, das einerseits von
möglichst wenigen unbeweisbaren Grund-Voraussetzungen ausgeht,
und andererseits zu der möglichst einfachen Anordnung der
beobachteten Tatsachen führt, den Ausschlag bei der Wahl einer
„Theorie“.
Geht man nun mit diesem Gesichtspunkte des Ökonomie-

Prinzipes der Wissenschaft an die theoretische Grundlegung eines
bestimmten Spezialgebietes, so muß man demgemäß vor allem
danach trachten, bei der Aufstellung jener „unbeweisbaren“ — das
heißt: nicht mehr auf bereits vorher Bekanntes zurückführbaren —
Grund-Voraussetzungen, die jede Spezial-Theorie benötigt, mit einer
m ö g l i c h s t g e r i n g e n Anzahl auszukommen.
Es handelt sich bei den vorliegenden Untersuchungen um
„ästhetische“ Probleme, das heißt um Fragen des Kunstschaffens
und des Kunstgenießens.
Zwei ganz allgemeine und für alle Teilgebiete geltende
Voraussetzungen haben wir dabei bereits gemacht: erstens das
tatsächliche Vorhandensein einer Außenwelt, und zweitens das
tatsächliche Vorhandensein einer auf die Eindrücke dieser
Außenwelt reagierenden lebendigen Substanz.
Nun wurden für das ästhetische Teilgebiet bisher meist von allem
Anfang an noch zwei weitere Grund-Voraussetzungen aufgestellt.
Es wurde erstens behauptet, daß der K ü n s t l e r sein Werk aus
einer „spezifischen“ Eignung erstelle, die die „gewöhnlichen“
Menschen n i c h t besitzen. Eine wesentliche und qualitativ besondere
„schöpferische“ Fähigkeit sollte dem „Künstler“ innewohnen, aus der
allein ein Kunstwerk erstehen könne. Sie sei dem Künstler, und bloß
diesem, „angeboren“, und Keiner, dem diese qualitative
Besonderheit fehle, könne irgendwie für den Bezirk des
Kunstschaffens in Anspruch genommen werden.
Zweitens aber sollte auch der A u f n e h m e n d e zum „wahren“
Kunstverständnis von einer spezifischen Eignung oder Veranlagung
abhängig sein, die keineswegs bei allen Menschen zu finden wäre.
Auch hier müsse eine qualitative Besonderheit als „angeboren“
vorliegen, wenn das Bemühen um das Erleben der Kunstwerke zum
„wirklichen“ ästhetischen Erleben und zum „wahren Verständnis“ der
Werke führen solle.
Wenn es nun gelingen würde, eine geschlossene ästhetische
„Theorie“ zu gewinnen, die zwar von der willkürlichen Voraussetzung
absieht, daß Künstler und Kunstverständige q u a l i t a t i v besonders
begabte Menschen seien, die aber dennoch zu einer „eindeutigen
Zuordnung eines in sich widerspruchslosen Begriffsystemes an die
ästhetischen Tatsachen“ gelangt, so wäre damit ein Resultat
erreicht, das „einfacher“ wäre, weniger unbeweisbare Prämissen
enthielte, und deshalb nach dem Ökonomie-Prinzipe der
Wissenschaft den Vorzug vor dem anderen System verdienen
würde. —
Ernst Mach war es, dem es im Laufe eines langen und an
wissenschaftlichen Ergebnissen reichen Lebens gelungen ist,
nachzuweisen, daß sich das alltägliche, jedem Menschen eignende
„Denken“ von dem reinsten wissenschaftlichen Denken der größten
intellektuellen Genies in keiner Weise qualitativ, sondern nur der
Schärfe und der Intensität nach unterscheide. Und so soll im
Folgenden versucht werden, die Tatsachen des ästhetischen
Bereiches unter der Voraussetzung anzuordnen, daß jene oben
erwähnte „spezifische“ Fähigkeit des Künstlers und des
Kunstgenießenden als etwas von den Eigenschaften des
„gewöhnlichen“ Menschen q u a l i t a t i v Unterschiedenes n i c h t
existiere; sondern daß sich das künstlerische Gestalten und Erleben
in nichts als in seiner Intensität und in seiner Tiefe von einem
bestimmten alltäglichen Verhalten aller Menschen unterscheidet.
Der Mensch wird in diese Welt, an die er sich in
jahrtausendelangem Prozeß angepaßt hat, hineingeboren und
nimmt mit seinen Sinnesorganen die Gegebenheiten dieser Welt in
einem weiten Erfahrungskreise auf. Der weiteste Umfang dieses
Kreises alles Sichtbaren, Hörbaren, Riechbaren, Schmeckbaren und
Tastbaren heiße, in der Ausdrucksweise von Ernst Mach, „groß U“.
Was außerhalb dieses weitesten Bezirkes alles Erfahrbaren U noch
liegen möge, wird wissenschaftlich n i c h t diskutiert. Jegliche
Metaphysik bleibt damit also von vornherein ausgeschlossen.
Innerhalb dieses weitesten Erfahrungsbezirkes U lebt nun der
erwachsene Mensch in dem kleineren Umfang seiner Leibeshülle,
die, wiederum in Machscher Terminologie, „klein u“ genannt werden
soll. Der Mensch stellt einen geschlossenen körperlichen Komplex
vor, der an einzelnen Punkten „Reiz-Stellen“ für Eindrücke jener
Außenwelt U besitzt. So sind die Augen für Licht-Reize, die Ohren
für Schallwellen, die Nase für Geruchsempfindungen, die Zunge für
Schmeck-Erfahrungen, die Haut für Tast- und Temperatur-
Einwirkungen empfindlich. Diese Reize werden durch die
Nervenbahnen zum Zentralsystem geleitet, das den Sitz des
Bewußtseins, der „Seele“ bildet.
Sind wir so weit gelangt, so fragen wir nach der, wiederum
unmittelbar gegebenen, nicht weiter „beweisbaren“
Zusammensetzung dieses Bewußtseins.
Beobachtung lehrt uns, daß wir drei seelische Grund-
Gegebenheiten besitzen, die nicht weiter auf „früher Bekanntes“
zurückführbar sind: das Erkennen = Denken, intellektuelle Funktion;
das Fühlen = emotionelle Funktion; und das Wollen =
voluntaristische Funktion.
Prüfen wir die Zueinanderordnung, die diese drei Funktionen des
menschlichen Bewußtseins untereinander einnehmen, so finden wir
Folgendes.
Sehen wir etwa einen Baum, so ist mit der rein sachlichen
Wahrnehmung des Gegenstandes, mit seinem „Erkennen“,
unmittelbar ein spezifisches „Begleit-Gefühl“ mit-gegeben. Man kann
sich, sollte man daran zweifeln, diese Tatsache unmittelbar
vergegenwärtigen, wenn man entweder das Objekt der Betrachtung
wechselt: also einmal eine „mächtige“ Eiche, dann eine „schlanke“
Birke, eine „Trauerweide“, eine „stolze“ Zypresse als Seh-Ding wählt.
Die in Anführungszeichen gesetzten Attribute geben dabei Hinweise
auf die unmittelbare Gefühls-Begleitung des intellektuell
Wahrgenommenen und „Erkannten“. Oder man denke daran, wie
außerordentlich stark die noch nicht abgestumpfte Gefühlsbegleitung
bei völlig neuartigen Erlebnissen ist. Sei es etwa, daß man in
zoologischen Gärten oder in Aquarien Tiere zum ersten Male sieht,
die man bisher noch nicht kannte; sei es, daß man sich die
unmittelbare Verknüpfung von erkennendem Wahrnehmen und
begleitender Gefühlserschütterung vergegenwärtigt, die man erfuhr,
als man zum erstenmal ein Luftschiff oder ein Flugzeug sah. — So
erweist vielfaches Erleben den unmittelbaren Dualismus, die
grundlegende Doppel-Gegebenheit des Erkennens und des Fühlens
im menschlichen Bewußtsein. Diese beiden Funktionen sind
offenbar nicht hintereinander geschaltet, sondern in unmittelbarem
Parallelismus einander nebengeordnet.
Anders ist es beim Wollen. Sieht man hier von den automatisch
gewordenen Reflex-Reaktionen ab, so kann man klar beobachten,
wie jedes „bewußte“ Wollen an das Vo r a u s g e h e n eines Erkennens
o d e r eines Fühlens geknüpft ist. Erst entweder wenn ich durch das
Thermometer „erkannt“ habe, daß es kalt geworden ist, kann ich die
schützende Hülle „wollen“; oder erst wenn ich „gefühlt“ habe, daß es
kalt zu werden beginnt, kann ich die Decke „wollen“. So ordnet sich
also die Dreiheit der Funktionen des erwachsenen menschlichen
Bewußtseins in der Form des folgenden Schemas teils
nebeneinander, teils hintereinander an.
Ist der Wissenschaftler nun dahin gelangt, eine Mehrheit von

Gegebenheiten als grundlegend bei einem äußeren oder bei einem
innerseelischen Vorgang zu konstatieren, so pflegt er weiterhin zu
versuchen, die einzelnen Teilstücke des Komplexes zu i s o l i e r e n .
Gelingt nämlich diese nach der „Methode der Variation“
durchzuführende Isolierung einzelner Elemente, so kann man für die
Beobachtung Aufschlüsse sowohl über die spezielle Artung der
Teilstücke, wie über die Rolle erwarten, die sie dem Gesamtkomplex
gegenüber spielen.
Wendet man diesen Gesichtspunkt der möglichen Isolierung der
Teilstücke eines Komplexes auf die obige schematische Zeichnung
an, so ist vorerst sofort ersichtlich, daß es ein „isoliertes Wollen“
nicht geben kann, sofern kein Zwang auf das lebendige Bewußtsein
ausgeübt wird. Zwar verlangt jedes „Du sollst“, daß man sein Wollen
ohne vorherige Betätigung des Erkennens oder des Fühlens, also
„isoliert“, in Funktion treten lasse. Doch derartige, seien es
erzieherische, seien es gesetzliche „Befehle“, die nicht v o r die
gewollte Handlung der Einzelnen die Einsicht oder das Gefühl
setzen, sind eben anormale Zwangs-Situationen, die für die
Beobachtung des frei lebenden Bewußtseins ausscheiden. Denn für
dieses ist jedes Wollen, das in seiner freien Auswirkung zu einer
Handlung führt, von einem vorausgehenden erkennenden oder von
einem vorausgehenden fühlenden Erlebnis bestimmt und
determiniert.
Anders aber als für das Wollen liegen die Gegebenheiten des
Bewußtseins für das Erkennen und für das Fühlen.
Wenn auch die innigen Verbundenheiten der drei Funktionen des
menschlichen Bewußtseins aus ihrem schon organisch bedingten
Zusammenhange letzten Endes niemals v ö l l i g isolierbar sind, so
kann doch die „Einstellung“, die „Tendenz“ der Betätigung sich
vornehmlich auf die Eine der Funktionen stützen oder richten. Und
zwar hier im Besonderen also entweder auf das Erkennen oder auf
das Fühlen.
Wir erleben vorerst vielfach, daß wir beim Aufnehmen der
Naturgegebenheiten unser Augenmerk hauptsächlich auf das
Wahrnehmen, das Erkennen der Tatsächlichkeiten richten können,
indem wir uns bemühen, unser Fühlen und unser Wollen möglichst
auszuschalten. Diese isolierende Einstellung der intellektuellen, der
erkennenden Funktion führt zu dem, was wir das „wissenschaftliche“
Verhalten nennen. Wenn wir also unser Augenmerk darauf richten,
unsere gefühlsmäßigen Reaktionen möglichst zurückzudrängen,
unser willentliches Verhalten nicht lebendig werden zu lassen:
sondern einzig und allein mit unserer intellektuellen Funktion die
Tatsachen unserer äußeren und inneren Erfahrung zu beobachten
und zu beschreiben, so befinden wir uns im Bezirke der reinen
W i s s e n s c h a f t . Oder, strenger formuliert: d a s m ö g l i c h s t e I s o l i e r e n
der intellektuellen Funktion des Bewußtseins führt zum
w i s s e n s c h a f t l i c h e n Ve r h a l t e n . —
Bevor wir uns nun der Frage nach der Isolierbarkeit des Fühlens
im menschlichen Bewußtsein zuwenden, wird eine kurze
Einschaltung notwendig. Seit Jahrhunderten mühen sich die
Menschen, jene vorhin erwähnte, angeblich q u a l i t a t i v besondere
Eignung des Künstlers und des „wahrhaft“ Kunst-Verständigen
definitorisch zu fassen. Die Streitfrage geht dabei um das Wesen
des „ästhetischen Schaffens“ und des „ästhetischen Genießens“. Da
nun bereits seit langem so weit Übereinstimmung erzielt ist, daß es
sich in beiden Fällen um spezifische G e f ü h l s -Verhaltungsweisen
handelt, wird andauernd danach gestrebt, das Wesen eben dieses
„ä s t h e t i s c h e n G e f ü h l e s “ als eine spezifisch besondere, von allen
anderen Gefühlen streng unterschiedene Klasse festzustellen. Und
das wichtigste Problem wird demgemäß, dieses „ästhetische“ Fühlen
von allen anderen Gefühlen, also vom „gewöhnlichen“ oder
„vulgären“ Fühlen, streng definitorisch zu unterscheiden.
Da es nun einerseits bisher noch niemals gelungen ist, die
spezifische Definition dieses „ästhetischen Gefühls“ einwandfrei zu
geben; und da andererseits die Theorie an Einfachheit, an
„Ökonomie“ gewinnen würde, falls es gelingen sollte, ohne eine
grundlegende Zweiteilung zwischen „ästhetischen“ und
„gewöhnlichen“ Gefühlen auszukommen, sei die folgende
Anschauung der Nachprüfung durch das Erleben empfohlen.
Diese Anschauung behauptet: e s g i b t ü b e r h a u p t k e i n
s p e z i f i s c h ä s t h e t i s c h e s G e f ü h l . Ein Wesensunterschied zwischen
den Gefühlen des Kunstschaffens oder des Kunstgenießens und
denen des „gewöhnlichen“ Lebens ist nicht aufzufinden, weil er nicht
vorhanden ist. Sondern so, wie das „vulgäre“ D e n k e n aller
Menschen dem „wissenschaftlichen“ Denken selbst der größten
intellektuellen Genies im W e s e n völlig gleich ist; und so, wie
zwischen diesen beiden nur ein Grad-Unterschied besteht, indem
das wissenschaftliche Genie a u f G r u n d e i n e r m i t g r ö ß e r e r
Vo l l k o m m e n h e i t d u r c h g e f ü h r t e n I s o l i e r u n g des Denkens vom
Fühlen und vom Wollen, sowie a u f Grund stärkerer
ursprünglicher quantitativer Ve r a n l a g u n g reichere oder
umfassendere Resultate erzielt: so soll der hier vertretenen
Anschauung nach auch das „vulgäre“ F ü h l e n dem „ästhetischen“
Fühlen im W e s e n völlig gleich sein. Und auch zwischen diesen
beiden Gefühlsarten soll nur ein Grad-Unterschied bestehen, indem
das künstlerische Genie a u f G r u n d e i n e r m i t g r ö ß e r e r
Vo l l k o m m e n h e i t d u r c h g e f ü h r t e n I s o l i e r u n g des Gefühls vom
Denken und vom Wollen, sowie a u f G r u n d s t ä r k e r e r
u r s p r ü n g l i c h e r q u a n t i t a t i v e r Ve r a n l a g u n g zu reicherem oder
tieferem ästhetischen Erleben gelangt.
In gleicher Weise also, wie man im obigen Bewußtseins-Schema
durch das möglichste Isolieren der intellektuellen, der erkennenden
Funktion des menschlichen Bewußtseins vom emotionellen und vom
voluntaristischen Bezirk zum wissenschaftlichen Verhalten gelangte;
in gleicher Weise führt, dieser Anschauung nach, die möglichste
Ausschaltung des erkennenden und des willentlichen Bezirkes, und
dabei die möglichst ausschließliche Einstellung auf das
G e f ü h l s e r l e b n i s in den „ästhetischen“ Bezirk. In strenger
Formulierung würde also der nicht weiter zurückführbare, sondern
nur durch das unmittelbare Erleben beweisbare Grundsatz lauten:
das möglichste Isolieren der emotionellen Funktion des
B e w u ß t s e i n s f ü h r t z u m ä s t h e t i s c h e n Ve r h a l t e n .
Diese Anschauung behauptet also: es gibt kein „spezifisch
ästhetisches“ Gefühl. Sondern j e d e s Gefühl, und sei es, welches
immer — sei es ein ethisches oder ein religiöses, ein Freude- oder
Trauergefühl, ein Zorn oder eine Hingebung, ein Schmerz oder eine
Wollust — kann ästhetisch w e r d e n . Und es „wird“ ästhetisch eben
dadurch, daß man es im Erleben seines funktionalen Ablaufes
möglichst strenge und möglichst rein von allem Intellektuellen und
von allem Willentlichen isoliert.
Während man also bisher entweder drei weiterhin unbeweisbare
Voraussetzungen — über das ästhetisch-produktive, über das
ästhetisch-rezeptive und über das „vulgäre“ Gefühl — machen
mußte, oder doch zumindest zwei unterschiedene Arten von
Gefühls-Erlebnissen: das „spezifisch-ästhetische“ Fühlen und das
„vulgäre“ Fühlen als im Wesen völlig verschieden behauptete, wäre
durch die obige Annahme der wissenschaftsökonomische Vorteil
gewonnen, an Stelle dreier oder zweier unbeweisbarer Grund-
Gegebenheiten bloß E i n e zu setzen, die — je nach dem Grade ihrer
Isolierung sowie nach dem Maße ihrer Auswirkung — das Gebiet der
Kunst-Gefühle wie das der „gewöhnlichen“ Gefühle einheitlich zu
umfassen imstande wäre.
Da nun derartige Grund-Behauptungen, die man früher „Axiome“
nannte, niemals im üblichen Sinne — wie bereits einige Male
zugestanden — durch weitere „Zurückführung auf bereits
Bekanntes“ unweigerlich feststellbar sind, sondern ihre „Wahrheit“,
ihre inner-seelische „Wirklichkeit“ nur im unmittelbaren Erleben
selbst erweisen oder eben nicht erweisen können, möge ein Beispiel
diese Anschauungsweise zu klären versuchen.
Man denke sich auf eine weite Gebirgsaussicht. Vor dem Blick
ein Kreis von Bergen mit Matten und Wiesen, Wäldern und Büschen,
höher hinauf Gletscher und Felszacken, Schneefelder und Firnen,
darüber der Himmel mit reichen Wolkenformen.
Nun isoliere man vor diesem Erlebnis vorerst die intellektuelle
Funktion, das Wahrnehmen und Erkennen: man verhalte sich also
„wissenschaftlich“. Man wird fragen, wie die Berge heißen, wie hoch
sie sind, zu welchem Staatsgebiete, zu welcher Bergkette sie
gehören, aus welchem geologischen Material sie bestehen, man
kann sich für ihre Geschichte interessieren, dafür, ob sie bereits alle
erstiegen, ob sie bewohnt sind, man kann ihre Vegetation, ihre
Bewässerungsverhältnisse, ihr Klima erkunden ...: kurz, man kann,
mit möglichster Ausschaltung alles gefühlsmäßigen und willentlichen
Erlebens, die reine „Wissenschaft“ der „Geographie“ im weitesten
Wortsinne treiben.
Und man kann sich anders verhalten. Man kann sich mit
möglichster Ausschaltung alles jenes oben aufgezählten
Intellektuellen, und mit Ausschaltung auch alles Willentlichen — das
etwa den besseren Aussichts-Punkt des Nebenstehenden „erstrebt“,
oder einen der gesehenen Berge ersteigen „will“, oder eine
erschaute vollfarbige Blüte pflücken und besitzen „will“ —:
möglichst rein der Gefühls-Begleitung der gesehenen Objekte
h i n g e b e n . Man „setzt sich auf das Gefühl zurück“, man läßt dieses
Begleitgefühl der Objekt-Konstellation mit möglichster Isolierung rein
und ungehemmt in sich schwingen. Tut man das, achtet man wirklich
auf nichts anderes als auf dieses isolierte Erleben des Gefühles-an-
sich, so kommt man in eine seelische Verhaltungsweise, die man
von starken Kunsterlebnissen her bereits kennt. Man ist intellektuell
völlig „abgeblendet“, und man ist willentlich völlig „passiv“, man gibt
sich rein dem Gefühle hin: und da erfährt man sich als im spezifisch
ästhetischen Verhalten lebend. —
Dieses ästhetische Verhalten steht also völlig parallel dem
wissenschaftlichen Verhalten. Entsteht das wissenschaftliche
Verhalten durch möglichst isolierendes Einstellen auf die
erkennende Funktion den Erlebnissen dieser Welt gegenüber; so
entsteht das ästhetische Verhalten durch möglichst isolierendes
Einstellen auf die fühlende Funktion den Erlebnissen dieser Welt
gegenüber. Und ist Wissenschaft keineswegs bloß die Lehre vom
„absolut Wahren“, das es „an sich“ ja nicht gibt, sondern ist
Wissenschaft die Lehre von der Erkenntnis u n d vom Irrtum im Laufe
der Menschheitsentwicklung: so ist Ästhetik keineswegs bloß die
Lehre vom „absolut Schönen“, das es ja gleichfalls nicht gibt,
sondern Ästhetik ist die Lehre vom „Schönen“ u n d vom „Häßlichen“
im Laufe der Menschheitsentwicklung. Denn so wie die ärztliche
Wissenschaft vom gesunden u n d vom kranken Körper handelt, so
wie die Volkswirtschaft die Wissenschaft von dem Vermögen u n d
von den Schulden, so wie die Elektrizitätslehre die Lehre von den
„positiven“ u n d von den „negativen“ elektrischen Vorgängen, so wie
die „Wärmelehre“ gleichzeitig auch „Kältelehre“, also die Lehre von
den Temperaturen über u n d unter der zufälligen menschlichen
Körpertemperatur ist: so müssen die „Gesetze der Ästhetik“, das
heißt die immer wieder beobachteten Abläufe dieses ästhetischen
Verhaltens ihre Gültigkeit bewahren, wie immer auch die
E r g e b n i s s e dieser Verhaltungsweise, sei es als positiv-schön, als
gleichgültig-nullwertig, oder als negativ-häßlich „gewertet“ werden.
Denn das Vorzeichen des R e s u l t a t e s kann niemals den
f u n k t i o n e l l e n A b l a u f bestimmen oder definieren, der zu gerade
diesem Resultat zufälligerweise geführt hat. So darf also eine
„Definition“, eine „Formel“, wenn sie allgemeine Gültigkeit besitzen
soll, niemals davon abhängig gemacht werden, mit welchem
Vorzeichen die einzelnen bestimmten Gegebenheiten an die Stelle
der allgemein begrifflichen Bestimmungsstücke eintreten. —
Nehmen wir also die Grund-These vorerst probeweise an:
ä s t h e t i s c h e s E r l e b e n i s t i s o l i e r t e s G e f ü h l s - E r l e b e n , so haben
wir eine allgemeine, für alle Gefühle geltende Definition des
Ästhetischen. Das Kunst-Ästhetische würde sich dann vom Natur-
Ästhetischen nur dadurch unterscheiden, daß in der N a t u r die
A n l ä s s e für die Gefühle, die wir erleben, o h n e u n s e r Z u t u n u n d
g a n z u n a b h ä n g i g v o m m e n s c h l i c h e n D a s e i n vorhanden sind —
auch wenn es überhaupt keine Menschen gäbe. W ä h r e n d d i e
Kunstwerke von einzelnen Menschen erst gemachte Gebilde
s i n d , die uns deren Gefühle vermitteln.
Es handelt sich demgemäß im Kunstbereiche ausschließlich um
Gefühlserlebnisse. So vage und unfaßbar nun Gefühlserlebnisse
sonst auch scheinen: innerhalb der Diskussion über Kunstwerke sind
sie auf das sicherste faßbar. Denn s i e h ä n g e n a n d e m z u
e r l e b e n d e n O b j e k t , am Kunstwerke, das als fertig Gestaltetes
vorliegt. Mag bei der Konzeption und während des Schaffens selbst
noch so viel Irrationales mitgehen: dies Irrationale bezieht sich bloß
auf den Schaffenden selbst w ä h r e n d des Werdeprozesses des
Kunstwerkes. Sowie der Künstler, der nichts Anderes und nichts
Mehreres ist, als ein Gestalter eines künstlichen Gefühls-Objektes,
sein Tun zu Ende geführt hat, sowie also das Kunstwerk als fertiges
Gebilde vorliegt, besteht nicht nur die Möglichkeit einer festen
Umgrenzung des Gefühls-Erlebnisses, sondern für jeden Geschulten
sogar die Sicherheit, nichts zu erfahren oder zu erleben, was nicht
durch seinen Anlaß in dem vorliegenden Kunstwerk bündig zu
erweisen wäre. Der Variationen, der Verflechtungen, der
Harmonieverschiebungen oder Linienkrümmungen mögen noch so
viele gebracht sein: in der festen Bindung des Erlebenden und des
Erlebten an das Kunst-Objekt, in der Verklammerung, mit der das
vermittelte Gefühl vom Gesamtbau bis zum Detail an der eben hier
und eben so gebrachten Formung des Gebildes haftet, darin liegt die
Sicherheit, nicht mit leeren Worten als Laut-Schällen zu jonglieren,
sondern deren erlebte B e d e u t u n g s - I n h a l t e als eindeutig bestimmte
Tatsachen zu erläutern und zur Diskussion zu stellen.
Ohne nun hier auf eine noch ausführlichere Ableitung
einzugehen, seien die Definitionen des „Künstlers“ und des
„Kunstwerkes“ gegeben, die aus dem Vorausgegangenen zu
erfolgern sind.
„Künstler“ vorerst ist jener, der imstande ist, auf Grund eines
isolierten tiefen Gefühles ein Gebilde zu erstellen, das dieses Gefühl
trägt. — Der Künstler unterscheidet sich also nach dieser
Auffassung vom „gewöhnlichen“ Menschen n i c h t durch sein G e f ü h l .
Hunderte von Menschen mag es geben, und gibt es auch sicherlich,
die ebenso tief empfinden, wie die großen und selbst die größten
Künstler, und die auch imstande sind, in ihrem ästhetischen
Verhalten dieses tiefe Fühlen vom Denken und vom Wollen
möglichst rein zu isolieren. Was sie vom Künstler unterscheidet, ist
bloß das Fehlen der zufälligen angeborenen Fähigkeit, sichtbare,
hörbare oder sprachliche Gebilde zu erstellen, die dieses Gefühl
tragen. Der K ü n s t l e r ist demgemäß ein mehr oder minder
„bedeutender“ Mensch, d e m e r s t e n s d i e G a b e b e s o n d e r s t i e f e n
und intensiven Fühlens eignet; und der zweitens die Fähigkeit
b e s i t z t , d i e s e s G e f ü h l a u c h i n e i n e n o b j e k t i v e n Tr ä g e r z u
fassen.
Ein K u n s t w e r k weiterhin ist e i n v o n d i e s e m M e n s c h e n a l s

Künstler aus isolierter Gefühls-Einstellung heraus gemachtes
Gebilde, das imstande ist, dem Erlebenden das Gefühl, aus
d e m e s e n t s t a n d u n d d a s a n i h m h ä n g t , z u v e r m i t t e l n . Und so
wie j e d e r Mensch, nicht etwa nur ein „spezifisch Begabter“, bei den
n a t u r h a f t vorhandenen Anlaß-Trägern für Gefühle zum ästhetischen
Erleben gelangt, indem er die Gefühls-Begleitung des eben
wahrgenommenen Objektes im Aufnehmen möglichst isoliert; so

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Mitonuclear ecology First Edition. Edition Hill

8. Detecting Prey Preferences and Prey Switching 89

© but could be translated to other scientific programming languages. The

Predators seem to be universally fascinating. Maybe that is because we

Steep/high functional response

Shallow/low functional response

Prey abundance or density

a particular habitat, so the functional response is really an emergent property

unicellular foragers of bacteria and phytoplankton (Roberts et al., 2010) to

1.1 Functional responses and food webs

Figure 1.2. A simplified food web. Functional responses characterize the

2.1 Types of functional responses

Functional responses are curves relating a predator’s foraging rate on a specific

Anchovies consumed d–1

Rotifers consumed h–1

Ants consumed d–1

The Basics and Origin of Functional Response Models 11

Box 2.1. Why a should be called “space clearance rate.”

The Basics and Origin of Functional Response Models 13

It is sometimes thought that the type I functional response increases linearly

Fpc = aRTs (2.2)

The traditional argument in the derivation of the type II functional response

Ttot = Ts + aRTs h (2.3)

The Basics and Origin of Functional Response Models 15

experiment actually may include an amount of time wasted on unsuccessful

The Basics and Origin of Functional Response Models 17

As with the Hill exponent, here the g is just a phenomenological parameter

2.2 Predator dependence of the functional response

The Basics and Origin of Functional Response Models 19

The mutual interference exponent m is negative to indicate that adding more

The a in equation (2.10/2.11) is the same as the a in equation (2.5), because

2.3 Relationship to alternative formulations

Many studies focused on aquatic invertebrates or microbes, as well as some

where I max is the maximum ingestion rate and K R is the half-saturation

which also clarifies that the half-saturation constant KR = 1/ah . We will

The Basics and Origin of Functional Response Models 21

2.4 The Rogers Random Predator equation

An important alternative expression for the functional response arises from

Re = Ro (1 − ea(hRe −t) ) (2.16)

Box 2.2. Derivation of the Rogers Random Predator equation.

ln Rt − ln Ro + ahRt − ahRo = −at

which we can rearrange as:

and then further to:

Rt = Ro ea(h(Ro −Rt )−t)

The Basics and Origin of Functional Response Models 23

−Ro ea(h(R0 −Rt )−t) = −Rt

Ro − Ro ea(h(Ro −Rt )−t) = Ro − Rt

Because Re occurs on both the left-hand side and right-hand side of

However, this is just an approximation and may produce slightly biased

Box 2.3. Derivation of the LambertW version of the Rogers