Remarks on antipredator behavior and food chain dynamics

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Theoretical Population Biology 66 (2004) 277–286

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Remarks on antipredator behavior and food chain dynamics

Sergio Rinaldia,b,�, Alessandra Gragnanic, Silvia De Monted

aCIRITA, c/o Department of Electronics and Information, Politecnico di Milano, Via Ponzio 34/5, 20133 Milano, ItalybAdaptive Dynamics Network, International Institute for Applied Systems Analysis, 2361 Laxenburg, AustriacDepartment of Electronics and Information, Politecnico di Milano, Via Ponzio 34/5, 20133 Milano, Italy

dDepartment of Physics, Danish Technical University, DK-2800 Kgs. Lyngby, Denmark

Received 8 September 2002

Available online 28 October 2004

Abstract

When consumers feeding on a resource spend time in avoiding high risks of predation, the predator functional response declines

with predator density. While this is well established, less attention has been paid to the dependence of the consumer functional

response on predator density. Here we show how the separation of behavioral and ecological timescales allows one to determine

both responses starting from an explicit behavioral model. Within the general set-up considered in this paper, the two functional

responses can tend toward Holling type II responses when consumers react only weakly to predation. Thus, the main characteristics

of the standard Rosenzweig–MacArthur tritrophic food chain (logistic resource and Holling type II consumer and predator) remain

valid also when consumers have weak antipredator behavior. Moreover, through numerical analysis, we show that in a particular

but interesting case pronounced antipredator behaviors stabilize the system.

r 2004 Elsevier Inc. All rights reserved.

Keywords: Antipredator behavior; Functional response; Tritrophic food chains; Population dynamics; Stabilization

1. Introduction

Antipredator behavior has been the subject of manyinvestigations ranging from the analysis of field ob-servations (e.g., Seghers, 1974; Milinski and Heller,1978; Lendrem, 1983; De Laet, 1985; Sih, 1986; Waiteand Grubb, 1987; Metcalfe et al., 1987; Rahel and Stein,1988; Holley, 1993; Schmitz et al., 1997; Anholt andWerner, 1998; Bednekoff and Lima, 1998; Hunter andSkinner, 1998; Lima and Bednekoff, 1999a; Schmitz andSuttle, 2001) to the study of mathematical models(Abrams, 1982, 1984 1989, 1990; Hassell and May, 1985;Parker, 1985; Ives and Dobson, 1987; Sih, 1987a,b;Lima and Dill, 1990; Sih, 1992; Matsuda and Abrams,1994; Ruxton, 1995; Abrams and Walters, 1996;

e front matter r 2004 Elsevier Inc. All rights reserved.

b.2004.07.002

ing author. CIRITA, c/o Department of Electronics

n, Politecnico di Milano, Via Ponzio 34/5, 20133

ax:+39-02-23993412.

ess: [email protected] (S. Rinaldi).

Abrams and Matsuda, 1997; Kats and Dill, 1998;Krivan, 1998; Lima, 1998).Some antipredator traits like crypsis and production

of special compounds are often not affected by predatordensity on a behavioral time scale, while other mechan-isms, like vigilance and habitat selection, can be moreactive at higher predator densities. The use of refuges,which is a typical example of these adaptive antipre-dator mechanisms, will be considered in this paper.Although some early studies considered the dynamic

implications of antipredator behavior in models whichcontained two trophic levels (e.g., Hassell and May,1985; Parker, 1985; Ives and Dobson, 1987; Sih, 1987a,1992; Ruxton, 1995), there has been a growing recogni-tion that a consideration of the trophic level below theprey species is required to understand the dynamicsproduced by adaptive antipredator behavior (Sih, 1980;Werner and Mittelbach, 1981; Abrams, 1982, 1984; Sih,1982; Werner and Peacor, 2003; Bolker et al., 2003 andreferences therein). We adopt this approach here by

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ARTICLE IN PRESSS. Rinaldi et al. / Theoretical Population Biology 66 (2004) 277–286278

considering tritrophic food chains composed of re-source, consumer and predator populations and byassuming that consumers have antipredator behavior.Unlike previous studies, we consider cases in which thetritrophic system can undergo cycles or chaotic fluctua-tions in the absence of antipredator behavior. Moreover,we assume that the dynamics of the behavioral options(e.g., foraging, hunting, escaping, and hiding), given thedensities of all relevant populations, are faster than birthand death processes and we do not consider theevolution of genetically transmitted antipredator traits.In the next section we show how the functional

responses of consumer and predator can be derived,once their behavioral dynamics have been explicitlyspecified. The method of analysis is the naturalextension to the case of tritrophic food chains of thetimescales separation method used by Ruxton et al.(1992) for ditrophic food chains.The tritrophic food chain model obtained with this

method can degenerate (in the absence of antipredatorbehavior) into the standard Rosenzweig–MacArthurfood chain model (logistic resource and Holling type IIconsumer and predator). Thus, one can a priori expectthat, within the context examined in this paper, the mainproperties of the Rosenzweig–MacArthur food chainremain qualitatively valid, at least for weak antipredatorbehaviors. This is confirmed by the analysis of a special,but realistic class of tritrophic food chains, for whichbifurcation analysis also shows that antipredatorbehavior is a stabilizing factor.

.

.

.3

1

φin y1

2

α12y1

α31xy3

φout y4

.1

.2β12z1 (β21y1+β21y3)z2

CONSUMERS

manipulatingand transporting

eatingand digesting

searching

PREDATORS

handling

searching

4 hidden withempty gut

.

α23y21

3φin y3

31

(a) (b)

Fig. 1. The flow-graphs describing the behavioral characteristics of

consumers (a) and predators (b).

2. From antipredator behavior to tritrophic food chain

models

In this section we first show how a tritrophic foodchain composed of logistic resource, consumer andpredator populations with densities x, y, and z,respectively, can be derived from an explicit behavioralmodel. This is done by subdividing consumers andpredators in various subpopulations to take intoaccount that there are various actions that individualsmust/can perform. Then, we eliminate the detailsconcerning the subpopulations and obtain an ecologicalmodel of the standard form

_x ¼ rx 1�x

K

� �� hyy;

_y ¼ cyhyy � dyy � hzz;

_z ¼ czhzz � dzz; ð1Þ

where r and K are net growth rate and carrying capacityof the resource, and hy, cy, and dy [hz, cz, and dz] arefunctional response, efficiency, and death rate ofconsumers [predators]. If consumers have antipredatorbehavior that also affects foraging, the consumer

functional response also depends on predator density,while the predator functional response also depends onresource density, but in a way that is dictated by thespecific behavioral rules that characterize consumersand predator.In the case we consider in this paper there are four

consumer and two predator subpopulations, withdensities y1, y2, y3, y4, and z1, z2, respectively.Consumers y1 are manipulating or transporting a unitof resource, consumers y2 are eating or digesting,consumers y3 are searching for food, and consumers y4are hidden with empty gut in a refuge. Thus, so-calledhandling consumers are split into two classes (y1 and y2).Consumers in the first and third stage are assumed to bevulnerable (in general at different rates), while con-sumers in stages 2 and 4 are invulnerable (i.e.,consumers eat and digest their food in a safe habitat).Similarly, z1 and z2 are predator in the handling andsearching stages.The graph of Fig. 1a specifies the flows among the

various stages. Consumers in stage 1 can either end up instage 2 where they will eat and digest their food or theycan abandon their already captured unit of resource(because disturbed by a predator) and enter (with emptygut) into a refuge inaccessible to predator, but alsocharacterized by the absence of resource. Consumers instage 2 enter into stage 3 when they have digested theirfood. Consumers in stage 3 enter into stage 1 when theydetect a unit of resource (this happens more frequently ifresource is more abundant) or into stage 4 when they aredisturbed by the presence of a predator. Finally,consumers in stage 4 switch to stage 3 when they leavethe refuge. The flows from stage i to stage j are assumed

ARTICLE IN PRESSS. Rinaldi et al. / Theoretical Population Biology 66 (2004) 277–286 279

to be proportional to yi, i.e., to the density of thepopulation in stage i, and the coefficients of proportion-ality aij are specified as follows (see Fig. 1a):

a12 ¼ a12; a21 ¼ 0; a31 ¼ a31x; a41 ¼ 0;

a13 ¼ 0; a23 ¼ a23; a32 ¼ 0; a42 ¼ 0;

a14 ¼ f1in; a24 ¼ 0; a34 ¼ f3

in; a43 ¼ fout;

where the a’s are constant parameters, the functions f1in

and f3in specify the flows of consumers which, for fear,

abandon their already captured prey or the field wherethey search for food, and the function fout specifies theflow of consumers leaving the refuges where theyremained hidden for a while with empty gut. The threefunctions f characterize the antipredator behavior ofthe consumer. The two functions f1

in and f3in are not

assumed to be equal because consumers might be morereluctant to abandon an already captured unit ofresource than the field where they search for food.The three characteristic functions can be derived

theoretically, by applying optimization arguments con-sistent with general evolutionary principles, or experi-mentally from direct observations in the field or in thelab. The functions fin have been assumed to beincreasing with predator abundance z by variousauthors (e.g., Ruxton, 1995; Krivan, 1998) and this isconsistent with observations reported in the literature(e.g., Sih, 1987a and reference therein). Of course,alternative assumptions for the functions fin are alsoplausible. For example, if consumers can distinguishbetween searching and handling predators (Lima andBednekoff, 1999b; Hamilton and Heithaus, 2001; Sihand McCarthy, 2002; Van Buskirk et al., 2002), then itwould be justified to assume that the functions fin

depend only on searching predators. Similarly, there arevarious alternative assumptions for fout: For example,fout should be constant if consumers remain hidden in arefuge without eating for a time which is roughlyindependent upon resource, consumer and predatorabundance, while it should be a decreasing function of z

if consumers remain hidden for longer periods whenpredators are more abundant (see Sih, 1992 for a criticaldiscussion on the validity of this hypothesis). We couldalso consider the case in which fout is increasing with y3,or better with y1 and y3, because the risk of beingcaptured by a predator can decrease with the number ofpotential victims (see Matsuda and Abrams, 1994).As for the graph of Fig. 1b concerning the predators,

we assume that the flows from stage i to j areproportional to zi, i.e., to the predator density in stagei, and the coefficients of proportionality bij are specifiedas follows:

b12 ¼ b12 b21 ¼ b121y1 þ b321y3;

where b121 is, in general, different from b321 becauseconsumers manipulating and transporting their cap-

tured resources can be more (or less) vulnerable thanconsumers searching for food.The graphs of Fig. 1 do not evidence the death and

birth processes, but only the flows among subpopula-tions due to individual activities. If we assume thatbehavioral dynamics are faster than ecological dynamics(timescales separation), we can freeze the variables x, y,z (but not y1, y2, y3, y4, z1, and z2) and use the imbalancebetween inflows and outflows at each node of the graphsto specify the time derivatives of y1, y2, y3, y4, z1, and z2.However, since _y1 þ _y2 þ _y3 þ _y4 ¼ 0 and _z1 þ _z2 ¼ 0;because y and z are constant, only four of thesedifferential equations are independent. Thus, inour case, disregarding node 2 of both flow-graphs, weobtain

_y1 ¼ a31xy3 � a12y1 � f1iny1; (2)

_y3 ¼ a23ðy � y1 � y3 � y4Þ þ fouty4 � f3iny3 � a31xy3;

(3)

_y4 ¼ f1iny1 þ f3

iny3 � fouty4; (4)

_z1 ¼ b121y1 þ b321y3� �

ðz � z1Þ � b12z1: (5)

Model (2–5), with x, y, z frozen, is called behavioralmodel, and is fully specified once the characteristicfunctions fin and fout are fixed. In principle, for somechoices of the functions fin and fout the behavioraldynamics can be quite complex. In particular, undersuitable assumptions on fin and fout; one can expectbistability (characterized by two stable equilibria andone unstable (saddle) equilibrium) as in Matsuda andAbrams (1994), or oscillatory behaviors (characterizedby an unstable equilibrium and a stable limit cycle (as inAbrams and Matsuda, 1997). However, for broadclasses of behavioral rules fin and fout; system (2–5) issimply characterized by a stable equilibrium. Forexample, if fin and fout depend upon the resource x

and the total consumer and predator densities y and z,as in the example considered in the next section, theJacobian matrix of system (2–5) is

and its triangular structure points out that one of itseigenvalues is negative (element (4,4) of J) while thethree remaining eigenvalues are the eigenvalues of the 3� 3 matrix J3 given by

J3 ¼

�a12 � f1in a31x 0

�a23 �ða23 þ f3in þ a31xÞ fout � a23

f1in f3

in �fout

��

��:


The characteristic polynomial of J3

D3ðlÞ ¼ detðlI � J3Þ ¼ l3 þ a1l2þ a2lþ a3

can be easily computed and the result is

a1 ¼ a23 þ a12 þ f1in þ fout þ a31x þ f3

in;

a2 ¼ a23fout þ a31xfout þ a23f3in þ a12a31x þ a31xf

1in

þ a23a31x þ a12a23 þ a12f3in þ a12fout

þ a23f1in þ f1

inf3in þ f1

infout;

a3 ¼ a12a23fout þ a12a31xfout þ a12a23f3in þ a23f

1infout

þ a23f1inf

3in þ a31a23xfout þ a31a23xf

1in:

From these expressions, it follows that the threecoefficients a1, a2, a3 of the characteristic polynomialof J3 are positive, since they are the sum of positiveterms. Moreover, each one of the seven terms compos-ing a3 is the product of one of the six terms of a1 by oneof the eleven terms of a2, so that

a1a24a3:

But this inequality, together with the positivity of a1 anda3, is the Hurwitz condition for the stability of J3. Thus,in conclusion, the Jacobian matrix J evaluated at thepositive equilibrium of the behavioral system (2–5) hasfour eigenvalues with negative real parts, i.e., thepositive equilibrium of the behavioral system remainsstable for all parameter values. In other words,transitions to bistability or cyclic behavior are excluded.If the behavioral model (2–5) has a single stable

equilibrium, as assumed from now on, the ecologicalmodel (1) can be easily derived, as shown in the rest ofthis section.Noticing that the total resource consumption rate at

the ecological timescale ðhyy in Eq. (1)) is proportionalto the flow of consumers from stages 3 to 1 in stationarybehavioral conditions (a31 x y3 in Fig. 1a), that the totalconsumer birth rate ðcyhyy in Eq. (1)) is proportional tothe total amount of resource eaten by consumers, which,in turn, is proportional to the flow of consumers fromstages 2 to 3 (a23 y2 in Fig. 1a), and that the totalpredation rate ðhzz in Eq. (1)) is proportional to theflow of predator from stages 2 to 1 ððb121y1 þ b321y3Þz2 inFig. 1b), we can write

hy � a31xy3y; cy � a23

y2hyy

hz � ðb121y1 þ b321y3Þz2

z:

(6)

Using Eqs. (2)–(6), we can explicitly derive the twofunctional responses hy and hz and all other parameters.In fact, Eqs. (2)–(4) with _y1 ¼ _y3 ¼ _y4 ¼ 0 is a system ofthree linear equations in three unknowns, namely y1, y3,and y4. Since the matrix of the coefficients of y1, y3, andy4 in Eqs. (2)–(4) is J3, that has already been proved tobe non-singular, Eqs. (2)–(4) can be solved uniquely wih

respect to y1, y3, and y4. From Eq. (5) with _z1 ¼ 0 it istherefore trivial to obtain the value of z1. Then, recallingthat y1+y2+y3+y4=y and z1+z2=z, it is also possibleto derive the values of y2 and z2. Finally, substituting theabove values into Eq. (6), after some easy, althoughcumbersome, computations one obtains

hy ¼ayx

x þ by

; ay ¼ a0y1þ

f1in

a12

1þf1

infout

a23a12þa23

264

375; by ¼ b0y

1þf1

ina12

� 1þ

f3in

fout

�

1þf1

infout

a23a12þa23

2664

3775;

(7)

cz ¼ c0y1

1þf1

ina12

264

375; (8)

hz ¼azy

y þ bz

; az ¼ a0z ; bz ¼ b0z

1þf1

inþ 1fout

f3in

a12þf1inð Þþf1

ina31xð Þ

a31x 1þa12a23

� �þa12

1þb321f1

in

b121a31xþb3

21a12

2664

3775;

(9)

cz ¼ c0z ; (10)

where a0, b0, c0 are the values of the parameters a, b, c inthe absence of antipredator behavior since the terms insquare brackets tend to unity when f1

in and f3in tend to

zero. Moreover,

a0y �a12a23

a12 þ a23; b0y �

a12a23a31 a12 þ a23ð Þ

; (11)

a0z � b12; b0z � b12a31x 1þ a12

a23

� �þ a12

b121a31x þ b321a12: (12)

Eqs. (7)–(12) support a number of interesting conclu-sions. As for consumers, we can notice that thefunctional response hy degenerates into the Holling typeII functional response in the absence of antipredatorbehavior ði:e:; for f1

in ¼ f3in ¼ 0Þ: Moreover, the attack

rate ay

�by is given by

ay

by

¼a0y

b0y

1

1þf3

infout

:

Thus, under the mild assumptions that fin and fout

depend only on z, with fin increasing and fout not, whenthe resource is scarce, consumers with antipredatorbehavior exploit smaller amounts of resource if there aremore predators. By contrast, when the resource isabundant, the result can be just the opposite, becauseay is greater than a0y if

1

fout

o1

a12þ

1

a23;

i.e., if the time ð1=foutÞ spent in refuges by consumersthat could be searching is smaller than the sum of thetime needed for manipulation ð1=a12Þ and that spent for


eating and digesting ð1=a23Þ: However, the efficiency cy

always decreases with predator density (see Eq. (8)),because consumers abandon more frequently the re-source they have just captured when predator are moreabundant. As a final remark, it is interesting to noticethat the consumer birth rate per capita (which isproportional to the amount of food ingested by eachconsumer in a unit of time) is decreasing with predatordensity because

cyhy ¼ a0yc0yx

1þf1

infout

a23a12þa23

� x þ b0y 1þ

f1in

a12

� 1þ

f3in

fout

� :

In conclusion, under the assumptions that fin isincreasing with z and fout not, it can happen thatconsumers with antipredator behavior destroy moreresource than they would destroy without antipredatorbehavior. However, they do not ingest a greater quantityof resource.As for the predator functional response hz, we can

notice that the maximum predation rate ay is constant(as in Ruxton, 1995), while the half-saturation constantbz depends both on x and z (trait-mediated interactions(Bolker et al., 2003)). Moreover, if the functions f1

in andf3

in increase linearly with z (at least at high density z),then bz also increases with z. This means thatantipredator behavior is somehow equivalent to inter-ference among predator (Beddington, 1975), a biologi-cally intuitive result pointed out by Charnov et al. (1976)and Abrams (1984). Formally, the predator functionalrepsonse given by (9) and (12) does not degenerate intothe Holling type II functional response in the absence ofantipredator behavior since b0z depends on x. However,hz is well approximated by a Holling type II functionalresponse if consumer spend definitely more time inmanipulating and transporting their prey than ineating and digesting it 1=a12 � 1=a23

� �and if

consumers manipulating and transporting their cap-tured resources are exposed roughly to the same riskthan those searching for food b121 ffi b321

� �: Indeed,

eqs. (9,12) for f1in ¼ f3

in ¼ 0; a12 a23; and b121 ¼ b321 ¼b21 give

hz ¼ b12y

y þb12b21

:

In conclusion, in the limit case of very weakantipredator behavior, our tritrophic food chain degen-erates into the Rosenzweig–MacArthur tritrophicfood chain if handling consumers spend most oftheir time in the stage where they are vulnerable andthe risk of predation is the same for all exposedconsumers. This in not a big surprise, since equalvulnerability of all handling consumers is the prerequi-site for the formal derivation of the Holling type IIfunctional response.

3. Analysis of a particular case

We now focus our attention on a special case ofantipredator behavior, namely that characterized by

f1in ¼ d1Dz; f3

in ¼ d3Dz; fout ¼ a43; (13)

where D is a measure of antipredator behavior, fromnow on called defense. This choice of the functions fin

and fout is perhaps the simplest one among those thatcan be intuitively justified and is consistent with someobservations reported in the literature (see Sih, 1987a,for a review).Since our tritrophic food chain model can degenerate,

for no antipredator behavior, into the Rosenzweig–MacArthur model, one can expect that all propertiesknown for this well studied model are valid also in ourmodel, at least for sufficiently low values of defense.Indeed, the simulation results reported below confirmthis guess, showing that for low defense, our modelenjoys three interesting properties concerning its chaoticbehavior, which hold also in the Rosenzweig–MacArthur model.

3.1. Chaotic behavior and tea-cup attractor

As first pointed out in Hastings and Powell (1991), theRosenzweig–MacArthur tritrophic food chain can havechaotic behavior. The geometry of the strange attractorrecalls that of a tea-cup in which the consumerpopulation varies aperiodically with high-frequencybursts (due to resource–consumer interactions), whilethe predator population varies at a lower frequency(associated with consumer–predator interactions). Allthis remains valid in our model, as shown in the examplepresented in Fig. 2 where D=0.02 (see caption for otherparameter values). These properties could formally beproved (as done in Muratori and Rinaldi (1992) for theRosenzweig–MacArthur model) by restricting theanalysis to the case of food chains in which consumersare much slower than resources but much faster thanpredator in growing and reproducing.

3.2. Peak-to-peak dynamics

The peak of any one of the three populations cansometimes be simply predicted from the previouspeak. When this happens the system is said to havepeak-to-peak dynamics (Rinaldi et al., 2001). Thefirst row of Fig. 3 shows the peak-to-peak maps ofconsumer and predator populations for no defense.The maps are continuous and very steep where theyintersect the 451 line, thus indicating that the sequenceof constant consumer and predator peaks is unstable.The second row of Fig. 3 reports the same plots forweak defense (D=0.05). The graphs show that weakantipredator behaviors do not destroy peak-to-peak

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(a)(b)

cons

umer

,y

0.0

0.4

0.8

6

8

10

12

0.00.5

1.0

0.00.4

0.8

pred

ator

,z

resource, xconsumer, y time, t

0 200 400 600 800 1000

pred

ator

,z

6

9

12

Fig. 2. Chaotic behavior of the tritrophic food chain model (1, 7�13): tea-cup attractor (a) and consumer and predator time series (b). Parameter

values are r=1.5, K=1.1, dy=0.4, dz=0.01, c0y ¼ 1; c0z ¼ 1; D=0.02, d1=0.1, d2=0.2, a12 ¼ 10=3; a31 ¼ 5; a23 ¼ 10=3; a43 ¼ 5;b12 ¼ 0:05; b121 ¼0:2; b321 ¼ 0:1:

yk

0.3 0.6 0.9

y k+1

0.3

0.6

0.9

yk

0.6 0.8

y k+1

0.6

0.8

zk

6 8 10 12

z k+1

6

8

10

12

zk

6 8 10 12 14

z k+1

6

8

10

12

14

1

2

3

1

2

3

Fig. 3. Peak-to-peak maps of consumers (first column) and predator (second column) in the absence of defense (first row) and with small defense

(D=0.05) (second row). Parameter values as in Fig. 2.

S. Rinaldi et al. / Theoretical Population Biology 66 (2004) 277–286282

dynamics but simply transform the continuous peak-to-peak maps into discontinuous maps. Actually, thetwo maps do not intersect the 451 line so that(even unstable) stationary sequences of peaks do not

exist. Indeed, the simplest regular sequences of peaks arethe unstable period-3 solutions indicated with thecontinuous lines in Fig. 3. Similar results holds alsofor larger values of D as well as for other extensions of


the Rosenzweig–MacArthur model (Candaten andRinaldi, 2003).

3.3. Predator mean density is maximum at the edge of

chaos

An important property of the Rosenzweig–MacArthur food chain, discovered by Abrams (Abrams,1993; Abrams and Roth, 1994), and further discussed inDe Feo and Rinaldi (1997), is that the mean value ofthe density of the top population is maximum at thetransition from cyclic to chaotic regimes where thedynamics are dominated by high-frequency resource–consumer oscillations. If the discussion is carried outwith respect to resource carrying capacity, this transitionoccurs at the right edge of the chaotic interval. Fig. 4shows that this intriguing property, which holds also ina number of variants of the same model (Gragnani et al.,1998; Rinaldi and De Feo, 1999; Lindstrom, 2002),is not destroyed by antipredator behavior (D=0.02 inFig. 4).An interesting problem in the context of antipredator

behavior is its impact on food chain stability. Even ifmany investigations support the idea that antipredatorbehavior is a stabilizing factor, there are many othercontributions, mainly by Abrams and coauthors,supporting the opposite idea (see, for instance, Abrams,1984, 1989, 1990; McNair, 1986; Matsuda and Abrams,1994; Abrams and Matsuda, 1997). We believe thatthese opposite conclusions are simply due to thedifferences in the behavioral rules (i.e., in the functionfin and fout) implicitly or explicitly used by variousauthors.Intuitively, one would expect that pronounced anti-

predator behaviors prevent predators from reachingvery high densities and, conversely, protect consumersfrom dropping to very low densities. In other words,antipredator behavior seems to favor stationary regimes

resource carrying capacity, K

0.9 1.0 1.1 1.2 1.3

mea

npr

edat

orde

nsit

y

8.5

9.0

9.5

chaos

Fig. 4. Mean predator density as a function of resource carrying

capacity (parameter values as in Fig. 2). Predator abundance is

maximum at the edge of chaos.

(against cyclic or chaotic regimes) and/or the dampingof population oscillations. We now show, throughbifurcation analysis, that this is, indeed, what happensin our food chain model when the characteristicfunctions fin and fout are as in (13). However, even ifour analysis points out a case in which antipredatorbehavior is stabilizing, this is certainly not always thecase.The numerical bifurcation analysis of the model with

respect to defense and any other second parametercan be performed through continuation techniques(Kuznetsov, 1995) starting with D=0 (Kuznetsov andRinaldi, 1996; Kuznetsov et al., 2001). Fig. 5 shows atypical result of the analysis: the parameter space (K, D)is subdivided into various regions where stationary,cyclic, and chaotic coexistence of the three populationsis possible. The chaotic region is a sort of narrow bandinside the cyclic region. The true boundaries of thechaotic region are the result of very complex bifurcationstructures (Kuznetsov et al., 2001), characterized byinfinite sequences of flip (period doubling) bifurcations,where a limit cycle of period T becomes a limit cycle ofperiod 2T. The gray region in Fig. 5 is only anapproximation of the chaotic region because its bound-aries are the first flip bifurcation curves of the above-mentioned sequences.Fig. 5 shows that for sufficiently high values of D, i.e.,

for pronounced antipredator behavior, steady coexis-tence of the three populations is the only possibleoutcome. Moreover, if the system is chaotic andantipredator behavior is gradually enhanced, thencoexistence becomes first cyclic and, finally, stationary.Thus, it is valid to say that antipredator behavior is astabilizing factor. However, Fig. 5 also shows that forquite high carrying capacities, cyclic coexistencecould become chaotic by enhancing antipredator beha-vior, a fact that would bring us to the oppositeconclusion, namely that antipredator behavior isdestabilizing. Nevertheless, a further increase of the

resource carrying capacity, K0.1 1 10 100

defe

nse,

D

0.01

0.1

1

10

100

pred

ator

exti

ncti

on

stationary

cycl

ic

cyclic

chaotic

Fig. 5. Bifurcation diagram of model (1, 7�13) showing the various

regions of coexistence in the parameter space (K, D) (notice that K and

D are in logarithmic scale). Parameter values as in Fig. 2.

ARTICLE IN PRESS

defense, D

0.01 0.1 1 10

tim

eof

conv

erge

nce,

T con

v

0

100

200

300

400

Fig. 6. Time of convergence toward equilibrium Tconv (see text for

definition) as a function of defense. Parameter values as in Fig. 2 and

K=0.55 (see dotted vertical line in Fig. 5).

S. Rinaldi et al. / Theoretical Population Biology 66 (2004) 277–286284

parameter D would again produce a cyclic regime and,finally, the stationary coexistence of the three popula-tions.Another way for judging if a parameter is stabilizing

or destabilizing is to find out if the speed of convergencetoward the attractor increases or decreases with theparameter. If the attractor is an equilibrium, this can bedone very easily. In fact, in that case, the Jacobianmatrix evaluated at the equilibrium has only eigenvalueswith negative real part (because the equilibrium isstable). If the smallest real part in module of alleigenvalues is a*, the state of the system convergestoward the equilibrium as expð�a�tÞ; so that thecharacteristic time of convergence is Tconv ¼ 1=a�:Therefore, the parameter D is stabilizing if Tconv

decreases with D. This is exactly what happens in ourmodel, as shown in Fig. 6 where Tconv hasbeen computed by varying D in the range indicated inFig. 5 (dotted line). In conclusion, if our food chain ischaracterized by stationary coexistence, antipredatorbehavior is a stabilizing factor, also in the sense thatan increase of D accelerates the transient towardequilibrium.

4. Conclusions

Tritrophic food chains composed of resource, con-sumer, and predator populations have been studied inthis paper under the assumption that consumers haveantipredator behavior. The method allows one to derivethe predator and consumer functional responses startingfrom rules for moving between discrete behavioralstates. The model assumes that both consumer feedingand risk of predation are affected by its behavioral state.The predator functional response is a simple extensionof previous results obtained with second-order prey–predator models (Ruxton, 1995) while the consumer

functional response is original. The resulting model iswell balanced because it points out explicitly the effectsof antipredator behavior on the bottom as well as on thetop of the food chain.The analysis performed in the paper shows that the

standard Rosenzweig–MacArthur model is not suitedfor describing tritrophic food chains where consumershave marked antipredator behavior. However, whenconsumers have weak antipredator behavior, some ofthe most relevant properties of the Rosenzweig–MacArthur model remain qualitatively valid.Predator and consumer functional responses are not

derived by adopting an optimization framework, butrather by assuming a particular set of behavioral rules ofthe two populations. The same method (namely time-scales separation of behavioral and ecological dynamics)can be applied to other food webs and other behavioralrules, provided behavioral dynamics are sufficiently fastrelative to ecological dynamics. For example, food webswith competing consumer subpopulations characterizedby different degrees of antipredator behavior could bestudied with the same approach used in this paper.The analysis of a particularly simple case is in

agreement with other studies pointing out the stabilizingnature of antipredator behavior (Ives and Dobson,1987; Sih, 1987a; Ruxton, 1995). It shares with thosestudies the assumption that consumers respond topredator density rather than to actual risk, and thatthe consumers’ rate of increase is a linear function of itsfood intake. Models in which consumers respond to risk(e.g., Abrams and Matsuda, 1997) and models in whichthe consumer’s rate of increase is a saturating functionof food intake (Abrams, 1995) show that anti-predatorbehavior can also be destabilizing. Fig. 5 shows that achaotic food chain becomes first cyclic and thenstationary if antipredator behavior (D in Fig. 5)becomes sufficiently pronounced. However, the samefigure points out a phenomenon that could not bedetected with second-order prey–predator models,namely that in some ranges antipredator behavior canbe destabilizing. Indeed, food chains with high resourcecarrying capacity can switch from a chaotic to a cyclicregime if antipredator behavior is reduced. However, forvery high antipredator behavior the food chain can onlybe stationary.We close the paper by pointing out three main

limitations of our analysis. First, the results have beenproved under the assumption that the behavioral modelhas a single stable equilibrium, while there are certainlycases of wild (e.g., bistable, cyclic or even chaotic)behavioral dynamics. In these cases, the time scaleseparation method can still be used, as described inRinaldi and Scheffer (2000), but becomes definitelymore complex. Second, the behavioral dynamics of thepopulations have been assumed to be structured as inFig. 1, while there are other important structures that


would be worth studying (e.g., consumers with alter-native habitats characterized by different degrees ofvulnerability and resource availability, less crude aggre-gations of the various behavioral stages, for example, bysplitting the first stage of Fig. 1 into four distinct stages,namely pursuit, killing, manipualting, and transport-ing). Finally, our conclusions on the stabilizing effect ofantipredator behavior are far from being general,because they have been derived through a specific choiceof behavioral rules. It is therefore not surprising thatother choices lead to opposite conclusions.

Acknowledgments

This work was partially supported by MIUR underProjects FIRB 2001—RBNE01CW3M and PRIN2002–2002098244 and by IEIIT, CNR. The authorsare very grateful to Peter Abrams and to twoanonymous reviewers for their constructive criticismson a previous version of this paper.

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