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Decoys in Predation and Parasitism
Wilkinson, Michael
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Comments on Theoretical Biology, 8: 321–338, 2003
Copyright # 2003 Taylor & Francis
0894-8550/03 $12.00 + .00
DOI: 10.1080/08948550390206830
Decoys in Predation and Parasitism
Michael H. F. Wilkinson
Institute for Mathematics and Computing Science,
University of Groningen, The Netherlands
Predator-prey or host-parasite dynamics can be altered by the presence of
other species through several mechanisms. One such mechanism is the
‘‘decoy effect,’’ which itself can take a variety of forms. In its simplest
form, the third species, which is inedible to the predator, nonetheless
interferes with predation because the predator spends time investigating
these decoys. The effect of this is to reduce instability in the ecosystem
by damping the Lotka-Volterra oscillations. This effect may be considerable in dense ecosystems such as the gut microflora. Other forms include
the prey species creating decoys by autotomy of appendages, such as tails,
in the case of certain lizards. Finally, decoys are created by various parasites to avoid elimination by the immune system. In this review the theoretical models and supporting observations for these forms of the decoy
effect are discussed, and some refinements on the theory are proposed.
In particular, a model for autotomy is extended, and the phenomenon of
signaling unpalatability or toxicity is discussed within the context of the
decoy effect. It is shown that such signaling will increase predation on
edible competitors.
Keywords: predator-prey dynamics, host-parasite dynamics, population dynamics, immune
evasion, ecological stability, decoy effect
Predator-prey and host-parasite systems are classical examples in mathematical modeling in ecology. Apart from the simplest two-species systems, several complications have been modeled involving other factors, such as the
presence of other species. The effect of only one species on the system can
be profound, either by preying on both species alike (Snyder and Ives
Address Correspondence to Michael H. F. Wilkinson, Institute for Mathematics and
Computing Science, University of Groningen, The Netherlands. E-mail: [email protected]
321
322
M. H. F. Wilkinson
2001), or by harming the prey, such as by causing increased predation
(Mallory et al. 1983; Harmon et al. 2000), or protecting the prey through a
variety of mechanisms. These mechanisms include protector species, as in
the case of nesting colonies, where smaller bird species benefit from the
aggressive behavior of larger birds in the same colonies toward predators
(Pius and Leberg 1998), or information parasitism, where a species may benefit from the alarm raised by others (Nuechterlein 1981), and the decoy effect
(Christensen et al. 1976; Vos et al. 2001; Wilkinson, 2001), which itself can
take a variety of forms. In its simplest form, the third species, which is
inedible to the predator (or unsuitable as host to the parasite), nonetheless
interferes with predation because the predator spends time investigating these
decoys. The effect of this is to increase stability in the ecosystem by damping
the Lotka-Volterra oscillations (Vos et al. 2001; Wilkinson 2001).
Usually, the effect requires a degree of specialization in the predator; that
is, it must not be able to prey on the other species, otherwise it serves as alternative prey (Mallory et al., 1983). Specialization is quite prevalent among
predators and parasites, whether microscopic (Drutz 1976; Christensen et al.
1976), or macroscopic (Vos et al. 2001). Besides, specialist predators are
often put forward as biological pest or pathogen controls (Smith and Huggins
1983; Fratamico and Whiting 1995; Vos et al. 2001), so good insight into the
population dynamics in more realistic (multispecies) ecosystems is important.
One case that has been well studied is that of arthropod predator-prey systems, precisely because of the potential for biological pest control (Vos et al.
2001). Often the damage caused by the herbivore to the plants causes the plant
to release mixtures of volatile substances, which act as attractants to predators
(Pels and Sabelis 2000). Predators are able to distinguish the type of prey on
the plant based on these mixtures. Therefore, the predators spend less time
searching for prey than they would if no attractant chemicals were present.
However, when multiple herbivore species infest plants, it becomes more difficult for the predator to identify the prey patches on the plant correctly, and
the predators spend more time searching for prey as the number of predatorprey couples in the ecosystem increases (Vos et al. 2001).
The most radical difference is in cases where there is no third species
involved at all. In those cases the prey species produces its own decoys,
which distract the predator long enough to effect an escape. Such cases
include autotomy of body parts, such as tails or limbs (Arnold 1988; Punzo
1997; Miller and Byrne 2000), or production of clouds of ink in the case
of squids. In the former case, the predator receives some reward for its efforts
in that it at least has a body part to eat (Harris 1989; Miller and Byrne 2000).
The cost to the prey may vary considerably, depending on the size of the
autotomized body part, and on the degree to which mobility and fecundity
are compromised (Guffey 1998; Brueseke et al. 2001; Miller and Byrne
2000; Downes and Shine 2001). Production of decoys is also used by certain
pathogens to elude the immune system (Williamson et al. 1996; Ramasamy
1998; Sibona and Condat 2002).
Decoys in Predation and Parasitism
323
In the following sections I discuss each of these different mechanisms
and the mathematical models put forward to describe them. I end with a
discussion of how the evolution of camouflage or warning coloration may
be viewed from the decoy effect point of view.
THE DECOY MECHANISM
Christensen et al. (1976) describe the decoy effect in Fasciola hepatica
(sheep liver fluke) miracidia, which infect the snail Lymnaea trunculata.
They describe how the presence of non-host snails inhibits the ability of the
parasite to find its host, depending in part on the non-host species (related=nonrelated). In a similar setting, Yousif et al (1998) found that Schistosoma mansoni (schistosomiasis parasite) miracidia were inhibited in their
ability to find their host snail Biomphalaria alexandrina by the presence of
several other snail species.
In a different setting, the decoy effect as described in Wilkinson (2001)
derives from observations of the lifestyle of Bdellovibrio bacteriovorus and
some attempts to use them and bacteriophages for pathogen control (Westergaard and Kramer 1977; Smith and Huggins 1983; Jackson and Whiting
1992; Fratamico and Whiting 1995; Sarkar et al. 1996). Like bacteriophages,
bacteria of the genus Bdellovibrio penetrate the outer envelope of their prey,
take over control of the cell, and replicate inside, with the new cells emerging
after lysis. Unlike phages, bdellovibrios are fast swimmers, but apparently do
not use chemotaxis to find their prey (Straley and Conti 1977). This can lead
them to waste time when encountering nonprey bacteria, as observed by
Drutz (1976). In this section I describe the mathematical model as described
for the bdellovibrio-bacterium predator prey system.
In the following discussion I use the Monod model for growth of both predator and prey (Canale 1969), with and without extensions for maintenance
energy costs (Nisbet et al. 1983). Note that the Monod model for the predator
corresponds to the Holling type II predator model (e.g., DeAngelis 1992). In
the case of bdellovibrios, maintenance energy must be taken into account, but
not in the case of bacteriophages. Thus, the growth of microbial predators Y
on prey species X1 is modeled as:
my X1
dY
¼
Y D þ dy Y
dt
KX þ X1
ð1Þ
in which my is the maximum specific growth rate, KX the saturation constant,
D the dilution rate of the chemostat, and dy the starvation rate. The differential equation for species X1 is
Vy X1
dX1
m1 X0
¼
X Y DX1
dt
K1 þ X0 1 KX þ X1
ð2Þ
324
M. H. F. Wilkinson
in which X0 is the concentration of the limiting substrate, m1 is the
maximum specific growth rate, K1 is the saturation constant, and Vy is the
maximum specific uptake rate of prey by predator. The differential equation
for the limiting substrate becomes
dX0
X0
¼ D ð S X0 Þ V1
X
dt
K1 þ X0 1
ð3Þ
in which S is the input substrate concentration and V1 is the maximum
specific uptake rate of X0 by X1 .
Now consider the addition of a nonprey species X2 , which we assume to be
present at a constant level, having no direct effect on either X0 by X1 . This
allows us to study the effect of the simple presence of a decoy species independently of any other competition effect. Disregarding starvation, the predator can be in three states: free, bound to X1 , and bound to X2 . These
complexes are denoted as ½X1 Y and ½X2 Y. Assume the rate of collisions is
r per unit of prey or nonprey species per unit of predator. Furthermore, the
prey=predator complex dissociates at a rate of k1 , and the nonprey=predator
complex dissociates at a rate of k2 . However, only the dissociation of the
first complex yields new predators, with a yield of yx þ 1. Since one phage
or predator is lost in the forming of this complex, the net yield per
unit of prey biomass is yx . This leads to the following set of differential
equations:
dYfree
¼ ðyx þ 1Þk1 ½X1 Y þ k2 ½X2 Y r ðX1 þ X2 ÞYfree
dt
ð4aÞ
d ½ X1 Y ¼ k1 ½X1 Y þ rX1 Yfree
dt
ð4bÞ
d ½ X2 Y ¼ k2 ½X2 Y þ rX2 Yfree
dt
ð4cÞ
At (quasi-)steady state we have
½X1 Y ¼
r
X Y
k1 1 free
and
½X2 Y ¼
r
X Y
k2 2 free
Summing the Eqs. (4a, b, c) we find a growth rate of:
my X1 Y
dY
yx k1 X 1 Y
¼
¼
dt
k1 =r þ X1 þ k1 X2 =k2 KX þ X1 þ Kinh X2
ð5Þ
Decoys in Predation and Parasitism
325
with KX ¼ k1 =r and Kinh ¼ k1 =k2 . In this form we can recognize that the
decoy effect is essentially a form of competitive inhibition (e.g., Vos et al.
2001; Wilkinson 2001).
The Consequences for the Ecosystem
In Wilkinson (2001), a detailed stability analysis is given based on the
work of Canale (1969). Four different ‘‘modes’’ of operation can be identified (Kooi and Kooijman 1994): (0) total washout of both species; (I)
stable prey population with washout of predator; (II) stable coexistence of
predator and prey; and (III) unstable coexistence (limit cycle behavior). If
predator and prey coexist, the (stable or unstable) focus is given by:
2
3
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!2
u
u
16
V1 KX
7
t S K1 þ4K1 S 5
X0 ¼ 4S K1 2
m y D dy
my D dy
V1 KX
ð6aÞ
D þ dy KX
X1 ¼
my D dy
Y¼
D my m1
ð S X0 Þ X1
D þ d y Vy V1
ð6bÞ
ð6cÞ
with KX ¼ KX þ Kinh X2 :
Thus it can be seen that the equilibrium concentration of prey is directly
proportional to saturation constant KX and should therefore vary linearly with
the concentration of decoys (of course, provided that the solution of Y is
positive). The boundaries between modes 0 and I and between I and II as a
function of the chemostat’s control parameters (dilution rate D and input
concentration of the limiting substrate S) can be obtained analytically
(Wilkinson 2001) and are shown in Figure 1. The parameter values used
are from Nisbet et al. (1983) and Kooi and Kooijman (1994) (i.e.,
K1 ¼ 8mg=L, KX ¼ 9mg=L, m1 ¼ 0:5h1 , my ¼ 0:2h1 , V1 ¼ 1:25h1 , and
Vy ¼ 0:3333h1 ). The zone boundary between zones II and III was obtained
by local stability analysis of the steady-state solution. Figures 1a and 1b
show the results for the double Monod model with metabolic costs
(dy ¼ 0:01h1 ). Figures 1c and 1d show the transient behavior of the
chemostat for the same metabolic parameters as 1a and 1b, respectively.
Figures 1b and 1d show the stabilizing effect of decoys, with KX ¼ 2KX . As
326
M. H. F. Wilkinson
FIGURE 1 Operating diagrams for chemostats containing microbial predator-prey
systems with different numbers of decoys: (a) no decoys, that is, KX ¼ KX ¼ 9; (b)
decoy present at a level that KX ¼ 18. In zone I, only prey survive, in zone II both predator and prey coexist in a stable equilibrium, and in zone III the system becomes
unstable. Increasing KX reduces zone III and increases zone I. The stabilizing effect
is also shown in (c) and (d), which show the transient behavior for the same parameter
values as in (a) and (b), respectively. In the absence of decoys, the prey is driven to near
extinction, whereas the presence of decoys leads to a reduction of only a factor of 3.6.
Decoys in Predation and Parasitism
327
can be seen, increasing KX means that, for a given D, the predator can only
be present in the ecosystem at all at a higher input substrate concentration
than in the absence of decoys.
PLANT SIGNALS AND THE DECOY EFFECT
An interesting variant of the decoy effect is described by Vos et al. (2001).
It is induced by volatile substances that are produced when certain plants are
infested by herbivorous insects in response to the damage caused (Pels and
Sabelis 2000). The volatile substances act as lures for predators that home
in on their prey (or at least prey-dense plants) using these plant signals. In
the model of Vos and coworkers (2001), there are N predator-herbivore
pairs in the ecosystem, with each predator Yi preying on just one prey species
Xi . The 2N differential equations describing this system are
!
Vy Yi Xi
dXi
Xi w X
P
¼r 1 Xj Xi dt
K K i6¼j
KX þ Xi þ Kinh Xj
ð7aÞ
my Yi Xi
dYi
P dy Yi
¼
dt
KX þ Xi þ Kinh Xj
ð7bÞ
i6¼j
i6¼j
in which r is the maximum relative growth rate of herbivores, K the carrying
capacity, and w the coefficient of competition between herbivores. Indices i
and j run from 1 to N inclusive. As in the previous case, the inhibitory
constant Kinh is attributed to time wasted inspecting patches containing
nonprey herbivores, and the same damping of predator-prey oscillations as
in the previous model occurs.
The main differences between the model of Vos et al. (2001) and the previous (Wilkinson 2001) are the presence of multiple weakly coupled predator-prey pairs and the use of logistic growth for the prey, instead of the
Monod model, which is more accurate for bacterial growth but is not used
for macroscopic prey species. Vos et al. do not provide an analytical treatment of stability analysis; instead, they run simulations with N ranging
from 1 to 8 for a number of different values of Kinh . Figure 2 shows the
pronounced damping of predator-prey oscillations they found, using r ¼
0:09, K ¼ 10; 000, KX ¼ 2000, w ¼ 0:1, my ¼ Vy ¼ 0:2, and dy ¼ 0:1, for
different numbers of species and different values of Kinh
AUTOTOMY: MAKING YOUR OWN DECOYS
Autotomy of body parts to escape predators has evolved in several groups
of both vertebrates (Arnold 1988) and invertebrates (Punzo 1997; Miller and
328
M. H. F. Wilkinson
FIGURE 2 The decoy effect in coupled host-parasitoid systems according to Vos
et al. (2001), in which only the parasitoid dynamics is shown: (a) a single host-parasitoid pair, with Kinh ¼ 0:1, showing large amplitude oscillations that almost drive
the prey to extinction; (b) increasing the number of host-parasitoid pairs to 4 shows
clear damping of oscillations; which is even more pronounced in (c), in which 7
pairs are present; finally, (d) shows a system with 4 pairs, in which Kinh ¼ 0:2, yielding a similar damping effect as in (c).
Byrne 2000). Autotomy allows the prey to escape while the predator is occupied with the wriggling appendage. The amount of motion of the autotomized
appendage appears to have some influence on the success of this strategy
(Dial and Fitzpatrick 1983), and some body parts even secrete large amounts
of mucus, presumably to enhance their effect as a decoy (Miller and Byrne
2000). Of course, there is a serious cost involved in the loss of a limb,
though certain arachnids seem to suffer few ill effects (Guffey 1998; Brueseke et al. 2001). Besides, the predator does gain from eating the sacrificed
body part (Miller and Byrne 2000).
Harris (1989) developed mathematical models to describe the effect of
autotomous defenses on predation. He starts by defining two groups of
Decoys in Predation and Parasitism
329
prey: (1) intact prey X1 and (2) damaged prey X2 . Let the reproduction and
death rates of Xi be denoted as mi and di , respectively. Note that the progeny
of X2 are added to X1 . The recovery rate by which X2 is transformed back to
X1 is r. Furthermore, assume that the predator Y grows by consuming only
the autotomized appendages, and damaged prey is not attacked (or recovers
before a second encounter). The set of differential equations governing this
system are
dX1
¼ ðm1 d1 ÞX1 vy X1 Y þ ðm2 þ rÞX2
dt
ð8aÞ
dX2
¼ vy X1 Y ðd2 þ rÞX2
dt
ð8bÞ
dY
¼ my X1 Y dy Y
dt
ð8cÞ
As can be seen, predator growth is modeled using the Lotka-Volterra model.
The steady-state solution is
dy
my
ð9aÞ
dy m 1 d1
my d2 m2
ð9bÞ
d2 þ r m 1 d1
v y d 2 m2
ð9cÞ
X1 ¼
X2 ¼
Y¼
For the solutions for X2 and Y to be positive, Harris derives that d2 > m2 ; that
is, the damaged prey must have a higher death rate than reproductive rate,
which is somewhat counterintuitive.
There are several comments that should be made regarding the above
model. First of all, no limits on growth of the prey other than predation are
modeled, which is clearly unrealistic. Therefore, some of the predictions
may have to be altered. Logistic growth would be an improvement.
Second, prey satiation should also be included, using, for example, the Holling Type II model. Third, and more seriously, the assumption that damaged
prey are not attacked is evidently not true. There is a great deal of literature
devoted to the increased risk of predation of prey damaged through autotomy
(Wilson 1992; Brueseke et al. 2000; Miller and Byrne 2000; Downes and
Shine 2001). Furthermore, there is ample evidence that predators do succeed
330
M. H. F. Wilkinson
in capturing prey despite the autotomous body part (Dial and Fitzpatrick
1983; Cooper and Vitt 1991; Punzo 1997; Brueseke et al. 2000). For example, Punzo (1997) reports that leg autotomy was only used successfully by the
wolf spider Schizocosa avida in 16% of attacks by the scorpion Centruroides
vittatus. Adding these effects we have
Vy X1 Y
dX1
X þ X2
¼ r1 1 1
X1 KX þ X1 þ X2
dt
K
X þ X2
þ r 2 1 þ d2 1
þ r X2
K
ð10aÞ
dX2 Vy ðpe X1 X2 ÞY
¼
ðd2 þ rÞX2
KX þ X1 þ X2
dt
ð10bÞ
dY my ½ð1 we pe ÞX1 þ we X2 Y
¼
dy Y
dt
KX þ X1 þ X2
ð10cÞ
with pe the probability of escape through autotomy and we the fraction of the
body mass of each prey individual that escapes successfully; that is 1 we
is the percentage of body mass lost through autotomy. Obviously, both pe and
we must be 1. Logistic growth is modeled as usual, with r1 and r2 the maximum net reproductive rates for X1 and X2 , respectively. It is assumed that as
the combined population approaches the carrying capacity K, that the
reproductive rates decrease, whereas the death rate d2 of X2 stays constant.
Furthermore, if the predator captures a damaged prey, it only obtains a
fraction we of the food value it would obtain if it had caught an intact prey
individual.
The stability analysis for Eqs. (10a, b, c) is rather more complicated
than that of (8a, b, c), and I am not aware of a discussion of (10a, b, c) or
similar equations in the literature. However, Figure 3 shows the change in
dynamics of the system for we ¼ 0:9, r1 ¼ 0:09, r2 ¼ 0:08, d2 ¼ 0:005,
and all other parameters as in Figure 2, when pe changes from 0 (no autotomy) to 0.4 (40%) chance of escape through autotomy (after Cooper and
Vitt 1991). The damping of predator-prey oscillation is again evident.
Cooper and Vitt (1991) note that autotomous body parts that are conspicuous in some way are more effective as decoys, because they cause the predator to grab the conspicuous part with increased probability. Examples given
by them are those of the lizards Eumeces fasciatus and E. laciteps, which
have brightly colored tails. Let Pd be the probability of discovery, and Pe the
probability of escape once detected. The probability of being captured Pc is
Pc ¼ Pd ð1 Pe Þ
ð11Þ
Decoys in Predation and Parasitism
331
FIGURE 3 The effect of autotomy in predator-prey systems: (a) system as in Eqs.
(10a,b,c) without autotomy strong oscillations occur; (b) given a 40% chance of
escape using autotomy of body part of 10% of total weight, the predator-prey oscillations are damped out.
Now suppose that within a population with cryptic coloration of the autotomizable body part a mutation occurs that increases the visibility of the body
part. This increased visibility means that the probability of detection
increases by an amount a, with 0 < a ð1 Pd Þ. Besides, the probability
of escape is assumed to increase, due to the increased probability that the predator grabs the prey by the autotomizable body part. Let this increase in the
conditional probability of escape be b, with 0 < b ð1 Pe Þ. The
probability of capture Pc0 of the mutant is
Pc0 ¼ ðPd þ aÞð1 Pe bÞ
ð12Þ
If Pc0 < Pc , or
b>a
1 Pe
Pd þ a
ð13Þ
the mutant is favored. This occurs when b or Pd is large, or a is small compared to either. Furthermore, if 1 Pe is small (i.e., the chance of escape is
large) the mutant is also favored, which can be explained by the fact that if
detected individuals usually escape, they can more easily afford to be conspicuous. By contrast, crypsis is favored if the prey is most often captured
once detected. Cooper and Vitt (1991) go on to postulate power-law relationships between b and a to derive equilibrium conditions, but do not provide
motivation for the relationship or stability analysis for the equilibrium
found, so the significance of this is unknown (as they themselves note).
They extend the model further to account for the effect of conspicuousness
in multiple stages of the predator-prey encounter. Let Pcjd be the probability
for being captured once detected, Pkjc the probability of being killed once
332
M. H. F. Wilkinson
captured, and bc and bk the improvements in avoiding capture and being
killed, respectively, obtained at the expense of an a increase in detectability.
In this case, it can be shown that conspicuousness is favored if
ðPd þ aÞ
bc Pkjc þ bk Pcjd bc bk
> Pcjd Pkjc
a
ð14Þ
Cooper and Vitt (1991) demonstrated the validity of their model with an
experiment in which tails of lizards were painted either blue or black. The
two groups were faced with scarlet king snakes, and the strikes at various
body parts and escapes were counted. When the tail was painted black, the
proportion of body strikes was 94.1%, compared to 45.0% in blue-tailed
lizards. The total proportion of escapes rose from 23.5% to 60% between
black- and blue-tailed lizards. They estimate the parameters of the model
as Pcjd ¼ 0:824, Pkjc ¼ 0:929, bc ¼ 0:124, and bk ¼ 0:358 (wild-state
values for Pd and a could not be estimated from the experiment). Given
these values, inequality (14) means that conspicuousness is selectively
favored if Pd > 1:09a. In other words, only if the detection rate approximately doubles due to the conspicuous autotomizable body part will cryptic
coloration be favored.
Of course, there are many ways in which an organism can avoid such a high
increase in detectability. The copious production of mucus from autotomized
cerata of the nudibranch Phidiana crassicornis (Miller and Byrne 2000) and
prolonged wriggling of autotomized body parts (Dial and Fitzpatrick 1983;
Miller and Byrne 2000) may be ways to increase the conspicuousness of the
autotomized body part in a way that simultaneously limits the detectability of
the prey itself, thereby increasing b without increasing a too much.
DECOYS AND EVASION OF THE IMMUNE SYSTEM
A variant of the mechanism just given is described by Sibona and Condat
(2002) where decoys are produced to counteract the immune response directed against a parasite. Pathogens and parasites have evolved various strategies
to avoid the actions of the immune system (e.g., Donelson et al. 1998).
Among these strategies is the production of decoys, such as the release of
immunodominant fractions of surface proteins into the blood stream in
Plasmodium falciparum (Williamson et al. 1996), or surface proteins with
extensive repeat regions, which may act as a sink for protective antibodies
(Ramasamy 1998). The model discussed by Sibona and Condat refers to
Chagas’ disease, caused by Trypanosoma cruzi, which also produces decoys.
Limiting the discussion to a single antibody species A, the parasite density X
is determined by the following set of differential equations:
Decoys in Predation and Parasitism
333
dX
¼ mX aAX
dt
ð15aÞ
dA
1
¼ gX aAX ðA A0 Þ
dt
t
ð15bÞ
in which a is the reaction rate constant for the antibody-parasite reaction, t
is the intrinsic lifetime of an antibody molecule, g determines the induced
antibody production rate, and A0 is the level of antibody A in the absence
of infection. These equations can lead to three outcomes: (1) chronic disease
if g > m and aA0 > m (X remains bounded as t ! 1); (2) healing if
g > m > aA0 (X ! 1 as t ! 1); and (3) host death if g < m (X goes to
infinity as t ! 1). They also note that if the initial inoculum is sufficiently
small, a separate case (3b) arises, in which the host survives despite the small
induction of antibody production, due to a relatively high aA0 .
Sibona and Condat then discuss two different forms of decoy effect: (1)
The parasite produces decoys that stimulate antibodies, which do not bind
to the actual parasite (similar to the case described by Williamson et al.
1996), and (2) decoys stimulate production of antibodies specific for the parasite itself. The first case is the simplest. Assuming the resources of the
immune system are limited, any immune response to the decoys draws
resources away from the immune response directed against the parasite
itself, so that g is reduced, increasing the risk of death. In the second case
an extra differential equation for the decoys D must be introduced,
dD
¼ md X ad AD
dt
ð16aÞ
and the equation for antibodies must be changed to
dA
1
¼ gX þ gd D aAX ad AD ðA A0 Þ
dt
t
ð16bÞ
in which md is the production rate of decoys, ad is the rate constant for the
antibody-decoy reaction, and gd determines the induction of the immune
response by decoys. For chronic disease, the steady-state situation becomes
m
a
ð17aÞ
md
m aA0
t ðg mÞad m þ ðgd a mad Þmd
ð17bÞ
A¼
D¼
334
M. H. F. Wilkinson
X¼
ad m
D
a md
ð17cÞ
They go on to show that the production of decoy is only beneficial to the
parasite if
m>
a
g
ad d
ð18Þ
which means that the decoy must not stimulate the antibody production too
much, but must be able to neutralize antibodies efficiently. Another interesting point discussed by Sibona and Condat is the appearance of a fourth case
in the phase diagram: limit-cycle behavior. Unfortunately, though they do
indicate that the existence of this region is determined by ‘‘a highly complex
inequality,’’ they do not give the exact boundary. The only examples they
give are in a region where inequality (18) does not hold, in which case
decoy production is not realistic from an evolutionary point of view. However, it is interesting that in this case decoys are capable of inducing oscillations, rather than damping them.
THE ‘‘INVERSE’’ DECOY EFFECT: SIGNALING TOXICITY
OR UNPALATABILITY
An interesting twist to the decoy effect may be the following. Consider
two species X1 and X2 that, unlike in the previous cases, compete for the
same resources. Assume that X1 is edible to predator Y, whereas X2 is
poisonous or at least unpalatable, which incurs a small cost c. If X1 and X2 are
indistinguishable to Y, or at least only distinguishable at short range, X2 will
act as a decoy (if present in any numbers) and therefore protect X1 . Assuming
logistic growth of X1 and X2 , and Holling Type II kinetics for the predator, we
obtain the following differential equations:
Vy X1 Y
dX1
X þ X2
¼r 1 1
X1 KX þ Kinh X2 þ X1
dt
K
ð19aÞ
dX2
X1 þ X2
¼r 1
X2 cX2
dt
K
ð19bÞ
my X1 Y
dY
¼
dy Y
dt
KX þ Kinh X2 þ X1
ð19cÞ
Decoys in Predation and Parasitism
335
Clearly, if no predators are present (Y ¼ 0) only X1 will survive, due to the
cost term cX2 , which is not balanced by the predation term in (19a), so we
will have X1 ¼ K. In general this equilibrium is unstable to invasion by Y. In
the absence of X2 , a two-species equilibrium will occur, in which
X1 ¼ dy KX =ðmy dy Þ, and Y ¼ Vy1 r½KX þ ð1 KX =KÞX1 X12 =K. When
all three species are present, the nontrivial steady-state solution to this set of
equations is
X1 ¼
KX þ Kinh Kð1 c=rÞ
Kinh þ my =dy 1
X2 ¼ Kð1 c=rÞ X1
Y¼
cX1
Vy d y
ð20aÞ
ð20bÞ
ð20cÞ
Under the condition that
Kð1 c=rÞ >
dy KX
m y dy
ð21Þ
that is, the carrying capacity of the ecosystem for X2 is larger than the twospecies equilibrium density for X1 , the three-species equilibrium density of X1
is an increasing function of the inhibition constant Kinh . Suppose that the
unpalatable species X2 can signal its unpalatability or toxicity. In that case
it can control the extent of the decoy effect by reducing the time spent by
the predator on fruitless tracking of individuals of X2 . This reduces Kinh , leading to a decrease of X1 and therefore an increase in X2 . A more thorough
study, including stability analysis, of the differential equations (20a,b,c)
has yet to be carried out.
This idea suggests that warning coloration may not only be useful to an
unpalatable or poisonous species to avoid damage caused by (abortive) predator attacks, but also may suppress the decoy effect, which might otherwise
benefit a competitor. The secretion of toxins in mucus by the nudibranch
Glossodoris pallida may serve such a purpose, especially if the predators
have poor eyesight (Avila and Paul 1997).
DISCUSSION AND CONCLUSIONS
Decoys occur in many forms, and the connecting theme is (dis)information
(Vos et al. 2001). Whenever the predator, parasite, or indeed immune system
is misinformed about the exact whereabouts of the prey, host, or pathogen,
a decoy effect is present. Seen this way, even the beneficial effect of
336
M. H. F. Wilkinson
camouflage may be considered as a variant of the decoy effect. In this case
the prey does not physically create decoys, but by mimicking something inedible in its environment that is common, it causes the predator to waste time
inspecting nonprey items. It therefore effectively does produce decoys, without the costs involved in autotomy. The molecular mimicry found in several
pathogens and parasites, and which is supposed to aid in immune evasion,
follows the same pattern (Ramasamy 1998).
The decoy effect causes an interaction between information webs and
food webs, and thus forms a ‘‘weak link’’ in the food web, which increases
stability (McCann 2000). This is supported by a number of observations. One
example is that chemotaxis toward the host was found to reduce the decoy
effect (Chipev 1993). The extra information obtained from chemoattractants
secreted by their hosts in part canceled the decoy effect caused by non-host
snails. This is given further support even in the earlier work of Christensen et
al. (1976), in which it was found that related (Lymnaea) species produced a
far more pronounced decoy effect than did a number of less related species.
They also showed that the F. hepatica miracidia were attracted to L. pereger,
a non-host Lymnaea.
Though the word ‘‘decoy’’ might be thought of as implying attraction of
the predator or parasite species, this need not be the case. In the case of B.
bacteriovorus described in Wilkinson (2001), the presence of the decoy
species simply leads to collisions with the predator, which is assumed to
move through the medium randomly. This is drastically different from the
case described by Vos et al. (2001), the F. hepatica–L. trunculata system of
Christensen et al. (1976), or the immune-evasion strategies of various
pathogens (Williamson et al. 1996; Ramasamy 1998; Sibona and Condat
2002). In these latter cases the information carried by the chemoattractants is
misleading, whereas in the former case the information is simply absent. Both
situations lead to the occurrence of a decoy effect, stabilizing the ecosystem.
A recurring problem is that of parameter estimation and model validation.
Though many studies do indeed provide estimates of parameters, and model
calculations with the estimated parameters are conducted, they are subsequently not compared to actual ecosystems (e.g., Dial and Fitzpatrick
1983; Cooper and Vitt 1991; Vos et al. 2001), usually due to the complexity
of carrying out such an experiment at the appropriate scale. In this respect,
the experiment using the bacterial model proposed in Wilkinson (2001) is
interesting, because of the small size of the organisms involved, combined
with a high reproductive rate. To the best of my knowledge, this experiment
has still not been carried out.
Finally, the word ‘‘decoy’’ seems to imply deception, and this is a bit
unfortunate, because it may lead to semantic confusion, in the same way
that the phrase ‘‘evolutionary stable strategy’’ can be misinterpreted as suggesting conscious decision making. No deception, and certainly no intentional deception, is present in the cases discussed in Christensen et al.
(1976), Vos et al. (2001), and Wilkinson (2001). The decoy species may
Decoys in Predation and Parasitism
337
not benefit at all from its decoy status. Indeed, if decoy and prey species are
competitors, the decoy species may suffer as a consequence, as shown in
Eq. (20b). In that case ‘‘honesty,’’ that is, signaling the inedible status to
the predator, may increase the probability of survival. This too shows how
important it is to evaluate the interplay between information and food
webs to fully understand ecosystems of even the most modest complexity.
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