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Journal of Theoretical Biology 310 (2012) 31–42
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Journal of Theoretical Biology
journal homepage: www.elsevier.com/locate/yjtbi
Optimal nutrient foraging strategy of an omnivore: Liebig’s law determining
numerical response
József Garay a,n, Zoltán Varga b, Tomás Cabello c, Manuel Gámez d
a
Research Group of Theoretical Biology and Evolutionary Ecology of the Hungarian Academy of Sciences and Department of Plant Systematics,
Ecology and Theoretical Biology, L. Eötvös University, Pázmány P. sétány 1/C, H-1117 Budapest, Hungary
b
+ Hungary
Institute of Mathematics and Informatics, Szent István University, Páter K. u. 1., H-2103 Gödöllo,
c
Department of Applied Biology, University of Almerı́a, Ctra. Sacramento s/n, ES-04120 Almeria, Spain
d
Department of Statistics and Applied Mathematics, University of Almerı́a, La Cañada de San Urbano s/n, 04120 Almeria, Spain
H I G H L I G H T S
c
c
c
c
We build a bridge between stoichiometric ecology and optimal foraging models.
Give a theoretical explanation of a bistable coexistence observed in omnivory.
Our model includes different biological mechanisms at a time.
May find application in nutritional ecology, intraguild predation and pest control.
a r t i c l e i n f o
abstract
Article history:
Received 5 May 2012
Received in revised form
14 June 2012
Accepted 15 June 2012
Available online 28 June 2012
The paper is aimed at a theoretical explanation of the following phenomenon. In biological pest control in
greenhouses, if an omnivore agent is released before the arrival of the pest, the agent may be able to colonize,
feeding only on plant and then control its arriving prey to a low density. If the pest arrives before the release
of the agent, then it tends to reach a high density, in spite of the action of the agent. This means that
according to the initial state, the system displays different stable equilibria, i.e. bistable coexistence is
observed. Based on the biological situation, the explaining theoretical model must take into account the
stoichiometry of different nutrients and the optimal foraging of the omnivore agent. We introduce an optimal
numerical response which depends on the optimal functional responses and on the ‘mixed diet–fitness’
correspondence determined by ‘egg stoichiometry’, in our case by Liebig’s Law; moreover we also study the
dynamical consequences of the latter when the plant is ‘‘inexhaustible’’. In our model, we found that under
Holling type II functional response, the omnivore–prey system has a unique equilibrium, while for Holling
type III, we obtained bistable coexistence. The latter fact also explains the above phenomenon that an
omnivore agent may control the pest to different levels, according to the timing of the release of the agent.
& 2012 Elsevier Ltd. All rights reserved.
Keywords:
Ecological stoichiometry
Imperfectly substitutable resources
Liebig’s Law
Numerical response
Omnivory
1. Inroduction
1.1. The theoretical problem
The basic experimental motivation of our present study is a
phenomenon observed in greenhouses, where an omnivore agent
feeding only on a plant, reached a lower population density than
in case of living on a mixed diet. If this agent is released before the
arrival of the pest, the agent may be able to colonize and then
control the arriving pest to a low density. If the pest arrives before
the release of the agent, then it tends to reach a high equilibrium
density, in spite of the action of the agent. At theoretical level this
means that according to the initial state, the system displays
different stable equilibria, i.e. bistable coexistence is observed.
In greenhouse tomato crops, for instance, the phytophagous
spider mites (Tetranychus urticae Kock) (Acari: Prostigmata: Tetranychidae) and its omnivore predator – Phytoseiulus persimilis AthiasHenriot (Acari: Mesostigmata: Phytoseiidae), also feeding on pollen –
are a good example for the above phenomenon (Gerson et al., 2003).
In the considered biological situation, for a general theoretical
model, the following three mechanisms must be taken into account:
stoichiometry, optimal foraging and population dynamics based on
functional response.
n
Corresponding author. Tel.: þ36 1 3722500/8798; fax: þ36 1 3812188.
E-mail addresses: [email protected] (J. Garay),
[email protected] (Z. Varga), [email protected] (T. Cabello),
[email protected] (M. Gámez).
0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jtbi.2012.06.021
1.1.1. Stoichiometry
In many cases mixed diet can improve the fitness of consumers
(see e.g. Raubenheimer et al., 2009; Groenteman et al., 2006; Bilde
32
J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42
and Toft, 1994). For instance, Shakya et al. (2009) found that
intraguild predation (IGP) was more intensive on pollen-breeding
than pollen-free flowers and the plant food (flowers, fruit, leaves)
was also important. Of course, the nutrient compositions of plant
and insect diet items are very complex (Romeis et al., 2005), but it is
observed in omnivore predators that when the usual prey diet is
supplemented with plant food, their development rate generally
increases and so do the other biological parameters such as survive
rate, longevity and/or fecundity of adults (Eubanks and Styrsky,
2005). These observations confirm that, the food intake has a strong
effect on the growth rate of omnivore and its prey. The importance
of stoichiometry was emphasized by Simpson et al. (2004) in
general: ‘‘There are growing evidences that, rather than maximizing
energy intake subject to constraints, many animals attempt to regulate
intake of multiple nutrients independently.’’ A well-known example
for that is the pollen collection by bumblebees (e.g. Rasheed and
Harger, 1997).
1.1.2. Optimal behavior
The consumer’s foraging strategy will depend on its stoichiometric constraints, (cf. Shakya et al. 2009). Moreover, Eubanks
and Denno (2000) found that the presence of high-quality plant
parts reduce the number of prey consumed by omnivore.
1.1.3. Population dynamics
The consumer foraging strategy depending on its stoichiometric constraints has an important effect on the dynamics of the
interacting species, e.g. van Rijn et al. (2002) found that addition
of pollen to a young mature leaf of male sterile cucumber plant
increased omnivore predator population growth and reduced
herbivore numbers. Clearly, the functional response also plays a
crucial role in the population dynamics of the system.
A basic aim of this paper is to give a theoretical explanation of
the phenomenon recalled at the beginning of Section 1, taking
into account the above three mechanisms.
1.2. Theoretical preliminaries: why must we build up a new model?
In classical optimal foraging theory, in most cases, the energy
intake is optimized (e.g. Stephens and Krebs, 1986). However, the
nutrient contents of the food may also be an important issue, see
e.g. León and Tumpson (1975) where, borrowed from classical
microeconomics, the concepts of ‘‘perfectly substitutable’’ resources
(where the difference between them is only quantitative) and
‘‘perfectly complementary’’ resources (where each one fulfils one of
the consumer’s essential needs) have been introduced. Clearly, the
nutrition compositions of two resources have an effect on the
coexistence of two consumers (Abrams, 1987a). This question also
was studied from evolutionary perspective: Abrams (1987b, c)
pointed out that when the resources are perfectly substitutable or
complementary, divergence away from or convergence towards the
competitor’s resource-acquisition traits happens on evolutionary
time scale. Furthermore, Fox and Vasseur (2008) also found that
competition generates convergence. Surprisingly, in optimal foraging
theory and population dynamics this topic was a little neglected,
except the following two basic papers: Abrams (1987d), starting
from the terminology of León and Tumpson (1975), gave a general
overview of adaptive functional responses. Recently, Vasseur and Fox
(2011) developed Abrams’s setup, studying its evolutionary consequences. In the present paper, we also elaborate Abrams’s concept,
calculate the optimal foraging strategy (determining optimal consumption rates) and study the population dynamic consequence of
it. As a matter of fact, bistable coexistence under nutrition constraints has not been found in the above theoretical models.
According to our motivating phenomenon in which omnivore
can live feeding only on plant and also feeding only on prey, we
have to consider ‘‘imperfectly substitutable resources, on any of
which alone the consumer can subsist, but if taken together would
improve the consumer fitness’’ (León and Tumpson, 1975).
Throughout the paper we will assume that the different foods
contain enough energy for the consumers to survive, thus a
consumer is only interested in nutrients that are essential for its
fitness.
We claim, that in the existing theoretical studies bistable
coexistence has not been observed, since they did not consider
imperfectly substitutable resources with Libieg’s Law and have
not considered Holling III functional response, at the same time.
1.3. Basic assumptions of the present work
Based on the above points, for the modeling of the motivating
phenomenon the above three mechanisms are indispensable
according to the greenhouse observations. But from the point of
view of applied ecology, neither stoichiometry nor optimal foraging of the agent can be omitted; we could only vary the
functional response. What we will demonstrate is that while in
case of Holling type II there was no bistable coexistence, in case of
Holling type III there was.
During our model building we will focus on the nutrient
composition of different diet items, based on the fact that there
are some experiments in which the food intake of the IG predator
does not maximize its energy income (Hailey et al., 1998), so our
approach is motivated by stoichiometrical ecological dynamics
(Andersen et al., 2004), in the sense that the IG predator growth
rate depends on the total intake of different nutrients from the
consumed food items. This view is in harmony with the observation
that the relative abundance of nitrogen and phosphor in biomass is
closely associated in insects (Elser et al., 2003); for nitrogen and
carbon, see Denno and Fagan (2003). We will consider a system,
where plant and prey are imperfectly substitutable resources, thus
we will consider the case when each food item is incomplete at
least for one nutrient, thus the omnivore predator has no unique
optimal food item for reproduction.
We assume that stoichiometric constraints can be described
by Liebig’s Law, which determines the optimal foraging strategy
of the omnivore. This idea is basically different from the usual
approach of optimal foraging theory, which assumes that according to the densities of different foods there is always only one
which is optimal (e.g. Křivan, 2000; Křivan and Sikder, 1999), and
it is usually assumed that for an omnivore it is only the energy
intake that counts, and there is no nutrient constraint. We assume
both food items are rich in energy and some nutrients are needed
for the survival of omnivore adults, thus the nutrient constraints
determine the fitness of the omnivore (cf. Pulliam, 1975). More
precisely, we assume that only the fitness depends on the mixed
diet, and the survival probability of adults does not.1
In this paper we will consider the simplest theoretical situation,
assuming that there is only one plant species, which produces food,
say pollen, for the omnivore; while the herbivorous prey feeds on
the leaves of the plant. The plant food is not exhausted2 by the
1
Moreover, we ignore the associative effect of different diet items, i.e. when
one diet item affects the digestion of another diet item (e.g. Bjorndal, 1991).
Although we are aware that the toxin dilution has an important rule in diet mixing
(e.g. Hirakawa, 1995; Singer et al., 2002); for the sake of simplicity, we assume
each food of the omnivore contains no toxin. This is particularly true in cultivated
plants, due to the loss of many of the above mentioned substances in the process
of domestication and transformation in agronomic varieties (Kogan, 1994).
2
This simplifying assumption is also used by Coll (2009), Srinivasu and Prasad
(2010a, b) and van Rijn and Sabelies (2005).
J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42
omnivore species, and herbivores do not feed on pollen. According
to our simplifying assumption we will not model the change of
plant biomass, we will assume there is always a fixed quantity of
pollen, the overflowing pollen falls to the ground.3 In other words,
we assume that plant food provided4 by the plant to the omnivore
is a non-exhaustible resource and there is no competition for the
plant food between the omnivore and its prey. Thus we will not
investigate the coexistence of the predator–consumer-plant IGP
system (e.g. Holt and Huxel, 2007; Holt and Polis, 1997; Křivan,
2000; Křivan and Diehl, 2005). We consider only the dynamics of
two interacting species: omnivore as predator and herbivore as
prey; omnivore has two kinds of food: pollen and prey.5
Our model building consists of the following steps: First step:
using a new method (see Appendix), for a fixed (arbitrary) foraging
strategy of the omnivore, we derive the functional response corresponding to fixed densities of different food items. Second step: we
calculate the nutrition intakes determined by the derived functional
response. Third step: according to the mixed diet–fitness function,
we calculate the optimal foraging strategy corresponding to the
nutrient constraints. Observe that in this way we actually calculate
the (optimal) numerical response. Fourth step: we consider the
dynamic consequence of the fact that functional and numerical
responses are connected by stoichiometric contraints.
In Section 2, for the case of Holling type II functional response,
the mixed diet–fitness mapping for the omnivore and the optimal
foraging strategy of omnivore under nutrition constraint are
calculated. As a consequence, the population dynamics corresponding to the mixed diet of the omnivore is presented, which
implies unique stable coexistence. In Section 3, it is shown that the
considered ecological conditions lead to Holling type III functional
response, and for the corresponding dynamics bistable coexistence is obtained. Sections 4 and 5 present a summary of our
results and their discussion with an outlook, respectively.
2. Case of a Holling type II functional response
In this section we will concentrate on the fact that the
increasing prey density must increase the number of satiated
omnivore predators.
2.1. Calculation of the functional response
For the derivation of the functional response, we have to
determine how the activity probabilities pi depend on the pollen
and prey densities. Let us denote by sA[0,1] the time proportion
dedicated to pollen by the omnivore, considering its foraging
strategy, constant for a short time period. Now our basic assumption
is that the food intake of the omnivore linearly depends on both the
density of prey and the pollen supply. Let p be the probability that
there is pollen in a perception range of the omnivore. If the prey
density (biomass) is y and N is the number of perception ranges
33
covering the considered area, the probability that there is a prey in
the perception range of the omnivore then is y/N. Given a perception
range, suppose that for the omnivore k1 and k2 are the probabilities
to find pollen, and find and kill a prey, respectively. Then for the
activity probabilities we have
p1 :¼ sð1k1 Þp þsð1pÞ ¼ s 1k1 p ,
y
y
y
¼ ð1sÞ 1k2
,
p2 :¼ ð1sÞð1k2 Þ þ ð1sÞ 1
N
N
N
p3 :¼ sk1 p,
y
p4 :¼ ð1sÞk2 :
N
Thus the average time dedicated to the considered activities is
y
EðTÞ ¼ T 1 þ ðT 3 T 1 Þk1 pT 2 ðT 4 T 2 Þk2 ðy=NÞ s þ T 2 þ ðT 4 T 2 Þk2 :
N
So we have the following functional responses of the omnivore
to pollen and prey, respectively:
Pðs,yÞ :¼
sk1 p
EðTÞ
and
Fðs,yÞ :¼
ð1sÞk2 ðy=NÞ
:
EðTÞ
We remind that s ¼1 means that the omnivore feeds only on
pollen, and in case of s ¼0 it feeds only on prey. Observe that at
fixed foraging strategy of the omnivore we get a Holling II type
functional response.
2.2. Nutrient intake and mixed diet–fitness mapping
For the sake of simplicity and more transparency, we minimize
our model, considering two chemical elements, say nitrogen and
carbon as nutrients, that determine the fitness of the omnivore.
This choice is only illustrative, since by its importance, e.g.
phosphorus can be also considered, see Perkins et al. (2004)
Acharya et al. (2004). We can assume the unit of each nutrient
is its quantity contained in an omnivore’s egg.6
Let ðn1 =c1 Þ and ðn2 =c2 Þ be the nutrient intake vectors resulting
from the consumption of pollen and prey, respectively. (For
example, n1 and c1 are the nitrogen and carbon absorbed from
the consumption of a unit quantity of pollen, respectively.)
For the fixed (s,y), the total nutrient intake function is
!
!
n1 pk1 s þ n2 k2 Ny ð1sÞ
n1 Pðs,yÞ þ n2 Fðs,yÞ
1
y
Mðs,yÞ :¼
:
¼
c1 Pðs,yÞ þ c2 Fðs,yÞ
EðTÞ c1 pk1 s þ c2 k2 N ð1sÞ
Now assume that pollen is abundant in the sense that p does
not depend on the consumption rate of pollen. According to
Liebig’s Law (Huisman and Weissing, 1999), the fitness (i.e. the
biomass of produced eggs) of the omnivore is determined by the
minimum coordinate of nutrient intake function M(s,y):
min M i ðs,yÞ ¼ min n1 Pðs,yÞ þ n2 Fðs,yÞ,c1 Pðs,yÞ þ c2 Fðs,yÞ :
i
3
In greenhouse crops (not in open air crops), which are closed and small
spaces with relatively low density of flowers, the intensive use of pollinator
(bumblebees or honeybees) can reduce the pollen available to the omnivore.
4
Relatively few theoretical studies have addressed the impact of plantprovided food on herbivore–carnivore dynamics, for a review, see van Rijn and
Sabelies (2005).
5
In our simplest situation, we can ignore other, non-nutritive factors, which
have effect on the diet mix (e.g. the omnivore may also be exposed to predation
risk in certain regions of its habitat, see Hailey et al. (1998) and Singer and Bernays
(2003). Finally, we also ignore the stage structure of the species, in other words we
do not deal with the dynamics of different stages of development of insects (van
Rijn et al., 2002). We also ignored the stage structure of the plant, because in the
particular conditions of greenhouses crops, the plants are transplanted from plant
nurseries after 1 or 2 months with more than 6 leaves, and then they start to
flower almost immediately.
Remark. Liebig’s Law allows us to consider in the same model
the following three cases, according to the terminology of León
and Tumpson (1975): 1. Perfectly substitutable resources that
satisfy the same requisite needs, e.g. n1 ¼n2 and c1 ¼c2; 2. Perfectly
6
Although all insects require the essential amino acids (methionine, threonine etc.), many of them can synthesize amino acids (Lundgren, 2009). Instead of
that, we only think of chemical elements (e.g. nitrogen and carbon) contained in
pollen and prey, since these are clearly essential. Of course, chemical elements
cannot be decomposed by any organism, even some carbohydrates (e.g. starch and
cellulose) are also not digestible by certain insects, see Cohen (2004).
34
J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42
complementary resources, where each of them fulfils one of the
consumer’s essential needs, e.g. n1 a0, n2 40 and c1 40, c2 ¼0. 3.
Imperfectly substitutable resources, on any of which alone the
consumer can subsist, but which if taken together would improve
the consumer fitness, e.g. n1 on2 and c1 4c2. In cases 1 and 3, the
consumer can grow on a single resource. In the case of perfectly
substitutable resources, fitness optimization means maximization
of the biomass intake, which is similar to the standard idea of
optimal foraging theory.
2.3. Optimal foraging strategy of omnivore under nutrition
constraint
Assume that there is no optimal food for omnivore, i.e. n1 on2
and c1 4c2. Now the optimal foraging strategy depends on two
different factors: the searching and handling times, and the
nutrition intakes from different food items. We will consider
two situations. 1. Prey is scarce, thus the corresponding average
searching time is too long in the sense that the omnivore is better
off by looking for pollen. 2. Prey population is large enough, so for
the omnivore it is remunerative to look for prey.
2.3.1. Switching prey density
When does omnivore start to feed on its prey? When y is very
small, nitrogen is scarce for the omnivore, thus the fitness of
omnivore is determined by n(s,y). Suppose that prey is very rare,
yE0. Then h ¼0, and there is a critical level y0 for the prey
density, below which the omnivore would not look for prey, since
the expected intakes from insects are too low compared with the
‘‘cost’’ of searching. This is the case when in time unit the
omnivore’s intake is greater, if it is looking only for pollen.
Since, the omnivore wants to maximize its nitrogen-intake
n1 k1 pn2 k2 ðy=NÞ s þ n2 k2 ðy=NÞ
nðs,yÞ :¼
EðT S Þ
by its optimal foraging strategy: an easy calculation shows that if
y oy0 :¼ N
n1 k1 pT 2
,
n1 k1 pk2 ðT 2 T 4 Þ þ n2 k2 T 1 þ ðT 3 T 1 Þk1 p
the omnivore would not eat prey. Here we assume that denominator is positive, which can be guaranteed by the simple assumption that the handling time of pollen, T4 T2 is shorter than that of
insect, T3 T1.7
2.3.2. Optimal mixed diet
Now assume that y 4y0. Then, according to Liebig’s Law the
growth rate of omnivore is determined by the maximum of the
minimum coordinate of nutrient intake function M(s,y). Actually,
at the optimal foraging strategy denoted by s(y), both total
nutrient intakes are equal. Indeed, if total intake of one nutrient
is less than the other one, then the omnivore can increase its
fitness by feeding more on the food containing more of
the missing nutrient. If y4 y0, the omnivore maximizes
min nðs,yÞ; cðs,yÞ at s(y). Let us denote c(s,y):¼c1P(s,y)þc2F(s,y),
then at the optimal foraging strategy we have
nðsðyÞ,yÞ ¼ cðsðyÞ,yÞ:
The above equation yields
sðyÞ ¼
k2 ðc2 n2 Þðy=NÞ
:
k1 ðn1 c1 Þp þ k2 ðc2 n2 Þðy=NÞ
7
If y0 o0, then there is no switching, since the prey is very rich in nitrogen.
In this case, we have to consider opportunism as well, since when prey is very
rare, the possible optimal behavior is this: when looking for abundant pollen, if a
very rare prey is occasionally found, then consume prey. In order to simplify the
mathematical discussion, we will not consider this case in the paper.
Clearly, if c2 on2 and n1 oc1, then s(y)A]0,1[, so there is an
optimal mixed diet of the omnivore.
Example 1. For an illustration, we consider parameters n1 ¼5;
c1 ¼10; n2 ¼20; c2 ¼ 9; k1 ¼0.06; k2 ¼0.46; p ¼ 0.9; T1 ¼T2 ¼1;
T3 ¼6; T4 ¼11 and N ¼ 30. Now, the switching prey density is
y0 ¼0.02, and for (s,y)A[0,1] [1,30] the total nitrogen and carbon
intakes n(s,y) and c(s,y) are plotted in Fig. 1. Moreover, the
intersection of the surfaces of Fig. 1 represents the optimal
foraging strategy s(y) which is also shown in Fig. 2.
Even in case of several nutrients (e.g. carbon:nitrogen:phosphor see Vrede et al., 2004), according to Liebig’s Law, the
optimal mixed diet is guaranteed by the maximum of the lower
hull of the essential nutrient income functions. In other words:
each foraging strategy determines several nutrient intakes, but
according to Liebig’s Law, the fitness of omnivore is given by the
minimal essential nutrient determined by its foraging strategy.
Thus the consumer tends to maximize the minimal essential
nutrient intake by changing its foraging strategy.
Now let us assume that diet change of the omnivore is faster
than the population density change (Krivan and Cressman, 2009).
We need this condition for the application of the method of Garay
and Móri (2010). Thus the functional responses determined by
the optimal numerical response are the following:
PðsðyÞ,yÞ :¼
k2 ðc2 n2 Þðy=NÞk1 p
,
T 1 1k1 p þ T 3 k1 p k2 ðc2 n2 Þðy=NÞ þ k1 ðn1 c1 Þp T 2 1k2 ðy=NÞ þ T 4 k2 ðy=NÞ
FðsðyÞ,yÞ
:¼
k1 ðn1 c1 Þpk2 ðy=NÞ
,
½T 1 ð1k1 pÞ þ T 3 k1 pk2 ðc2 n2 Þðy=NÞ þ k1 ðn1 c1 Þp T 2 ð1k2 ðy=NÞÞ þ T 4 k2 ðy=NÞ
Moreover, the numerical response determined by the optimal
functional responses is the following:
nðsðyÞ,yÞ ¼ n1 PðsðyÞ,yÞ þ n2 FðsðyÞ,yÞ:
2.4. Population dynamics when change in the optimal foraging is
immediate
Under our simplifying assumption that developmental stages
of the omnivore are neglected and only the fitness depends on the
mixed diet, and the survival probability of adults does not; we can
follow the standard dynamical setup well-known in the literature
(Arditi et al., 2004; Danger et al., 2008; Miller et al., 2004), let us
consider the following population dynamics:
x_ ¼ xðmin nðs,yÞ; cðs,yÞ k1 xÞ
Fðs,yÞ
x :
y_ ¼ y rk2 y
y
For y4y0, we obtain the optimal mixed diet dynamics:
x_ ¼ xðnðsðyÞ,yÞk1 xÞ
FðsðyÞ,yÞ
x :
y_ ¼ y rk2 y
y
Here the conversion efficiency for the predator is 1, since we
assumed that the survival probability of adults does not depend
on the diet mix, and the unit of each nutrient is equal to its
quantity contained in an omnivore’s egg.
An easy calculation shows that this system has at most two
equilibria8, and at most one asymptotically stable equilibrium,
where both species coexist, as illustrated in the following.
8
Using the index theory of planar vector fields, the following can be proved:
for bistable coexistence the existence of at least three interior equilibria is
necessary.
J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42
35
Fig. 1. Total nitrogen and carbon intakes depending on two factors: density of prey (y) and foraging strategy of predator (s). Since the different nutrient intakes are
measured in the nutrient contents of an egg of IG predator as a unit, the vertical axis corresponds to the fitness of IG predator. According to Liebig’s Law, the realized fitness
of IG predator is determined by the minimum of both nutrient income functions, at each fixed prey density.
Fig. 2. Optimal foraging strategy s(y) corresponding to prey density y. Observe that
in our model, the optimal forging strategy is not a fixed ratio of different foods, and it
does not follow the well-known one-zero rule of optimal foraging theory.
Example 2. We consider the mixed diet dynamics with the
parameters of Example 1, and r ¼3, k1 ¼0.46 and k2 ¼1. We have
a positive equilibrium E0 ¼(1.32, 0.98), which is locally asymptotically as shown in Fig. 3.
3. Holling type III functional response: a two-patch model
with density dependent perception
The Holling type II functional response can take account of the
fact that the increasing prey density must increase the number of
satiated omnivore predators. However, when the density of prey
is very small, there may be another effect, namely, the increase of
the prey population implies a dramatic increase in the encounter
rate, and at low prey density some satiation of the omnivores
takes place, but it does not decrease dramatically the number of
active omnivores. This interaction can be described by a Holling
type III functional response. In this section we will follow the
model-building scheme of the previous one for a different predation process, therefore some details will be skipped.
3.1. Functional response
Let us consider the following simple two-patch model: Inside
the flower: there is only pollen, and outside the flower there is
prey. The omnivore diet strategy is the time h it spends outside
the flower (then it spends time 1 h inside the flower). Since now
the strategy of the IG predator is spacing we use a notation
different from that of Section 2. In the flowers the pollen
‘‘density’’ is fixed, i.e. the omnivore finds pollen there with certain
probability p0 (then it finds no pollen with probability 1 p0).
Outside of flower: Assume the perception range contains only two
home ranges of prey. Thus there are three types of perception
ranges: empty, containing one prey and containing two prey, with
respective relative frequencies ð1ðy=NÞÞ2 , 2ð1ðy=NÞÞðy=NÞ and
ðy=NÞ2 . Now assume that the omnivore only observes its prey if
there are at least two prey in its perception range. Furthermore,
once a prey is observed by the omnivore, it will be killed. Thus the
predator omnivore will observe a perception range ‘‘empty’’ with
probability ð1ðy=NÞÞ2 , and observe a perception range with prey
with probability ðy=NÞ2 (see Table 2).
Here TS is the searching time, TP the handling time of pollen
and TI the handling time of insect. Now the average time is
Tðh,yÞ ¼ T S þ ð1hÞT P p0 þ hT I
y 2
N
:
Using the methodology of Garay and Móri (2010), for a fixed
pair (h,y), we obtain that the omnivore’s respective functional
responses to pollen and prey are
Pðh,yÞ : ¼
ð1hÞp0
hðy=NÞ2
and Fðh,yÞ :¼
:
Tðh,yÞ
Tðh,yÞ
Observe that the latter function response is Holling type III.
3.2. Optimal foraging strategy
Let us assume that the omnivore can optimize its nutrient
intake instantaneously, in particular, during the time of finding its
optimal foraging strategy h, the prey density does not change.
According to Liebig’s Law, the growth rate of the omnivore is
determined by the minimum nutrient intake, thus for a fixed prey
density y, the fitness of the omnivore using foraging strategy h(y)
as the value of h the following minimum is attained:
min n1 Pðh,yÞ þ n2 Fðs,yÞ; c1 Pðh,yÞ þ c2 Fðh,yÞ :
If the pollen supply is fixed, there are two possibilities.
36
J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42
3.2.1. Switching density of prey
Similar to that in Section 3.1, the omnivore starts feeding on
the herbivore when the herbivore density is greater than
3.2.2. Optimal mixed diet
If the prey is abundant, i.e. y4y0, the minimum condition
defining h(y) we obtain
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 p0 T S
y0 ¼ N
:
ðT S þ T P p0 Þn2 n1 p0 T I
hðyÞ :¼
ðn1 c1 Þp0
2 :
ðn1 c1 Þp0 þðc2 n2 Þ y=N
Now, the functional responses determined by the optimal
numerical responses are the following:
PðhðyÞ,yÞ ¼
ðc2 n2 Þðy=NÞ2 p0
,
T S ðn1 c1 Þp0 þ ðc2 n2 ÞðT S þ T P p0 Þ þ ðn1 c1 Þp0 T I ðy=NÞ2
and
FðhðyÞ,yÞ ¼
ðn1 c1 Þp0 ðy=NÞ2
:
T S ðn1 c1 Þp0 þ ðc2 n2 ÞðT S þT P p0 Þ þ ðn1 c1 Þp0 T I ðy=NÞ2
Moreover, now the optimal numerical response is the following:
nðhðyÞ,yÞ :¼
Fig. 3. Vector field and isoclines for the mixed diet dynamics with Holling type II
functional response. There is only one stable coexistence.
Table 1
Activity distribution (two-patch model).
Empty flower
Flower with
pollen
TS
TP þTS
p2 ¼ (1 h)(1 p0) p1 ¼ (1 h)p0
No insect
observed
Insect observed and
handled
TS
TI þTS
2
p3 ¼ h 1 Ny
p4 ¼ h
ðn1 c2 n2 c1 Þp0 ðy=NÞ2
:
T S ðn1 c1 Þp0 þ ðc2 n2 ÞðT S þT P p0 Þ þðn1 c1 Þp0 T I ðy=NÞ2
3.3. Population dynamics
Now the dynamics reads as
x_ ¼ xðmin nðh,yÞ; cðh,yÞ k1 xÞ
Fðh,yÞ
x :
y_ ¼ y rk2 y
y
For y4y0, the above dynamics reads as
x_ ¼ xðnðhðyÞ,yÞk1 xÞ
FðhðyÞ,yÞ
x :
y_ ¼ y rk2 y
y
y 2
N
Table 2
Activity distribution.
Search for pollen
without finding it
Search for insect
without finding it
Search and
eat pollen
Search and
eat insect
T1
p1
T2
p2
T3
p3
T4
p4
Example 3. Let us consider now our model with parameters
n1 ¼5; c1 ¼10; n2 ¼20; c2 ¼9; p0 ¼0.9; r¼ 3; TP ¼7; TS ¼5; TI ¼10;
N ¼30; k1 ¼0.0044; k2 ¼0.1 (some of them are the same as in
Examples 1 and 2). The intersection of the surfaces n(h,y) and
c(h,y) in Fig. 4 represents the optimal foraging strategy h(y) which
is also shown in Fig. 5.
Fig. 4. Total nitrogen and carbon intakes, for Holling type III functional response. According to Liebig’s Law, the realized fitness of IG predator is again determined by the
minimum of both nutrient income functions, at each fixed prey density.
J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42
Now for the dynamics corresponding to the optimal foraging
strategy h(y), the only positive equilibrium E0 ¼(163.49, 28.68) is
locally asymptotically stable. The isoclines of the system are
shown in Fig. 6.
Example 4. If in Example 3 we change only parameter TS ¼5 to
TS ¼5/900, the new model have 3 positive equilibria E1 ¼ (75.33,
0.27); E2 ¼(274.05, 4.12); and E3 ¼(277.15, 25.77); and it can be
proved that E1 and E3 are locally asymptotically stable; E2 is
unstable. The isoclines and the equilibria are shown in Fig. 7.
Now, with the parameters of Example 4, we show in Fig. 8 how
the equilibrium coexistence depends on which species arrives first. In
case (a), the omnivore is released with population size 50 only when
the prey from its initial value 2 already has reached a size 20, which
implies coexistence at equilibrium E3. In case (b), the omnivore is
released first with population size 50, reaching 70 by the time t¼0
when the prey arrives with the same initial value 2, as in case (a). In
the latter case the omnivore is capable to control the prey and keep at
a low level corresponding to equilibrium E1. This result also shows
how an omnivore can be applied as agent in biological pest control.
37
4. Summary of our results on bistable coexistence
The basic motivation of this paper was to shed a light on
bistable coexistence of omnivore and its prey observed in greenhouses. First we emphasize that our model does not depend on
environmental factors, displays no regime shift (Scheffer and
Carpenter, 2003; Móré et al., 2009). It is well-known that the
existence of alternative stable states depends on the type of
functional response. In general, only Holling III guarantees bistability for the single-species dynamics (May, 1977; Ludwig et al.,
1978); the novelty of present work is that we combine optimal
foraging theory with stoichiometric and nutritional ecology
(Raubenheimer et al., 2009) for the case when the foods are
imperfectly substitutable. We would like to emphasize that our
model can be considered as a generalization of the standard
optimal foraging model, since in the case of perfectly substitutable resources we get back that. Furthermore, in our model setup
we also use a new general method for the derivation of functional
responses (Garay and Móri, 2010), which makes it possible to take
account of the details of the consumption process, incorporating
in this way the foraging strategy into the functional response.
Fig. 5. Optimal foraging strategy h(y). Here again IG predator consumes at a rate depending on the density of its prey.
Fig. 6. Vector field and isoclines for the mixed diet dynamics with Holling type III functional response, with parameters of Example 3. Here again there is a unique stable
coexistence.
38
J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42
Fig. 7. Vector fields and isoclines for the mixed diet dynamics with Holling type III functional response, with parameters of Example 4. (a) General picture; (b) locally
asymptotically stable equilibrium E3; (c) locally asymptotically stable equilibrium E1; (d) unstable equilibrium E2. In this case there are two stable coexistence states.
We found that while Holling type II response always implies
only one equilibrium, with Holling type III (when at low densities
of prey, consumption grows at increasing rate) the systems may
also display two equilibria, with high and low prey densities. It is
also observed that the reached equilibrium depends on which
species arrives first.
1. If omnivore predators arrive first, they soon reach a low
equilibrium density, since certain nutrients are scarce in
pollen. When the prey arrive, the fitness of the omnivore
starts to increase rapidly, and once its density achieved a
certain level, it would establish the prey density at a low level.
This effect is very similar to apparent competition (cf. Holt,
1977; Shakya et al., 2009) in the sense that the plant produces
pollen feeding the omnivore, which in its turn reduces the
density of the pest damaging the plant. However, in our model
two resources together dramatically increase the fitness of the
omnivore predator, since the two resources are imperfectly
substitutable.
2. If prey arrive first and reach a high density, then this kind of
apparent competition-like effect does not work, since at high
prey density the Holling II and III functional responses are
close to the saturation of omnivore.
Summing up: when Liebig’s Law determines the mixed diet–
fitness mapping, for bistability two things are needed: first, the
two resources are imperfectly substitutable, second, at low prey
density the functional response to prey is convex, so it dramatically increases with prey density.
This result raises the following open question: in the case of
Holling II functional response, does there exist a mixed diet–
fitness mapping other then that determined by Liebig’s Law,
which can guarantee that the numerical response increases
dramatically at low density of prey? (For e.g. a mixed diet may
dramatically increase the absorption of the limiting nutrient.) We
think, if there exists such mixed diet–fitness mapping, then
bistable coexistence is also possible.
One of our basic aims was to demonstrate when bistable
coexistence is possible. Our model dynamics is in fact much more
complicated than the results presented above. But in general we
claim that for our model, in mathematical terms, using the index
theory of planar vector fields, the following can be proved: for
J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42
39
Fig. 8. The equilibrium coexistence depends on which species arrives first. (a) Pest arrives first, x0 ¼50, y0 ¼10, the system tends to E3; (b) omnivore arrives first, x0 ¼ 70,
y0 ¼ 2, the system tends to E1. Now our model describes the phenomenon observed in biological control, which was the experimental starting point of our model building.
bistable coexistence the existence of at least three interior equilibria is necessary. Here bistable coexistence means that either there
are two locally asymptotically stable equilibria, or there are two
locally asymptotically stable limit cycles.
We mention that van Rijn et al. (2002), in a different setup,
found that herbivore number depends on the initial numbers of
predator and prey. Moreover, since the alternative stable states
depend on the arrival sequence of IG predator and prey, our
model can describe a minimal succession process (Beisner et al.,
2003). We feel that, if a system of several interacting species
having nutritional constraints are considered, in which there are
alternative stable states, and the attraction basins of these locally
stable states are near the origin, then according to the arrival
sequence of species (assembly history), the system can reach
different stable states, thus the nutritional constraints may have
an effect on the community assembly, as well.
5. Discussion
In this section, first we compare our model to existing ones.
Then we relate our results to those on the coexistence in IGP
systems. Finally, we mention some possible applications and
generalizations.
5.1. Comparison of existing models
There are theoretical papers in which the quality of food is also
taken into account (Abrams, 1987a, b, c, d; Fox and Vasseur, 2008;
van Langevelde et al., 2008; Vasseur and Fox, 2011). There are the
following differences between them and the present paper. First,
we consider multidimensional nutrient intake. In the literature
there are theoretical works, in which it is assumed that only one
nutrient limits the fitness of each consumer (Loladze et al., 2004).
Second, we build up an optimal foraging model based on
numerical response of the omnivore determined by the mixeddiet–fitness mapping (described by Liebig’s Law). Third, we
consider imperfectly substitutable resources, thus it is not necessary to consume different foods at a given ratio, as in the case of
perfectly complementary resources (cf. Abrams, 1987a, b, c, d; Fox
and Vasseur, 2008; Vasseur and Fox, 2011). Furthermore, in the
present approach, in the first step we calculate function responses
at fix prey density, for all possible foraging strategies. These interrelated functional responses determine the two essential nutrient
incomes for all possible foraging strategies. As a consequence of
Liebig’s Law, we claim that optimal foraging strategy provides the
same fitness from different nutrients. Abrams (1987d) also
considered our biological case, but he, with a phenomenological
approach, concentrated on the optimal consumption rates. Our
model can be considered as a mechanism based variant of his
model, since we focused on the foraging strategy which determines the consumption rates.
5.2. About coexistence in IGP systems
Our result may contribute to the solution of the problem of
coexistence of IG predator, IG prey and their common resource
(Holt and Huxel, 2007; Křivan 2000), although we have assumed
that the plant density does not change. IGP food web, on the one
hand, is a competitive system, since omnivore and its prey feed on
the same plant; on the other hand, it is like an apparent competitive
system, as well, since both plant and herbivore are consumed by
the same omnivore. If we assume that the competition for the plant
is very weak, then we get a system near to ours. In this sense our
case will be a particular (or ‘‘degenerate’’) case of IGP, namely when
a plant provides food to an IG predator. We emphasize that IG
predator also feeds at more than one trophic level, see e.g. Arim and
Marquet (2004), Coll and Guershon (2002), Diehl and Feissel (2001),
Holt and Polis (1997), Jordan and Scheuring (2004) and Wäckers
et al. (2005). IGP also gained major interest in biological control, not
only for conservation purpose but also for pest control in open-field
and greenhouse crops (see e.g. the book by Brodeur and Boivin
(2006) on this issue).
Since the resource in a traditional IGP model never goes
extinct, the question arises: why do we get stable coexistence of
prey and omnivore predator in the greenhouse (with Holling II
and III as well), when the plant is not a limiting factor? More
precisely: why does the rare prey not die out in our system in the
light of the results on IGP coexistence?
In our model we use indirectly the following assumptions
which can guarantee the coexistence in traditional IGP models: 1.
‘‘IG prey should be superior in competition.’’ In general, if at least
one of the nutrients is scarce in plant, then IG predator population
is established at a low level without its prey, thus IG predator
40
J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42
population has a weak effect on the plant biomass, implying that
the competition for plant is low when the prey is rare. 2. ‘‘IG
predator should gain significant fitness benefits from consumption of
the IG prey.’’ In our model, IG prey contains such a mineral which
is not enough in pollen. Thus, the nutrient constraints may help to
solve the theoretical problems of existence of three species IG
systems. This opinion is in harmony with the fact that stable
coexistence of IG predator and its prey becomes possible if both
are limited by the nutrient content of the resource (Diehl, 2003).
We emphasize that in our model the above two assumptions of
traditional IGP model are consequences of the Libieg’s Law and of
the assumption that recourses are imperfectly substitutable.
Furthermore, there are optimal foraging models to study the
coexistence in IGP systems (Abrams and Fung, 2010; Křivan and
Diehel, 2005). The basic idea is that the optimal foraging strategy
(e.g. switching) may guarantee coexistence (Abrams, 2006;
Carnicer et al., 2008). Krivan and Diehel (2005) used a classical
optimal foraging model based on ‘‘diet rule’’: IG predator feeds on
the less profitable resources only if the more profitable one is too
rare. They pointed out that ‘‘y3-species coexistence requires that
the intermediate consumer is a more profitable prey than the basal
resource’’. In spite of the fact that in our case ‘‘less and more
profitable resources’’ make no sense (each of them is scarce in one
of the nutrients), the less abundant resource is always more
profitable. Abrams and Fung (2010) considered another optimization model of a 3-species IGP system, in which functional
responses were of Holling type II, the resources are perfectly
complementary and the optimal foraging strategy of IG predator
is imperfect (it does not reach its extrema, 0 and 1). They found
that in this model the coexistence of IGP system is more likely.
Based on the above well-known results on IGP, in our case when
IG predator is intentional forager (strictly follows its optimal
foraging strategy), there exist prey densities below which IG
predator does not consume its prey (see Sections 2.3.1 and
3.2.1). In the case, if the switching density can guarantee the
maintenance of prey population (it is needed that the plant
biomass is high enough for the prey below its switching density),
then the intentional foraging can stabilize the IGP system
(cf. Abrams and Fung, 2010; Krivan and Diehel, 2005). However,
when the IG predator is opportunistic (consumes occasionally
found rare prey while looking for plant food, see Cressman and
Garay (2010) and Garay and Móri (2010), the coexistence of the
IGP system seems a harder question. We suspect that when the
level of opportunism is not too high, then the coexistence remains
possible (cf. Abrams and Fung, 2010). There is another related
open question, whether imperfectly substitutable resources alone
can stabilize the IGP system when predator is not an optimal
forager (cf. HilleRisLambers et al., 2006; Huisman and Weissing,
1999, 2001). However, in all these three questions the appropriate
dynamics should take into account the change in plant biomass,
as well.
In summary: in the light of the present paper, the coexistence
of IGP system is more likely, when the IG predator is the optimal
forager with stochiometric constraints, and IG prey and its plant
are imperfectly (or perfectly) substitutable resources.
Finally, we mentioned that there are two differences between
our results and those of HilleRisLambers et al. (2006), who propose
a few mechanisms for coexistence of omnivore and its prey, and
one of them is the case when prey is an essential food source for the
omnivore. We consider imperfectly substitutable resources and our
omnivore is an optimal forager.
5.3. About possible applications
The developed model may have an important application in
the real case of the biological control in greenhouses, which is
done fundamentally with omnivorous species, as it was mentioned earlier; it is also based on the fact that the majority of the
species display a type-III functional response (Cabello et al., 2007,
2011; Garcia-Martin et al., 2008; Tellez et al., 2009).
Our result may be used not only in biological control in greenhouse, but may be also applied in the case when plants provide food
items for natural enemies of herbivores (Wäckers et al., 2005; Young
et al., 1997). In this kind of mutualism it is an important point for
the plant that the cost of defense is less than its benefit (Szilágyi
et al., 2009). Here, the cost for the plant is its biomass consumed by
the omnivore (with or without the presence of the herbivore), and
its benefit is the biomass the herbivore would have consumed in
absence of the predator. In our present model this is what happens.
In the light of our results, we can formulate the following hypothesis: the food provided by the plant for its mutual omnivore is poor at
least in one of the essential nutrients, and the pest of the plant is rich in
this nutrient. This hypothesis could be chemically tested. We already
found some facts in the literature which are in harmony with this
hypothesis (Matras et al., 1998; Muszynska, 2001; Wäckers et al.,
2008; Wäckers, 2005; Sidhu and Patton, 1970).
From theoretical point of view, for the study of the evolution of
the origin of plant which provided food, the tools evolutionary
game theory will be needed (cf. Cressman et al., 2004; Garay and
Varga, 2000; Urbani and Ramos-Jiliberto, 2010).
5.4. Possible generalizations
Not only omnivores, but also many carnivores require a range of
different prey to obtain a full complement of essential amino acids.
For instance, wolf spider tends to prey on three species in proportions of the essential amino acids they provide in the diet
(Greenstone, 1979), and the protein and lipid ratio is also important
(Jensen et al., 2011). Moreover, herbivores also balance nutrient
intake (Behmer, 2009; Felton et al., 2009). For instance, moose eats
energy reach terrestrial vegetation, which is rather low in sodium,
thus moose generally needs to consume sodium rich aquatic plants
(Belovsky, 1978). Based on that, we can introduce ‘‘stochiophagous’’
animals consuming imperfectly substitutable and/or perfectly complementary resources, thus their mixed diet is determined by the
nutrients absorbable from the resources and by the stochiometric
constraints of ‘‘stochiophagous’’ animal. For ‘‘stochiophagous’’ animals, we think that a generalization of Abrams’s and our models
can connect in a synergic way optimal foraging theory, nutritional
ecology and population dynamics (Raubenheimer et al., 2007). For
instance, in this paper we defined the numerical response based on
Liebig’s Law. In general, the physiology of the consumer determines
the mixed diet–fitness mapping. The lipid, protein and carbohydrate ratios are also important, see e.g. Raubenheimer et al. (2007);
associative effects occur when one diet item affects the digestion of
another diet item, see e.g. Bjorndal (1991); the body–mass ratio of
the omnivore and its prey is also important, see e.g. Otto et al.
(2007); toxin dilution has an important rule in diet mixing, see e.g.
Hirakawa (1995) and Singer et al. (2002); dilution of allelochemicals contained in the food plants plays an important role, see e.g.
Hagele and Rowle-Rahier (1999). This mapping is not necessarily as
described by Liebig’s Law (cf. Logan et al., 2004; Loladze et al.,
2004). If we have a mathematically treatable mixed diet–fitness
function based on biological facts (e.g. on feeding experiment in
labs), then our model can be generalized in the future. In this
generalized model, nutrition physiology can be connected to
population dynamics and optimal foraging theory, as well.
The predation process is a quite complex phenomenon (cf.
Abrams and Ginzburg, 2000), depending at least on functional
responses (on the details of the predation actions) and on
numerical responses (on energy and nutrient intake according
to the diet choice of the predators). These functions yield different
J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42
population dynamics. In this paper we have introduced a general
model building method which can deal with the general predation phenomenon in the case when the population change is slow
in comparison with the behavior change of predator.
41
difference is more important, in our new derivation method,
although the i-th activity probability (pi) is fixed, it is not specified
during the derivation. This gives a freedom for us to take account of
the biological details of the considered situation, as we have done in
the main text.
Acknowledgments
References
The authors acknowledge Prof. A.P. Abrams and the anonymous referees for their helpful suggestions to improve the paper.
The research has been supported by the Hungarian Scientific
Research Fund OTKA (K81279) and the Scientific and Technological Innovation Fund of Hungary (ES-17/2008), the Ministry of
Education and Sciences of Spain (HH2008-0023).
Appendix. General derivation of functional response
In order to build up a population dynamics, we have to take
account of the functional response and the foraging strategy of
the omnivore. Now, from Garay and Móri (2010) we recall a new
method of derivation of functional responses. The basic assumption of this method is that during a given short time period the
density of prey does not change, thus the omnivore activity
distribution is constant during this time period. Let us consider
now the following probability distribution of activities of the
omnivore (see Table 1). Suppose an omnivore is staying in a
perception range and looking for pollen. Then it either finds
pollen or it does not, which occur with probabilities p3 and p1,
respectively. Denote by T1 the average time of searching, if there
is no pollen in the perception range, and let T3 be the average time
of searching and handling pollen, if there is any.
Similarly, if the omnivore staying in a perception range is out
for prey, then it will find a prey with probability p4, and it will not
with probability p2. Now T2 is the average time of searching if
there is no prey in the perception range, and T4 the average time
of searching and handling prey, if there is. Clearly, Ti 40 for all
i¼1, 2, 3, 4 will be assumed.
Then, from Garay and Móri (2010) we know that, for the fixed
activity distribution of omnivore p ¼(p1,p2,p3,p4), in terms of the
average time dedicated to the considered activities EðTÞ :¼
P4
i ¼ 1 pi T i , the functional response of the omnivore to the pollen
(i.e. the pollen intake by a unit quantity of omnivore) is
PðpÞ :¼
p3
:
EðTÞ
Similarly, for the prey intake of the omnivore, corresponding
to its activity distribution p, is
FðpÞ :¼
p4
:
EðTÞ
Clearly, E(T)40.
Now the question arises, in which way pi depends on the
optimal foraging strategy of omnivore, as well as on the density
of prey and pollen supply. In the forthcoming sections we will use
this general form of functional responses under different conditions.
At first glance, the reader may think that this method is not
quite new, since renewal theory gives the same result: functional
response is the amount of food items consumed on average, divided
by the average duration time (see e.g. Houston and McNamara,
1999 p. 56, and the references therein). However, there are two
main differences: the first one is rather technical; we used Wald’s
equality, which is a description of ‘‘vertical’’ in time (considering a
given time period in which the density change of prey can be
neglected); unlike renewal theory, which is a description of
‘‘horizontal’’ in time (assuming that duration time tends to infinity,
and the prey density does not change, since it renews). The second
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