Richards digger and grazer paper
vol. 155, no. 2 the american naturalist february 2000
Grazers and Diggers: Exploitation Competition and
Coexistence among Foragers with Different
Feeding Strategies on a Single Resource
Shane A. Richards,1,* Roger M. Nisbet,2 William G. Wilson,1 and Hugh P. Possingham3
1. Department of Zoology, Duke University, Durham, North was then expanded, and the competitive exclusion prin-
Carolina 27708-0325; ciple (Hardin 1960) was proposed, which stated that n
2. Department of Ecology, Evolution and Marine Biology,
species could not coexist on !n resources. In order to apply
University of California, Santa Barbara, California 93106;
this principle, a clear way in which to identify distinct
3. Department of Applied and Molecular Ecology, University of
resources and consumers is required (Haigh and Maynard
Adelaide, Waite Campus, PB 1 Glen Osmond, South Australia
Smith 1972; Schoener 1974; Levins 1979), as well as a clear
5064, Australia
definition of coexistence (Koch 1974; Hsu et al. 1978;
Submitted January 25, 1999; Accepted August 23, 1999 Armstrong and McGehee 1980; Smith and Waltman 1995).
There have been numerous theoretical studies where mod-
els have apparently violated the competitive exclusion
principle. Competing species can coexist if mechanisms
are present that effectively increase the number of re-
abstract: A mathematical model is presented that describes a sys-
tem where two consumer species compete exploitatively for a single sources present and/or allow populations to exhibit stable
renewable resource. The resource is distributed in a patchy but homo- cyclic behavior. Examples include resource partitioning of
geneous environment; that is, all patches are intrinsically identical.
the same prey item (Haigh and Maynard Smith 1972;
The two consumer species are referred to as diggers and grazers,
Schoener 1974), interspecific variation during the re-
where diggers deplete the resource within a patch to lower densities
source’s life cycle (Briggs 1993; Briggs et al. 1993), life-
than grazers. We show that the two distinct feeding strategies can
history variation among the consumers (McCann 1998),
produce a heterogeneous resource distribution that enables their co-
existence. Coexistence requires that grazers must either move faster temporal fluctuations in environmental conditions (Koch
than diggers between patches or convert the resources to population 1974; Levins 1979; Turelli 1981; Abrams 1984; Chesson
growth much more efficiently than diggers. The model shows that
1990), disturbance (Hastings 1980), interference compe-
the functional form of resource renewal within a patch is also im-
tition (Vance 1985; Fishman 1997), and spatial structuring
portant for coexistence. These results contrast with theory that con-
of the habitat (Tilman 1994).
siders exploitation competition for a single resource when the re-
In this article we investigate whether two species (or
source is assumed to be well mixed throughout the system.
phenotypes) that differ in their strategies for resource ex-
Keywords: exploitation competition, foraging, coexistence, invasion ploitation can coexist in a system where they both utilize
analysis.
the same resource. We assume the resource is distributed
in a patchy but homogeneous environment; that is, all
patches are intrinsically identical. The two species differ
Understanding the mechanisms that allow species co-
in the degree to which they deplete resources within a
existence remains a key topic in community ecology. The
patch and hence differ in their feeding strategy. The species
mathematical model of Volterra (1926) was the first to
that depletes the resource to lower levels is termed the
suggest that the indefinite coexistence of more than one
“digger” species; the other is termed the “grazer” species.
species on the same resource was impossible. This result
An important assumption we make is that the two species
interact only through exploitation competition (Milinski
* Present address: Population Biology Section, University of Amsterdam,
and Parker 1991). There is no direct interference between
Kruislaan 320, 1098 SM Amsterdam, The Netherlands; e-mail: sarichar@
individuals, all interactions occur through each species’
duke.edu.
influence on a shared food resource. Schmitt (1996) has
Am. Nat. 2000. Vol. 155, pp. 266–279. 2000 by The University of Chicago.
studied an example of such a system, where two species
0003-0147/2000/15502-0009$03.00. All rights reserved.
Coexistence of Feeding Strategies 267
of benthic marine snails, Tegula aureotincta and Tegula the contrasting feeding strategies of the two competing
eiseni, compete for microalgae. Differences in the foraging species and the dynamics of resource renewal. Coexistence
morphologies of the two snails have been shown to have can occur because each consumer does not encounter a
different effects on the distribution of the algal resource. fixed amount of resource when it visits a patch but an
Tegula eiseni was found to be capable of reducing algal amount that depends on the time since the last consumer’s
densities to lower levels than T. aureotincta. Nectarivores visit and its type. Hence, consumers encounter resource
are another example of a group of consumers that often levels described by a probability density function, which
appear to interact through exploitation competition. Lav- effectively increases the number of resource types. The
erty and Plowright (1985) studied a system where two digger species can often persist because it can exploit re-
species of bumblebee and a hummingbird compete for sources that the grazer species cannot reach. The grazer
nectar in jewelweed. In this example the consumer species species can also persist if it moves faster among patches
were found to differ in the depth to which they could than the digger species and encounters patches that have
drain the nectar spur and also the rate at which they visited not been recently visited by diggers. In doing so the grazer
flowers (Laverty and Plowright 1985). species can often reduce the mean resource abundance
Possingham (1987) constructed a mathematical model within the system so that it stops the diggers from taking
of nectarivore competition (which was assumed to be ex- over. We also find that the dynamics of resource renewal
ploitative) and showed that two consumer species, which play an important role in determining the outcome of
differed in their ability to deplete nectar from a flower, exploitation competition.
could coexist on a single flower species. Coexistence was
dependent on the competitive ability of both species,
The Model
which was defined as the mean net energy gained per
calorie extracted from a flower, divided by the cost of using We consider a habitat that contains a large number of
each flower. Wilson et al. (1999) fitted a similar mathe- identical small patches each containing a resource of den-
matical model, which described the dynamics of two ben- sity x. There are two species of consumers, which we refer
thic grazers competing for algae, to data collected by to as diggers and grazers. Grazers (G) can only consume
Schmitt (1996). Although the model required a number resources on a patch whose density exceeds xG, and when
of parameters, the data were sufficient to give estimates a grazer visits such a patch, its density drops to xG. Diggers
to all parameters but one. The best fit was found to lie (D) consume resources in a similar manner, reducing the
very close to a region of coexistence. Given the uncertainty resources in a patch to level xD. They may eat resources
in the fitted parameters, the model predictions were not on patches that have a lower density than that accessible
inconsistent with observed coexistence in the field to the grazers so that x D ! x G. We assume that patch size
(Schmitt 1996). In this article we have reviewed, gener- is sufficiently small so the timescale at which patches are
alized, and extended the work of Possingham (1987) and depleted of resources is fast compared with the timescale
Wilson et al. (1999) by investigating how the model of of resource renewal (Possingham 1988). A patch can be
resource renewal affects coexistence. in one of two states, depending on the amount of resource
We present a mathematical model that describes the it contains. The state of a patch is dependent on the time
dynamics of the two consumer species and the shared since it was last visited by a consumer and the species of
resource. The habitat is assumed to be made up of a large the last visitor. Patches that have a resource density 1xD
number of identical patches, each containing a renewing and ! xG are referred to as low-density patches (L-patches),
resource. Each patch is sufficiently small so that its re- and patches that have a resource density 1xG are referred
source drops rapidly whenever a consumer visits and then to as high-density patches (H-patches). Consumers are
recovers relatively slowly between visits. Both consumer assumed to move randomly among patches. The resource
species visit patches in a random manner. This form of in a patch renews according to the following equation:
consumer-resource interaction has been shown in previous
dx
work to have a stabilizing effect on consumer-resource
= r(x). (1)
dynamics (Nisbet et al. 1997, 1998). We use invasibility dt
analyses to identify the outcome of exploitation compe-
tition. In most cases one of the two species is predicted Hence, renewal is a continuous process that depends on
to displace the other; however, coexistence can occur over the current resource density within the patch. Renewal may
a relatively small range of parameter values. It is important be due to local processes (e.g., regrowth or resource pro-
to note that coexistence in our model does not occur duction) or resource immigration from sources that are
because of any intrinsic patch heterogeneity. Differences external to the habitat.
in the resource abundance among patches are created by As well as their state, patches are also characterized by
268 The American Naturalist
their age, a. The age of an L-patch is the time that has nL nL
= m(t)n L , (7)
elapsed since the patch was last visited by a digger. The t a
age of an H-patch is the time since its density was xG. This
nH nH
density occurs either when the patch is visited by a grazer = (m(t) n(t))n H , (8)
t a
or when the density on an L-patch renews and reaches xG.
Denote by t the time it takes for the resource on an L- dx L
= r(x L ), (9)
patch to renew from xD to xG. This time interval is obtained dt
by solving
dx H
= r(x H), (10)
dt
t
r(x(a))da = x G x D. where m(t) and n(t) are the rates that all patches are visited
(2)
by the digger and grazer populations at time t, respectively.
0
These four equations are associated with the following
boundary conditions:
Note that x(0) = x D, x(t) = x G, and all L-patches have an
n L(0, t) = m(t), (11)
age !t.
In order to describe resource and consumer dynamics,
n H(0, t) = n L(t, t) n(t)NH(t), (12)
we keep track of the age distribution of patches. The frac-
tion of the L-patches at time t that are aged between a x L(0) = x D , (13)
and a da, is nL(a, t) da. Similar notation is used to rep-
x H(0) = x G. (14)
resent the age distribution of the H-patches. The fraction
of all patches that are L-patches and the fraction of all
Equation (11) expresses the fact that the rate of creation
patches that are H-patches at time t are denoted NL(t) and
of age-0 L-patches at any time is equal to the rate that all
NH(t), respectively. These fractions can be calculated using
patches are visited by diggers. Equation (12) arises because
the rate of creation of age-0 H-patches is equal to the rate
that L-patches change their state (which occurs when they
t
have survived to age t) plus the rate that grazers visit H-
NL(t) = n L(a, t)da, (3)
patches.
0
The variables D(t) and G(t) represent the density of
diggers and grazers, respectively. Because all individuals in
NH(t) = n H(a, t)da. (4) the system move randomly from patch to patch and move-
0
ments are independent of the presence of other individ-
uals, patches are visited by the consumers at a rate that is
proportional to the number of consumers. When individ-
Note that NL(t) NH(t) = 1 for all time t.
ual diggers and grazers are searching for patches, they
The resource density within L-patches and H-patches,
encounter them at rates jD and jG, respectively. Both spe-
which are of age a, are denoted xL(a) and xH(a), respec-
cies exhibit a Holling Type II functional response when
tively. The average resource density within L-patches and
seeking and handling the resource. The parameter h is the
H-patches at time t can be calculated from the following
average time a consumer takes to handle a unit of resource,
equations:
which we assume is the same for both consumer species.
If the number of patches is large compared to the number
of consumers, then the patch-encounter rates exerted by
t
the digger and grazer populations can be approximated
XL(t) = n L(a, t)x L(a)da/NL(t), (5)
by the following (Nisbet et al. 1997, 1998):
0
jD D(t)
XH(t) = n H(a, t)x H(a)da/NH(t). (6) m(t) = , (15)
1 jD h[NL(t)XL(t) NH(t)XH(t) x D]
0
jGG(t)
n(t) = . (16)
Resource dynamics are governed by the following equa- 1 jGhNH(t)[XH(t) x G]
tions:
Coexistence of Feeding Strategies 269
Note that in this model grazers may visit patches that the distribution of patch ages is described by an expo-
contain a resource density too low for them to consume. nential density function (Nisbet et al. 1997). The associated
We assume constant conversion efficiencies of resource steady state distributions for nL(a, t) and nH(a, t) are de-
to consumer numbers, D and G. Both species have con- fined and given by
stant per capita death rates, dD and dG. Consumer dynamics
0 ≤ a ≤ t,
n ∗(a) = m∗ exp ( m∗a)
come from the two ordinary differential equations (ODEs), (19)
L
a ≥ 0,
n H(a) = m∗ exp [ m∗(a
∗
(20)
t)]
[ ]
dD jD(NLXL NHXH x D)
D
= dD D, (17)
dt 1 jD h(NLXL NHXH x D) where m∗ is the equilibrium patch visitation rate of the
digger population. Later we will show how this rate can
[ ]
dG j NH(XH x G) be calculated.
GG
= dG G. (18)
dt 1 jGhNH(XH x G) The average resource density in patches at steady state,
∗
which we denote XD , can be calculated from the above
Note that we have assumed there is no direct intraspecific steady state distributions:
or interspecific competition for the resource in these equa-
tions. Consumer growth rates are only directly regulated t
∗
n ∗(a)x L(a)da n ∗ (a)x H(a)da,
by the distribution of the resource, which is regulated by XD = (21)
L H
the two populations. 0 0
m∗ exp ( m∗a)x(a)da.
= (22)
Invasion Analysis 0
We also know from equation (17) that at steady state,
In this section we examine under what conditions, if any,
the two consumer species can coexist. To do this we use
NL∗XL
∗ ∗∗
the technique termed “invasion analysis” (MacArthur and NHXH = Q D xD, (23)
Levins 1967; Turelli 1981). We first let one species establish
itself within the environment and come to equilibrium. where Q D = dD /[jD( D hdD)] and the star notation in-
We next examine whether the population size of the second dicates steady state values. The left-hand side of equation
species will increase when it is placed within the environ- (23) is simply the average resource density among all
∗
ment at low densities (i.e., we see whether it can invade). patches, XD . This density must be 1xD, which means QD
We then repeat this with the role of both species reversed. must be positive or equivalently D 1 hdD. If this condition
Coexistence occurs when both species can invade each is not true, the digger population cannot persist. When
other when the resident is well established. we combine equation (22) with equation (23) we get an
implicit formula for calculating m∗, namely,
To show coexistence is possible, we need to show that
both species can become well established in the absence
of the other (Turelli 1981). To show this is indeed true for
m∗ exp ( m∗a)x(a)da = Q D
the model presented here, we rewrite the model in terms xD. (24)
of coupled time-dependent ODEs and use this formulation 0
to show that there exists a unique globally stable nontrivial
Equation (23) tells us that the average density of resource
equilibrium population size and resource distribution (see
among all patches, when only diggers are in the system,
app. A). Stability of the equilibrium with only one con-
is independent of the assumptions on resource renewal.
sumer present is assumed in the analysis to follow.
However, the fraction of patches that are L-patches and
the fraction that are H-patches, at equilibrium, is depen-
Grazers Invading Diggers dent on the assumption about resource renewal. This de-
pendence can be seen by noting that the patch-visiting rate
First, we assume that only diggers are present and the
of the diggers, m∗, is dependent on x(a) in equation (24).
system is at equilibrium. We can calculate resource den-
Grazers can invade if, in equation (18), dG/dt 1 0, when
sities within patches using x L(a) = x(a) and x H(a) =
the resource is in the steady state distribution associated
x(a t), where x(a) is the solution to dx/da = r(x) subject
with the digger-only state. This is true when
to x(0) = x D. Because movement among patches is ran-
dom, all patches, irrespective of their age or state, expe-
∗ ∗
rience the same risk of a visit from a digger. As a result, Q G ! NH (XH x G), (25)
270 The American Naturalist
where Q G = dG /[jG( G hdG)]. We can expand the previous x(a) = x D r0 a. (31)
equation to give the following condition for invasion:
The time for the resource on a patch to renew from level
xD to level xG is t = (x G x D)/r0. Substituting the above
m∗ exp ( m∗a)(x(a)
QG ! x G)da. (26)
renewal function into equation (24) gives m∗ = r0 /Q D.
t
When this rate is then substituted into equation (26), we
get the following condition for grazers to invade diggers,
In order to determine whether the grazer species can in-
vade the digger species when resources renew according
to some function r, we first evaluate the steady state patch- Q G ! Q D exp [ (x G x D)/Q D]. (32)
visitation rate of the digger population, m∗, using equation
(24). Next, we use this visitation rate in equation (26) and Note that this condition is independent of the renewal
see whether the invasion condition is satisfied. rate, r0. The condition for diggers to invade grazers is given
by equation (29). It can be easily shown that for all
x G 1 x D, there exist pairs (Q D , Q G) that satisfy both in-
Diggers Invading Grazers
vasion conditions. Hence, if resource renewal is linear and
When only grazers are present and the population has unbounded, then diggers and grazers may coexist. An ex-
reached steady state, all patches have a resource density ample of this model is presented in figure 1A, which shows
of at least xG, so NL∗ = 0 and NH = 1. From equation (18),
∗
when coexistence occurs and when either the digger or
∗ ∗
we know that NH (XH x G) = Q G; hence, the grazer excludes the other. We have confirmed the lo-
cations of the boundaries in figure 1A using an explicit
∗
XH = Q G xG (27) numerical solution to the dynamic equations.
is the average resource density among patches. Diggers can
invade the system if
Free-Space Renewal
Q D ! NL∗XL
∗ ∗∗
NHXH xD. (28)
An alternative formulation for the rate of renewal is
Substituting equation (27) into equation (28) gives the
following invasion condition:
() x
r(x) = r0 1 . (33)
K
QD ! QG xG xD, (29)
This can describe a number of situations where, in the
which is independent of the assumption on resource re-
absence of consumers, the resource density approaches
newal. The important feature for invasion is the difference
some density, K. One example of such a situation is
in the depletion levels of the two competitors.
when resources renew at a constant rate, as in the pre-
vious model, but now resources become nonviable at
Results
some constant rate. An example of this process is aerial
insects that have fallen onto water being washed up
In this section we investigate three models that describe
along a riverbank at a constant rate and then washed
resource renewal. For each of these models, we determine
away at some constant per capita rate (Davies and
whether coexistence of the grazer and digger species is
Houston 1981). Alternatively, this model can be used
possible using the invasion conditions derived in the pre-
to describe the process where resource particles enter a
vious section.
patch at a fixed rate but only establish within the patch
if they happen to land on a section of the patch that is
Linear Renewal
not currently occupied by another resource particle.
This process is often referred to as free-space recruit-
The simplest assumption about resource renewal is that it
ment and has been applied to models that describe pop-
occurs at some constant rate, r0, and resources remain
ulation dynamics of benthic marine invertebrates (e.g.,
viable until they are consumed. The renewal function is
Roughgarden 1997 and references within). In this case
r(x) = r0 . (30) the parameter K is often referred to as the patch-car-
rying capacity.
Suppose there are only diggers present, then the resource
This renewal model gives the following:
density on a patch of age a is
Coexistence of Feeding Strategies 271
Again, note that this condition is independent of the pa-
rameter r0. The rate at which the resource approaches the
carrying capacity has no influence on whether the digger
population will be invaded; however, the success of in-
vasion is influenced by the carrying capacity. The presence
of a carrying capacity means we have extra conditions
about when the grazers and diggers can persist in the
system. From equation (17) it can be shown that the
growth rate of the digger population is never positive and,
hence, cannot persist if Q D 1 K x D. Similarly, the grazer
population cannot persist if Q G 1 K x G. Thus, for any
given K, there now exists a region in the (Q D , Q G) plane
where neither the digger nor the grazer can persist. As
with the previous model on resource renewal, it can be
shown that for this model, pairs (Q D , Q G) exist that satisfy
both invasion conditions; therefore, coexistence is possible.
An example of the coexistence region for this model is
presented in figure 1B.
Three examples of consumer-resource dynamics are
presented in figure 2. The grazer species is characterized
by the same parameter values in each example (table 1)
and is associated with a Q value of Q G = 0.15. Grazers
encounter patches twice as fast as diggers, and both diggers
and grazers live, on average, one time unit (table 1). The
three examples differ in the Q value associated with the
diggers: (A) Q D = 0.4, (B) Q D = 0.5, and (C) Q D = 0.6. In
all three examples, the grazer species is assumed to be well
established before an inoculum of diggers is introduced.
The dynamics are qualitatively different in each of the three
examples, and they match the predictions (fig. 1B). The
steady state resource density when only grazers are present
∗
is X G = 0.65 resource units. The steady state resource den-
sities for the three examples, when only diggers are present,
∗ ∗ ∗
are (A) XD = 0.5, (B) XD = 0.6, and (C) XD = 0.7 resource
∗ ∗
units. Although X G 1 XD in figure 2B, the grazers clearly
Figure 1: Coexistence regions when resource renewal is (A) linear and
coexist with the diggers. The dynamics presented in figure
(B) free space (carrying capacity, K = 1 ). In both examples, the digger
2 were generated by numerically solving a set of six coupled
and grazer resource thresholds are given by xD = 0.1 and xG = 0.5.
delay differential equations (app. B).
x(a) = K (K x D) exp ( r0 a/K ), (34)
Logistic-Type Renewal
() A more general model that can be used to describe resource
KK xD
renewal is
t= ln , (35)
r0 K xG
j
r0
()x
∗
m= (K xD Q D). (36) i
r(x) = r0 x 1 , (38)
QD K K
The condition for grazers to invade diggers can be shown where i and j are nonnegative constants. The linear renewal
to be model is recovered when i = j = 0, and the free-space
model is recovered when i = 0 and j = 1. When i = j = 1,
(K xD)/QD
()
K xD resource renewal is logistic, which is a commonly adopted
QG ! QD . (37)
K xG model to describe the growth of a wide range of prey
272 The American Naturalist
so must be evaluated numerically. Figure 3 shows four
examples of the renewal function, equation (38), and
the form of the age-dependent change in resource den-
sity that results. In figure 4 we present the coexistence
regions associated with the renewal functions presented
in figure 3. Coexistence is possible for all resource-re-
newal functions presented, and the region of coexistence
varies depending on the assumption of resource re-
newal. The region is smallest when renewal is described
by the free-space model and is largest when renewal is
described by the third model. Numerical results again
show that the condition for grazers to invade diggers
is independent of the renewal parameter, r0.
Discussion
In this article we have shown that in some situations it is
possible that species that compete exploitatively for a com-
mon resource may coexist. This result is in contrast to
previous studies of exploitation competition that assume
the resource is well mixed within the habitat (e.g., Fishman
1997). The fact that we found the possibility of coexistence
in our model is not all that surprising. Our model is a
subtle example of resource partitioning, which, as was
noted in our introduction, has been shown to promote
coexistence. The two consumers are essentially competing
within a system that contains two types of resources,
namely L-patches and H-patches. An important result
from our model is that spatial and temporal changes in
the resource distribution and, hence, the two types of patch
are created by the foraging behaviors of the consumers
and not from any intrinsic differences among resource
patches. Although the abundance of L-patches and H-
patches are correlated at any time, knowledge of the abun-
Figure 2: Consumer-resource dynamics for the three cases presented in
dance of one does not necessarily specify the abundance
figure 1B (solid circles). In all examples K = 1, xD = 0.1, xG = 0.5, and
QG = 0.15. The Q-values associated with the digger species are (A) of the other; hence, coexistence may occur in this system
QD = 0.4, (B) QD = 0.5, and (C) QD = 0.6. Each example shows the mean
(Haigh and Maynard Smith 1972).
resource level, NLXL NHXH (dotted line); digger density, D (solid thick
The situation presented here corresponds to the idea of
line); and grazer density, G (solid thin line).
an included niche (Miller 1964, 1967), where in this case
organisms. There are two important properties of this
Table 1: Parameter values used in all the simulations
model. When j 1 0 the resource density within a patch will
presented
tend toward a carrying capacity K, and when i 1 0 the
Parameter Description Value
resource does not renew after it has been completely de-
pleted from a patch. From the previous model, we have K Patch carrying capacity 1
seen that the presence of a carrying capacity means that r0 Maximum renewal rate 20
species characterized by high Q values cannot persist in h Resource handling time .01
the system regardless of the presence or absence of other xD Digger resource threshold .1
xG Grazer resource threshold .5
species. The second property is particularly important if
Digger mortality rate 1
dD
consumers deplete all resources from a patch because even-
Grazer mortality rate 1
dG
tually they will drive themselves to extinction.
Digger patch encounter rate 100
jD
An analytic solution to the invasion condition, equa-
Grazer patch encounter rate 200
jG
tion (26), is generally not possible for this model and
Coexistence of Feeding Strategies 273
Figure 3: Examples of the logistic-type resource-renewal model as described by equation (38). Also shown are the associated renewal trajectories,
x(a). Parameter values for each example are as follows: (A) i = 0 , j = 1 ; (B) i = 1 , j = 1 ; (C) i = 2 , j = 1; and (D) i = 1, j = 2. In all examples, the
digger and grazer resource thresholds are given by xD = 0.1 ( filled dots) and xG = 0.5 (open dots), and the carrying capacity K = 1.
the niche of the grazer (i.e., the densities of resource that within its restricted niche). We have shown that if the
can be accessed) is a subset of the niche of the digger. consumer with the restricted niche is sufficiently efficient,
Coexistence may be possible, provided the grazer species then it can exclude the consumer that has the greater niche.
is more efficient than the digger species at utilizing high- The parameter r0 that appears in all three renewal mod-
density resource patches (i.e., it is a better competitor els considered in this article has no influence on the region
274 The American Naturalist
then their persistence depends on the presence of high-
density patches, which may become sufficiently rare in the
presence of grazers such that diggers are driven to
extinction.
A similar mechanism that shares many features with
ours was studied by Briggs (1993) in the context of insect
parasitoids. In her system two parasitoid species attack
different developmental stages of a single host species.
Briggs (1993) presents an age-structured model of the sys-
tem and shows that the two parasitoids could coexist if
the parasitoid that utilizes the later host stage could suc-
cessfully attack hosts that had been attacked earlier by the
other parasitoid. Like the results presented here, it was
found that the outcome of competition was dependent on
the consumer efficiency of both parasitoid species, which
was defined in a similar manner as the Q value of the
diggers and grazers in this article.
In this article we have shown that the consumer’s effect
on the distribution of resources, and not necessarily the
mean abundance of resources, is important for coexis-
tence. This result was also shown by Mittler (1997), who
Figure 4: Coexistence regions when resource renewal is described by the
investigated a model of predator-prey dynamics that over-
logistic-type functional form. The Digger only and Digger and Grazer
laps to some extent with the model presented in this article.
boundary correspond to the four cases presented in figure 3: (A) solid
thick line, (B) long-dashed line, (C) short-dashed line, and (D) solid thin Predators did not necessarily consume prey in their en-
line.
tirety, leaving smaller prey items that may be more effi-
ciently consumed by other predators. Mittler (1997)
of coexistence. The effect of r0 is to alter the timescale of showed that a rich range of dynamics could occur in a
resource renewal, which affects population sizes but not two-predator system. In some cases, depending on certain
the essential qualities of the dynamics that influence co- competitive trade-offs, frequency-dependent dynamics oc-
existence. Thus, the rate at which a resource renews has curred, where the first predator species to occupy a habitat
no bearing on which species are expected to persist in a could exclude the other predator species. In other cases
habitat. What is important for determining species com- coexistence of the two predator species was predicted. Re-
position is the manner in which the resource renews within sources, which may vary in size, were assumed to enter
patches. The functional form of resource renewal defines the habitat at some deterministic rate. Once a prey item
the conditions under which the grazer is able to persist in entered the habitat it did not change its size until it was
a habitat that is occupied by diggers. The region of co- attacked by a predator. Our work differs from Mittler
existence is increased when the relative rate of renewal in (1997) because we assume the state of the resource changes
the H-patches compared to the L-patches is increased. as it ages.
When a free-space renewal function is assumed, the re- One important issue in models of coexisting consumers
source increases much faster in the L-patches compared is how the density of the resource compares with the sit-
to the H-patches, and in this case the region of coexistence uation when only one consumer is present. This question
is small. However, if the rate of renewal is high in H- is of course important in biological control, where the
patches (e.g., fig. 3C), then the region of coexistence is resource is the item that we wish to regulate. W. W. Mur-
significantly increased (fig. 4). Although in this case it may doch (private communication) noted that in the simple
take some time for patches to become accessible to grazers models of coexistence discussed in our introduction, the
after they have been visited by a digger, if diggers move addition of a second coexisting consumer (e.g., predator,
slowly or have a low population size, then grazers can herbivore, parasite) to a system never decreases the re-
persist if they are sufficiently efficient (i.e., they make bet- source level below that which would arise with the most
ter use of the high resource-renewal rates of the H- “effective” consumer present alone. The dynamics of the
patches). By either moving quickly through the habitat or current model are consistent with this pattern. When co-
having high population growth rates, the grazers persist existence occurs the digger is always the most effective
even though they may encounter patches that they cannot consumer, and in these circumstances the mean equilib-
utilize. However, if diggers are not efficient consumers, rium resource density among patches with both diggers
Coexistence of Feeding Strategies 275
and consumers present is the same as when only diggers but variability in the resources encountered may be par-
are present (fig. 2). This minimum density is simply ticularly important for risk-sensitive foragers (Bulmer
∗
XD = x D Q D. 1994). Consumers may be expected to alter their foraging
Previous theoretical studies have shown that when mul- behavior depending on whether they are risk averse or risk
tiple species compete exploitatively for the same limiting prone. This work could be extended and applied to risk-
resource, it is the species with the lowest equilibrium re- sensitive foragers by incorporating the variance of the en-
source requirement that eventually displaces all other spe- countered resource in the equation that describes con-
cies (Armstrong and McGehee 1980; Tilman 1982). An sumer growth.
important result from our model is that one cannot predict A final assumption that needs further investigation is
which species will persist by only looking at the equilib- that age-0 H-patches renew at the same rate regardless of
rium resource level when each species is in isolation. We whether they were just attacked by a grazer or whether
have shown that grazers that have a higher equilibrium they were an L-patch that reached the age t. We may expect
than the diggers may still coexist with the diggers (fig. 2B). for some resource types that renewal may differ depending
In order to allow analytic tractability for much of our on whether it was just attacked by a grazer (e.g., vegetative
analysis, we have had to make a number of simplifying regrowth). The rate at which a patch is grazed may be
assumptions with regards to consumer and resource dy- important for modeling how a resource renews. This is
namics. An important assumption we have made that certainly true for resource depression where the resource
needs further investigation is a lack of resource-dependent itself may exhibit predator avoiding behavior (e.g., insect
consumer behavior. Consumers are assumed to move ran- prey). Resource responses could potentially alter the
domly through the habitat; they do not make systematic model’s predictions.
movements or modify their movements based on recent The model presented in this article has shown that two
resource encounters. When visits to resource patches are species, which compete exploitatively, can coexist on a
not random the distribution of resources that a forager single resource in a homogeneous environment, given that
encounters is not necessarily the same as the overall dis- they satisfy certain conditions with regard to the level at
tribution of resources (Possingham 1989; Abrams 1999). which they deplete resources (x) and their associated char-
This may affect the region of coexistence. Because we have acter value (Q). However, this model does not tell us
assumed random movements, both consumer species will whether another species could invade and exclusively take
be scattered within the habitat at any time, but systematic over the system and not be invadable by any other po-
foraging may partition the distributions of the two species. tential species. If there exists some trade-off between x and
Laverty and Plowright (1985) observed that humming- Q (Schmitt 1996), then future work could use the model
birds (diggers) and bumblebees (grazers) partition the re- to investigate whether selection would favor some inter-
source distribution in both space and time. Part of the mediate species or allow coexistence of a suite of species.
spatial segregation may be explained by differences in each
species’ ability to access the flowers (the inner flowers of
a patch were protected from hummingbirds by vegetative Acknowledgments
cover). Temporal segregation may be due to differences in
the metabolic cost of foraging. Here we have assumed that We thank C. J. Briggs, J. S. Brown, M. A. Gilchrist, W. W.
consumers are always foraging and renewal is a continuous Murdoch, K. J. O’Keefe, C. W. Osenberg, R. J. Schmitt,
process. Despite our simple assumptions on foraging be- L. Stone, and two anonymous referees for their helpful
havior, the model does suggest that multiple species that comments. The research was supported in part by the
exhibit little or no apparent systematic foraging may still Office of Naval Research (grant N00012-93-10952), the
coexist even if they consume the same resource. National Science Foundation (grants DEB-93-19301 and
Another potentially important assumption we have DEB-98-73650), and the Department of Zoology, Duke
made is that the habitat is homogeneous. In a real system University. R.M.N. and H.P.P. performed part of the work
we would expect variability among patches with regard to as sabbatical fellows at the National Center for Ecological
renewal rates and their carrying capacity (Possingham Analysis and Synthesis, a center funded by the National
1988, 1989). We have also assumed that the population Science Foundation (grant DEB-94-21535); the University
growth of the consumer is related to the mean intake rate, of California, Santa Barbara; and the state of California.
276 The American Naturalist
APPENDIX A
Stability Analysis of the Single Consumer–Single Resource Model
Here we present an alternative way to describe resource and consumer dynamics when only one consumer species is
present. Suppose only the grazer species is present. It can be shown that the dynamics of the mean patch density (XH)
and the grazer population size (G) are governed by the following coupled ODEs:
dXH jG(XH(t) x G)G(t)
= FH(t) ,
dt 1 jGh(XH(t) x G)
[ ]
dG j (XH(t) x G)
GG
= dG G(t),
dt 1 jGh(XH(t) x G)
where
FH(t) = n H(a, t)r(x H(a, t))da.
0
This formulation requires that the renewal function r is such that x(a) is not exponential for large a. When resource
renewal is linear, FH(t) = r0; when resource renewal obeys the free-space model, FH(t) = r0(1 XH(t)/K ) . A simple analytic
form for FH(t) is not possible for logistic-type renewal because the function r(x) is nonlinear. Note that the age
distribution of patches may not need to be explicitly modeled in this system.
Now we examine the stability of this model formulation. The associated Jacobian is
[ ]
jGG ∗/(1 jGhQ G)2 FH / XHFX∗ jGQ G /(1 jGhQ G)
J= ,
H
jGG ∗/(1 jGhQ G)2 0
G
where G ∗ and XH are equilibrium values for the system. There exist two equilibrium points for this model. One is the
∗
trivial solution where there are no consumers at all, G ∗ = 0 , and XH is the carrying capacity (this is infinite for the
∗
linear renewal model). There also exists a nontrivial equilibrium point, XH = Q G x G and G ∗ = FH (1
∗ ∗
hjGQ G)/(jGQ G). If we let D and T denote the determinant and trace of the Jacobian matrix J, then an equilibrium
point (G ∗, XH ) is locally stable if D 1 0 and T ! 0; otherwise, it is unstable (Bulmer 1994). Using these criteria, it can
∗
be easily shown that for both the linear and the free-space renewal models, the trivial equilibrium point is unstable
and the nontrivial point is stable. Hence, the grazer will always become well established when introduced into the
system when it is free of the digger. The same reasoning can be applied to the digger-only situation. Although we
cannot prove analytically that consumers can become well established when resources renew according to the more
general logistic-type model, numerical simulations suggest this is true also (see Nisbet et al. 1998).
APPENDIX B
Resource Dynamics Expressed as Ordinary Differential Equations
When one wishes to generate numerical approximations to partial differential equations (PDEs), it is sometimes easier
to convert them to ODEs and approximate the solutions to the ODEs instead. Here we transform equations (7)–(10),
which describe the dynamics of nL(a, t ), nH( a, t ), xL(a), and xH(a), into equivalent equations that describe the dynamics
of NL(t) and the products NL(t)XL(t) and NH(t)XH(t). Hence, we show that resource dynamics can be described by
equations that do not explicitly keep track of the age distribution of patches.
First, we integrate equation (7) over the duration it takes an L-patch to become an H-patch,
Coexistence of Feeding Strategies 277
t t
nL nL
da = n L(a, t)da,
m(t)
t a
0 0
which reduces to the following:
dNL
= m(t)NL(t) n L(t, t) n L(0, t). (B1)
dt
The last term is simply m(t) (see eq. [11]). The second to last term represents the fraction of patches that were last
visited by a digger at time t t . The probability a patch is not visited by a digger from time t t to time t is given
by
( )
t
S(t) = exp m(y)dy . (B2)
tt
Hence,
n L(t, t) = S(t)n L(0, t t) = S(t)m(t (B3)
t).
Substituting equations (11) and (B3) into (B1) gives the following delay differential equation (DDE) for NL:
dNL
= m(t)(1 NL(t)) S(t)m(t (B4)
t).
dt
Differentiating equation (5) with respect to t gives
t
d nL
(NL(t)XL(t)) = x Lda,
dt t
0
t
( )
nL
= m(t)n L x Lda,
a
0
t t t
( )
nL dx L dx L
= xL nL da nL da n Lx Lda, (B5)
m(t)
a da da
0 0 0
t t
= (n Lx L )da n Lr(x L )da m(t)NL(t)XL(t),
a
0 0
= n L(0, t)x L(0, t) n L(t, t)x L(t, t) FL(t) m(t)NL(t)XL(t),
= m(t)x D S(t)m(t FL(t) m(t)NL(t)XL(t),
t)x G
where
t
FL(t) = n L(a, t)r(x L(a))da.
0
When resource renewal is described by the free-space model (eq. [33]),
FL(t) = r0(NL(t) NL(t)XL(t)/K ).
Similarly, we can differentiate equation (6) with respect to t, which, after a little algebra, gives
278 The American Naturalist
d
(N (t)XH(t)) = [S(t)m(t XL(t))]x G FH(t) (m(t) n(t))NH(t)XH(t), (B6)
t) n(t)(1
dt H
where
FH(t) = n H(a, t)r(x H(a))da.
0
For the case of free-space renewal,
FH(t) = r0(1 NL(t) NH(t)XH(t)/K ).
Differentiating equation (B2) with respect to t gives the final DDE that closes the system:
dS
= [m(t (B7)
t) m(t)]S(t).
dt
Consumer-resource dynamics can be generated by numerically approximating solutions to the following sets of coupled
equations: (B4)–(B7), (17), and (18). Subtleties related to initializing DDEs are discussed by Nisbet (1997).
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Schmitt, R. J. 1996. Exploitation competition in mobile Associate Editor: Lewi Stone
Grazers and Diggers: Exploitation Competition and
Coexistence among Foragers with Different
Feeding Strategies on a Single Resource
Shane A. Richards,1,* Roger M. Nisbet,2 William G. Wilson,1 and Hugh P. Possingham3
1. Department of Zoology, Duke University, Durham, North was then expanded, and the competitive exclusion prin-
Carolina 27708-0325; ciple (Hardin 1960) was proposed, which stated that n
2. Department of Ecology, Evolution and Marine Biology,
species could not coexist on !n resources. In order to apply
University of California, Santa Barbara, California 93106;
this principle, a clear way in which to identify distinct
3. Department of Applied and Molecular Ecology, University of
resources and consumers is required (Haigh and Maynard
Adelaide, Waite Campus, PB 1 Glen Osmond, South Australia
Smith 1972; Schoener 1974; Levins 1979), as well as a clear
5064, Australia
definition of coexistence (Koch 1974; Hsu et al. 1978;
Submitted January 25, 1999; Accepted August 23, 1999 Armstrong and McGehee 1980; Smith and Waltman 1995).
There have been numerous theoretical studies where mod-
els have apparently violated the competitive exclusion
principle. Competing species can coexist if mechanisms
are present that effectively increase the number of re-
abstract: A mathematical model is presented that describes a sys-
tem where two consumer species compete exploitatively for a single sources present and/or allow populations to exhibit stable
renewable resource. The resource is distributed in a patchy but homo- cyclic behavior. Examples include resource partitioning of
geneous environment; that is, all patches are intrinsically identical.
the same prey item (Haigh and Maynard Smith 1972;
The two consumer species are referred to as diggers and grazers,
Schoener 1974), interspecific variation during the re-
where diggers deplete the resource within a patch to lower densities
source’s life cycle (Briggs 1993; Briggs et al. 1993), life-
than grazers. We show that the two distinct feeding strategies can
history variation among the consumers (McCann 1998),
produce a heterogeneous resource distribution that enables their co-
existence. Coexistence requires that grazers must either move faster temporal fluctuations in environmental conditions (Koch
than diggers between patches or convert the resources to population 1974; Levins 1979; Turelli 1981; Abrams 1984; Chesson
growth much more efficiently than diggers. The model shows that
1990), disturbance (Hastings 1980), interference compe-
the functional form of resource renewal within a patch is also im-
tition (Vance 1985; Fishman 1997), and spatial structuring
portant for coexistence. These results contrast with theory that con-
of the habitat (Tilman 1994).
siders exploitation competition for a single resource when the re-
In this article we investigate whether two species (or
source is assumed to be well mixed throughout the system.
phenotypes) that differ in their strategies for resource ex-
Keywords: exploitation competition, foraging, coexistence, invasion ploitation can coexist in a system where they both utilize
analysis.
the same resource. We assume the resource is distributed
in a patchy but homogeneous environment; that is, all
patches are intrinsically identical. The two species differ
Understanding the mechanisms that allow species co-
in the degree to which they deplete resources within a
existence remains a key topic in community ecology. The
patch and hence differ in their feeding strategy. The species
mathematical model of Volterra (1926) was the first to
that depletes the resource to lower levels is termed the
suggest that the indefinite coexistence of more than one
“digger” species; the other is termed the “grazer” species.
species on the same resource was impossible. This result
An important assumption we make is that the two species
interact only through exploitation competition (Milinski
* Present address: Population Biology Section, University of Amsterdam,
and Parker 1991). There is no direct interference between
Kruislaan 320, 1098 SM Amsterdam, The Netherlands; e-mail: sarichar@
individuals, all interactions occur through each species’
duke.edu.
influence on a shared food resource. Schmitt (1996) has
Am. Nat. 2000. Vol. 155, pp. 266–279. 2000 by The University of Chicago.
studied an example of such a system, where two species
0003-0147/2000/15502-0009$03.00. All rights reserved.
Coexistence of Feeding Strategies 267
of benthic marine snails, Tegula aureotincta and Tegula the contrasting feeding strategies of the two competing
eiseni, compete for microalgae. Differences in the foraging species and the dynamics of resource renewal. Coexistence
morphologies of the two snails have been shown to have can occur because each consumer does not encounter a
different effects on the distribution of the algal resource. fixed amount of resource when it visits a patch but an
Tegula eiseni was found to be capable of reducing algal amount that depends on the time since the last consumer’s
densities to lower levels than T. aureotincta. Nectarivores visit and its type. Hence, consumers encounter resource
are another example of a group of consumers that often levels described by a probability density function, which
appear to interact through exploitation competition. Lav- effectively increases the number of resource types. The
erty and Plowright (1985) studied a system where two digger species can often persist because it can exploit re-
species of bumblebee and a hummingbird compete for sources that the grazer species cannot reach. The grazer
nectar in jewelweed. In this example the consumer species species can also persist if it moves faster among patches
were found to differ in the depth to which they could than the digger species and encounters patches that have
drain the nectar spur and also the rate at which they visited not been recently visited by diggers. In doing so the grazer
flowers (Laverty and Plowright 1985). species can often reduce the mean resource abundance
Possingham (1987) constructed a mathematical model within the system so that it stops the diggers from taking
of nectarivore competition (which was assumed to be ex- over. We also find that the dynamics of resource renewal
ploitative) and showed that two consumer species, which play an important role in determining the outcome of
differed in their ability to deplete nectar from a flower, exploitation competition.
could coexist on a single flower species. Coexistence was
dependent on the competitive ability of both species,
The Model
which was defined as the mean net energy gained per
calorie extracted from a flower, divided by the cost of using We consider a habitat that contains a large number of
each flower. Wilson et al. (1999) fitted a similar mathe- identical small patches each containing a resource of den-
matical model, which described the dynamics of two ben- sity x. There are two species of consumers, which we refer
thic grazers competing for algae, to data collected by to as diggers and grazers. Grazers (G) can only consume
Schmitt (1996). Although the model required a number resources on a patch whose density exceeds xG, and when
of parameters, the data were sufficient to give estimates a grazer visits such a patch, its density drops to xG. Diggers
to all parameters but one. The best fit was found to lie (D) consume resources in a similar manner, reducing the
very close to a region of coexistence. Given the uncertainty resources in a patch to level xD. They may eat resources
in the fitted parameters, the model predictions were not on patches that have a lower density than that accessible
inconsistent with observed coexistence in the field to the grazers so that x D ! x G. We assume that patch size
(Schmitt 1996). In this article we have reviewed, gener- is sufficiently small so the timescale at which patches are
alized, and extended the work of Possingham (1987) and depleted of resources is fast compared with the timescale
Wilson et al. (1999) by investigating how the model of of resource renewal (Possingham 1988). A patch can be
resource renewal affects coexistence. in one of two states, depending on the amount of resource
We present a mathematical model that describes the it contains. The state of a patch is dependent on the time
dynamics of the two consumer species and the shared since it was last visited by a consumer and the species of
resource. The habitat is assumed to be made up of a large the last visitor. Patches that have a resource density 1xD
number of identical patches, each containing a renewing and ! xG are referred to as low-density patches (L-patches),
resource. Each patch is sufficiently small so that its re- and patches that have a resource density 1xG are referred
source drops rapidly whenever a consumer visits and then to as high-density patches (H-patches). Consumers are
recovers relatively slowly between visits. Both consumer assumed to move randomly among patches. The resource
species visit patches in a random manner. This form of in a patch renews according to the following equation:
consumer-resource interaction has been shown in previous
dx
work to have a stabilizing effect on consumer-resource
= r(x). (1)
dynamics (Nisbet et al. 1997, 1998). We use invasibility dt
analyses to identify the outcome of exploitation compe-
tition. In most cases one of the two species is predicted Hence, renewal is a continuous process that depends on
to displace the other; however, coexistence can occur over the current resource density within the patch. Renewal may
a relatively small range of parameter values. It is important be due to local processes (e.g., regrowth or resource pro-
to note that coexistence in our model does not occur duction) or resource immigration from sources that are
because of any intrinsic patch heterogeneity. Differences external to the habitat.
in the resource abundance among patches are created by As well as their state, patches are also characterized by
268 The American Naturalist
their age, a. The age of an L-patch is the time that has nL nL
= m(t)n L , (7)
elapsed since the patch was last visited by a digger. The t a
age of an H-patch is the time since its density was xG. This
nH nH
density occurs either when the patch is visited by a grazer = (m(t) n(t))n H , (8)
t a
or when the density on an L-patch renews and reaches xG.
Denote by t the time it takes for the resource on an L- dx L
= r(x L ), (9)
patch to renew from xD to xG. This time interval is obtained dt
by solving
dx H
= r(x H), (10)
dt
t
r(x(a))da = x G x D. where m(t) and n(t) are the rates that all patches are visited
(2)
by the digger and grazer populations at time t, respectively.
0
These four equations are associated with the following
boundary conditions:
Note that x(0) = x D, x(t) = x G, and all L-patches have an
n L(0, t) = m(t), (11)
age !t.
In order to describe resource and consumer dynamics,
n H(0, t) = n L(t, t) n(t)NH(t), (12)
we keep track of the age distribution of patches. The frac-
tion of the L-patches at time t that are aged between a x L(0) = x D , (13)
and a da, is nL(a, t) da. Similar notation is used to rep-
x H(0) = x G. (14)
resent the age distribution of the H-patches. The fraction
of all patches that are L-patches and the fraction of all
Equation (11) expresses the fact that the rate of creation
patches that are H-patches at time t are denoted NL(t) and
of age-0 L-patches at any time is equal to the rate that all
NH(t), respectively. These fractions can be calculated using
patches are visited by diggers. Equation (12) arises because
the rate of creation of age-0 H-patches is equal to the rate
that L-patches change their state (which occurs when they
t
have survived to age t) plus the rate that grazers visit H-
NL(t) = n L(a, t)da, (3)
patches.
0
The variables D(t) and G(t) represent the density of
diggers and grazers, respectively. Because all individuals in
NH(t) = n H(a, t)da. (4) the system move randomly from patch to patch and move-
0
ments are independent of the presence of other individ-
uals, patches are visited by the consumers at a rate that is
proportional to the number of consumers. When individ-
Note that NL(t) NH(t) = 1 for all time t.
ual diggers and grazers are searching for patches, they
The resource density within L-patches and H-patches,
encounter them at rates jD and jG, respectively. Both spe-
which are of age a, are denoted xL(a) and xH(a), respec-
cies exhibit a Holling Type II functional response when
tively. The average resource density within L-patches and
seeking and handling the resource. The parameter h is the
H-patches at time t can be calculated from the following
average time a consumer takes to handle a unit of resource,
equations:
which we assume is the same for both consumer species.
If the number of patches is large compared to the number
of consumers, then the patch-encounter rates exerted by
t
the digger and grazer populations can be approximated
XL(t) = n L(a, t)x L(a)da/NL(t), (5)
by the following (Nisbet et al. 1997, 1998):
0
jD D(t)
XH(t) = n H(a, t)x H(a)da/NH(t). (6) m(t) = , (15)
1 jD h[NL(t)XL(t) NH(t)XH(t) x D]
0
jGG(t)
n(t) = . (16)
Resource dynamics are governed by the following equa- 1 jGhNH(t)[XH(t) x G]
tions:
Coexistence of Feeding Strategies 269
Note that in this model grazers may visit patches that the distribution of patch ages is described by an expo-
contain a resource density too low for them to consume. nential density function (Nisbet et al. 1997). The associated
We assume constant conversion efficiencies of resource steady state distributions for nL(a, t) and nH(a, t) are de-
to consumer numbers, D and G. Both species have con- fined and given by
stant per capita death rates, dD and dG. Consumer dynamics
0 ≤ a ≤ t,
n ∗(a) = m∗ exp ( m∗a)
come from the two ordinary differential equations (ODEs), (19)
L
a ≥ 0,
n H(a) = m∗ exp [ m∗(a
∗
(20)
t)]
[ ]
dD jD(NLXL NHXH x D)
D
= dD D, (17)
dt 1 jD h(NLXL NHXH x D) where m∗ is the equilibrium patch visitation rate of the
digger population. Later we will show how this rate can
[ ]
dG j NH(XH x G) be calculated.
GG
= dG G. (18)
dt 1 jGhNH(XH x G) The average resource density in patches at steady state,
∗
which we denote XD , can be calculated from the above
Note that we have assumed there is no direct intraspecific steady state distributions:
or interspecific competition for the resource in these equa-
tions. Consumer growth rates are only directly regulated t
∗
n ∗(a)x L(a)da n ∗ (a)x H(a)da,
by the distribution of the resource, which is regulated by XD = (21)
L H
the two populations. 0 0
m∗ exp ( m∗a)x(a)da.
= (22)
Invasion Analysis 0
We also know from equation (17) that at steady state,
In this section we examine under what conditions, if any,
the two consumer species can coexist. To do this we use
NL∗XL
∗ ∗∗
the technique termed “invasion analysis” (MacArthur and NHXH = Q D xD, (23)
Levins 1967; Turelli 1981). We first let one species establish
itself within the environment and come to equilibrium. where Q D = dD /[jD( D hdD)] and the star notation in-
We next examine whether the population size of the second dicates steady state values. The left-hand side of equation
species will increase when it is placed within the environ- (23) is simply the average resource density among all
∗
ment at low densities (i.e., we see whether it can invade). patches, XD . This density must be 1xD, which means QD
We then repeat this with the role of both species reversed. must be positive or equivalently D 1 hdD. If this condition
Coexistence occurs when both species can invade each is not true, the digger population cannot persist. When
other when the resident is well established. we combine equation (22) with equation (23) we get an
implicit formula for calculating m∗, namely,
To show coexistence is possible, we need to show that
both species can become well established in the absence
of the other (Turelli 1981). To show this is indeed true for
m∗ exp ( m∗a)x(a)da = Q D
the model presented here, we rewrite the model in terms xD. (24)
of coupled time-dependent ODEs and use this formulation 0
to show that there exists a unique globally stable nontrivial
Equation (23) tells us that the average density of resource
equilibrium population size and resource distribution (see
among all patches, when only diggers are in the system,
app. A). Stability of the equilibrium with only one con-
is independent of the assumptions on resource renewal.
sumer present is assumed in the analysis to follow.
However, the fraction of patches that are L-patches and
the fraction that are H-patches, at equilibrium, is depen-
Grazers Invading Diggers dent on the assumption about resource renewal. This de-
pendence can be seen by noting that the patch-visiting rate
First, we assume that only diggers are present and the
of the diggers, m∗, is dependent on x(a) in equation (24).
system is at equilibrium. We can calculate resource den-
Grazers can invade if, in equation (18), dG/dt 1 0, when
sities within patches using x L(a) = x(a) and x H(a) =
the resource is in the steady state distribution associated
x(a t), where x(a) is the solution to dx/da = r(x) subject
with the digger-only state. This is true when
to x(0) = x D. Because movement among patches is ran-
dom, all patches, irrespective of their age or state, expe-
∗ ∗
rience the same risk of a visit from a digger. As a result, Q G ! NH (XH x G), (25)
270 The American Naturalist
where Q G = dG /[jG( G hdG)]. We can expand the previous x(a) = x D r0 a. (31)
equation to give the following condition for invasion:
The time for the resource on a patch to renew from level
xD to level xG is t = (x G x D)/r0. Substituting the above
m∗ exp ( m∗a)(x(a)
QG ! x G)da. (26)
renewal function into equation (24) gives m∗ = r0 /Q D.
t
When this rate is then substituted into equation (26), we
get the following condition for grazers to invade diggers,
In order to determine whether the grazer species can in-
vade the digger species when resources renew according
to some function r, we first evaluate the steady state patch- Q G ! Q D exp [ (x G x D)/Q D]. (32)
visitation rate of the digger population, m∗, using equation
(24). Next, we use this visitation rate in equation (26) and Note that this condition is independent of the renewal
see whether the invasion condition is satisfied. rate, r0. The condition for diggers to invade grazers is given
by equation (29). It can be easily shown that for all
x G 1 x D, there exist pairs (Q D , Q G) that satisfy both in-
Diggers Invading Grazers
vasion conditions. Hence, if resource renewal is linear and
When only grazers are present and the population has unbounded, then diggers and grazers may coexist. An ex-
reached steady state, all patches have a resource density ample of this model is presented in figure 1A, which shows
of at least xG, so NL∗ = 0 and NH = 1. From equation (18),
∗
when coexistence occurs and when either the digger or
∗ ∗
we know that NH (XH x G) = Q G; hence, the grazer excludes the other. We have confirmed the lo-
cations of the boundaries in figure 1A using an explicit
∗
XH = Q G xG (27) numerical solution to the dynamic equations.
is the average resource density among patches. Diggers can
invade the system if
Free-Space Renewal
Q D ! NL∗XL
∗ ∗∗
NHXH xD. (28)
An alternative formulation for the rate of renewal is
Substituting equation (27) into equation (28) gives the
following invasion condition:
() x
r(x) = r0 1 . (33)
K
QD ! QG xG xD, (29)
This can describe a number of situations where, in the
which is independent of the assumption on resource re-
absence of consumers, the resource density approaches
newal. The important feature for invasion is the difference
some density, K. One example of such a situation is
in the depletion levels of the two competitors.
when resources renew at a constant rate, as in the pre-
vious model, but now resources become nonviable at
Results
some constant rate. An example of this process is aerial
insects that have fallen onto water being washed up
In this section we investigate three models that describe
along a riverbank at a constant rate and then washed
resource renewal. For each of these models, we determine
away at some constant per capita rate (Davies and
whether coexistence of the grazer and digger species is
Houston 1981). Alternatively, this model can be used
possible using the invasion conditions derived in the pre-
to describe the process where resource particles enter a
vious section.
patch at a fixed rate but only establish within the patch
if they happen to land on a section of the patch that is
Linear Renewal
not currently occupied by another resource particle.
This process is often referred to as free-space recruit-
The simplest assumption about resource renewal is that it
ment and has been applied to models that describe pop-
occurs at some constant rate, r0, and resources remain
ulation dynamics of benthic marine invertebrates (e.g.,
viable until they are consumed. The renewal function is
Roughgarden 1997 and references within). In this case
r(x) = r0 . (30) the parameter K is often referred to as the patch-car-
rying capacity.
Suppose there are only diggers present, then the resource
This renewal model gives the following:
density on a patch of age a is
Coexistence of Feeding Strategies 271
Again, note that this condition is independent of the pa-
rameter r0. The rate at which the resource approaches the
carrying capacity has no influence on whether the digger
population will be invaded; however, the success of in-
vasion is influenced by the carrying capacity. The presence
of a carrying capacity means we have extra conditions
about when the grazers and diggers can persist in the
system. From equation (17) it can be shown that the
growth rate of the digger population is never positive and,
hence, cannot persist if Q D 1 K x D. Similarly, the grazer
population cannot persist if Q G 1 K x G. Thus, for any
given K, there now exists a region in the (Q D , Q G) plane
where neither the digger nor the grazer can persist. As
with the previous model on resource renewal, it can be
shown that for this model, pairs (Q D , Q G) exist that satisfy
both invasion conditions; therefore, coexistence is possible.
An example of the coexistence region for this model is
presented in figure 1B.
Three examples of consumer-resource dynamics are
presented in figure 2. The grazer species is characterized
by the same parameter values in each example (table 1)
and is associated with a Q value of Q G = 0.15. Grazers
encounter patches twice as fast as diggers, and both diggers
and grazers live, on average, one time unit (table 1). The
three examples differ in the Q value associated with the
diggers: (A) Q D = 0.4, (B) Q D = 0.5, and (C) Q D = 0.6. In
all three examples, the grazer species is assumed to be well
established before an inoculum of diggers is introduced.
The dynamics are qualitatively different in each of the three
examples, and they match the predictions (fig. 1B). The
steady state resource density when only grazers are present
∗
is X G = 0.65 resource units. The steady state resource den-
sities for the three examples, when only diggers are present,
∗ ∗ ∗
are (A) XD = 0.5, (B) XD = 0.6, and (C) XD = 0.7 resource
∗ ∗
units. Although X G 1 XD in figure 2B, the grazers clearly
Figure 1: Coexistence regions when resource renewal is (A) linear and
coexist with the diggers. The dynamics presented in figure
(B) free space (carrying capacity, K = 1 ). In both examples, the digger
2 were generated by numerically solving a set of six coupled
and grazer resource thresholds are given by xD = 0.1 and xG = 0.5.
delay differential equations (app. B).
x(a) = K (K x D) exp ( r0 a/K ), (34)
Logistic-Type Renewal
() A more general model that can be used to describe resource
KK xD
renewal is
t= ln , (35)
r0 K xG
j
r0
()x
∗
m= (K xD Q D). (36) i
r(x) = r0 x 1 , (38)
QD K K
The condition for grazers to invade diggers can be shown where i and j are nonnegative constants. The linear renewal
to be model is recovered when i = j = 0, and the free-space
model is recovered when i = 0 and j = 1. When i = j = 1,
(K xD)/QD
()
K xD resource renewal is logistic, which is a commonly adopted
QG ! QD . (37)
K xG model to describe the growth of a wide range of prey
272 The American Naturalist
so must be evaluated numerically. Figure 3 shows four
examples of the renewal function, equation (38), and
the form of the age-dependent change in resource den-
sity that results. In figure 4 we present the coexistence
regions associated with the renewal functions presented
in figure 3. Coexistence is possible for all resource-re-
newal functions presented, and the region of coexistence
varies depending on the assumption of resource re-
newal. The region is smallest when renewal is described
by the free-space model and is largest when renewal is
described by the third model. Numerical results again
show that the condition for grazers to invade diggers
is independent of the renewal parameter, r0.
Discussion
In this article we have shown that in some situations it is
possible that species that compete exploitatively for a com-
mon resource may coexist. This result is in contrast to
previous studies of exploitation competition that assume
the resource is well mixed within the habitat (e.g., Fishman
1997). The fact that we found the possibility of coexistence
in our model is not all that surprising. Our model is a
subtle example of resource partitioning, which, as was
noted in our introduction, has been shown to promote
coexistence. The two consumers are essentially competing
within a system that contains two types of resources,
namely L-patches and H-patches. An important result
from our model is that spatial and temporal changes in
the resource distribution and, hence, the two types of patch
are created by the foraging behaviors of the consumers
and not from any intrinsic differences among resource
patches. Although the abundance of L-patches and H-
patches are correlated at any time, knowledge of the abun-
Figure 2: Consumer-resource dynamics for the three cases presented in
dance of one does not necessarily specify the abundance
figure 1B (solid circles). In all examples K = 1, xD = 0.1, xG = 0.5, and
QG = 0.15. The Q-values associated with the digger species are (A) of the other; hence, coexistence may occur in this system
QD = 0.4, (B) QD = 0.5, and (C) QD = 0.6. Each example shows the mean
(Haigh and Maynard Smith 1972).
resource level, NLXL NHXH (dotted line); digger density, D (solid thick
The situation presented here corresponds to the idea of
line); and grazer density, G (solid thin line).
an included niche (Miller 1964, 1967), where in this case
organisms. There are two important properties of this
Table 1: Parameter values used in all the simulations
model. When j 1 0 the resource density within a patch will
presented
tend toward a carrying capacity K, and when i 1 0 the
Parameter Description Value
resource does not renew after it has been completely de-
pleted from a patch. From the previous model, we have K Patch carrying capacity 1
seen that the presence of a carrying capacity means that r0 Maximum renewal rate 20
species characterized by high Q values cannot persist in h Resource handling time .01
the system regardless of the presence or absence of other xD Digger resource threshold .1
xG Grazer resource threshold .5
species. The second property is particularly important if
Digger mortality rate 1
dD
consumers deplete all resources from a patch because even-
Grazer mortality rate 1
dG
tually they will drive themselves to extinction.
Digger patch encounter rate 100
jD
An analytic solution to the invasion condition, equa-
Grazer patch encounter rate 200
jG
tion (26), is generally not possible for this model and
Coexistence of Feeding Strategies 273
Figure 3: Examples of the logistic-type resource-renewal model as described by equation (38). Also shown are the associated renewal trajectories,
x(a). Parameter values for each example are as follows: (A) i = 0 , j = 1 ; (B) i = 1 , j = 1 ; (C) i = 2 , j = 1; and (D) i = 1, j = 2. In all examples, the
digger and grazer resource thresholds are given by xD = 0.1 ( filled dots) and xG = 0.5 (open dots), and the carrying capacity K = 1.
the niche of the grazer (i.e., the densities of resource that within its restricted niche). We have shown that if the
can be accessed) is a subset of the niche of the digger. consumer with the restricted niche is sufficiently efficient,
Coexistence may be possible, provided the grazer species then it can exclude the consumer that has the greater niche.
is more efficient than the digger species at utilizing high- The parameter r0 that appears in all three renewal mod-
density resource patches (i.e., it is a better competitor els considered in this article has no influence on the region
274 The American Naturalist
then their persistence depends on the presence of high-
density patches, which may become sufficiently rare in the
presence of grazers such that diggers are driven to
extinction.
A similar mechanism that shares many features with
ours was studied by Briggs (1993) in the context of insect
parasitoids. In her system two parasitoid species attack
different developmental stages of a single host species.
Briggs (1993) presents an age-structured model of the sys-
tem and shows that the two parasitoids could coexist if
the parasitoid that utilizes the later host stage could suc-
cessfully attack hosts that had been attacked earlier by the
other parasitoid. Like the results presented here, it was
found that the outcome of competition was dependent on
the consumer efficiency of both parasitoid species, which
was defined in a similar manner as the Q value of the
diggers and grazers in this article.
In this article we have shown that the consumer’s effect
on the distribution of resources, and not necessarily the
mean abundance of resources, is important for coexis-
tence. This result was also shown by Mittler (1997), who
Figure 4: Coexistence regions when resource renewal is described by the
investigated a model of predator-prey dynamics that over-
logistic-type functional form. The Digger only and Digger and Grazer
laps to some extent with the model presented in this article.
boundary correspond to the four cases presented in figure 3: (A) solid
thick line, (B) long-dashed line, (C) short-dashed line, and (D) solid thin Predators did not necessarily consume prey in their en-
line.
tirety, leaving smaller prey items that may be more effi-
ciently consumed by other predators. Mittler (1997)
of coexistence. The effect of r0 is to alter the timescale of showed that a rich range of dynamics could occur in a
resource renewal, which affects population sizes but not two-predator system. In some cases, depending on certain
the essential qualities of the dynamics that influence co- competitive trade-offs, frequency-dependent dynamics oc-
existence. Thus, the rate at which a resource renews has curred, where the first predator species to occupy a habitat
no bearing on which species are expected to persist in a could exclude the other predator species. In other cases
habitat. What is important for determining species com- coexistence of the two predator species was predicted. Re-
position is the manner in which the resource renews within sources, which may vary in size, were assumed to enter
patches. The functional form of resource renewal defines the habitat at some deterministic rate. Once a prey item
the conditions under which the grazer is able to persist in entered the habitat it did not change its size until it was
a habitat that is occupied by diggers. The region of co- attacked by a predator. Our work differs from Mittler
existence is increased when the relative rate of renewal in (1997) because we assume the state of the resource changes
the H-patches compared to the L-patches is increased. as it ages.
When a free-space renewal function is assumed, the re- One important issue in models of coexisting consumers
source increases much faster in the L-patches compared is how the density of the resource compares with the sit-
to the H-patches, and in this case the region of coexistence uation when only one consumer is present. This question
is small. However, if the rate of renewal is high in H- is of course important in biological control, where the
patches (e.g., fig. 3C), then the region of coexistence is resource is the item that we wish to regulate. W. W. Mur-
significantly increased (fig. 4). Although in this case it may doch (private communication) noted that in the simple
take some time for patches to become accessible to grazers models of coexistence discussed in our introduction, the
after they have been visited by a digger, if diggers move addition of a second coexisting consumer (e.g., predator,
slowly or have a low population size, then grazers can herbivore, parasite) to a system never decreases the re-
persist if they are sufficiently efficient (i.e., they make bet- source level below that which would arise with the most
ter use of the high resource-renewal rates of the H- “effective” consumer present alone. The dynamics of the
patches). By either moving quickly through the habitat or current model are consistent with this pattern. When co-
having high population growth rates, the grazers persist existence occurs the digger is always the most effective
even though they may encounter patches that they cannot consumer, and in these circumstances the mean equilib-
utilize. However, if diggers are not efficient consumers, rium resource density among patches with both diggers
Coexistence of Feeding Strategies 275
and consumers present is the same as when only diggers but variability in the resources encountered may be par-
are present (fig. 2). This minimum density is simply ticularly important for risk-sensitive foragers (Bulmer
∗
XD = x D Q D. 1994). Consumers may be expected to alter their foraging
Previous theoretical studies have shown that when mul- behavior depending on whether they are risk averse or risk
tiple species compete exploitatively for the same limiting prone. This work could be extended and applied to risk-
resource, it is the species with the lowest equilibrium re- sensitive foragers by incorporating the variance of the en-
source requirement that eventually displaces all other spe- countered resource in the equation that describes con-
cies (Armstrong and McGehee 1980; Tilman 1982). An sumer growth.
important result from our model is that one cannot predict A final assumption that needs further investigation is
which species will persist by only looking at the equilib- that age-0 H-patches renew at the same rate regardless of
rium resource level when each species is in isolation. We whether they were just attacked by a grazer or whether
have shown that grazers that have a higher equilibrium they were an L-patch that reached the age t. We may expect
than the diggers may still coexist with the diggers (fig. 2B). for some resource types that renewal may differ depending
In order to allow analytic tractability for much of our on whether it was just attacked by a grazer (e.g., vegetative
analysis, we have had to make a number of simplifying regrowth). The rate at which a patch is grazed may be
assumptions with regards to consumer and resource dy- important for modeling how a resource renews. This is
namics. An important assumption we have made that certainly true for resource depression where the resource
needs further investigation is a lack of resource-dependent itself may exhibit predator avoiding behavior (e.g., insect
consumer behavior. Consumers are assumed to move ran- prey). Resource responses could potentially alter the
domly through the habitat; they do not make systematic model’s predictions.
movements or modify their movements based on recent The model presented in this article has shown that two
resource encounters. When visits to resource patches are species, which compete exploitatively, can coexist on a
not random the distribution of resources that a forager single resource in a homogeneous environment, given that
encounters is not necessarily the same as the overall dis- they satisfy certain conditions with regard to the level at
tribution of resources (Possingham 1989; Abrams 1999). which they deplete resources (x) and their associated char-
This may affect the region of coexistence. Because we have acter value (Q). However, this model does not tell us
assumed random movements, both consumer species will whether another species could invade and exclusively take
be scattered within the habitat at any time, but systematic over the system and not be invadable by any other po-
foraging may partition the distributions of the two species. tential species. If there exists some trade-off between x and
Laverty and Plowright (1985) observed that humming- Q (Schmitt 1996), then future work could use the model
birds (diggers) and bumblebees (grazers) partition the re- to investigate whether selection would favor some inter-
source distribution in both space and time. Part of the mediate species or allow coexistence of a suite of species.
spatial segregation may be explained by differences in each
species’ ability to access the flowers (the inner flowers of
a patch were protected from hummingbirds by vegetative Acknowledgments
cover). Temporal segregation may be due to differences in
the metabolic cost of foraging. Here we have assumed that We thank C. J. Briggs, J. S. Brown, M. A. Gilchrist, W. W.
consumers are always foraging and renewal is a continuous Murdoch, K. J. O’Keefe, C. W. Osenberg, R. J. Schmitt,
process. Despite our simple assumptions on foraging be- L. Stone, and two anonymous referees for their helpful
havior, the model does suggest that multiple species that comments. The research was supported in part by the
exhibit little or no apparent systematic foraging may still Office of Naval Research (grant N00012-93-10952), the
coexist even if they consume the same resource. National Science Foundation (grants DEB-93-19301 and
Another potentially important assumption we have DEB-98-73650), and the Department of Zoology, Duke
made is that the habitat is homogeneous. In a real system University. R.M.N. and H.P.P. performed part of the work
we would expect variability among patches with regard to as sabbatical fellows at the National Center for Ecological
renewal rates and their carrying capacity (Possingham Analysis and Synthesis, a center funded by the National
1988, 1989). We have also assumed that the population Science Foundation (grant DEB-94-21535); the University
growth of the consumer is related to the mean intake rate, of California, Santa Barbara; and the state of California.
276 The American Naturalist
APPENDIX A
Stability Analysis of the Single Consumer–Single Resource Model
Here we present an alternative way to describe resource and consumer dynamics when only one consumer species is
present. Suppose only the grazer species is present. It can be shown that the dynamics of the mean patch density (XH)
and the grazer population size (G) are governed by the following coupled ODEs:
dXH jG(XH(t) x G)G(t)
= FH(t) ,
dt 1 jGh(XH(t) x G)
[ ]
dG j (XH(t) x G)
GG
= dG G(t),
dt 1 jGh(XH(t) x G)
where
FH(t) = n H(a, t)r(x H(a, t))da.
0
This formulation requires that the renewal function r is such that x(a) is not exponential for large a. When resource
renewal is linear, FH(t) = r0; when resource renewal obeys the free-space model, FH(t) = r0(1 XH(t)/K ) . A simple analytic
form for FH(t) is not possible for logistic-type renewal because the function r(x) is nonlinear. Note that the age
distribution of patches may not need to be explicitly modeled in this system.
Now we examine the stability of this model formulation. The associated Jacobian is
[ ]
jGG ∗/(1 jGhQ G)2 FH / XHFX∗ jGQ G /(1 jGhQ G)
J= ,
H
jGG ∗/(1 jGhQ G)2 0
G
where G ∗ and XH are equilibrium values for the system. There exist two equilibrium points for this model. One is the
∗
trivial solution where there are no consumers at all, G ∗ = 0 , and XH is the carrying capacity (this is infinite for the
∗
linear renewal model). There also exists a nontrivial equilibrium point, XH = Q G x G and G ∗ = FH (1
∗ ∗
hjGQ G)/(jGQ G). If we let D and T denote the determinant and trace of the Jacobian matrix J, then an equilibrium
point (G ∗, XH ) is locally stable if D 1 0 and T ! 0; otherwise, it is unstable (Bulmer 1994). Using these criteria, it can
∗
be easily shown that for both the linear and the free-space renewal models, the trivial equilibrium point is unstable
and the nontrivial point is stable. Hence, the grazer will always become well established when introduced into the
system when it is free of the digger. The same reasoning can be applied to the digger-only situation. Although we
cannot prove analytically that consumers can become well established when resources renew according to the more
general logistic-type model, numerical simulations suggest this is true also (see Nisbet et al. 1998).
APPENDIX B
Resource Dynamics Expressed as Ordinary Differential Equations
When one wishes to generate numerical approximations to partial differential equations (PDEs), it is sometimes easier
to convert them to ODEs and approximate the solutions to the ODEs instead. Here we transform equations (7)–(10),
which describe the dynamics of nL(a, t ), nH( a, t ), xL(a), and xH(a), into equivalent equations that describe the dynamics
of NL(t) and the products NL(t)XL(t) and NH(t)XH(t). Hence, we show that resource dynamics can be described by
equations that do not explicitly keep track of the age distribution of patches.
First, we integrate equation (7) over the duration it takes an L-patch to become an H-patch,
Coexistence of Feeding Strategies 277
t t
nL nL
da = n L(a, t)da,
m(t)
t a
0 0
which reduces to the following:
dNL
= m(t)NL(t) n L(t, t) n L(0, t). (B1)
dt
The last term is simply m(t) (see eq. [11]). The second to last term represents the fraction of patches that were last
visited by a digger at time t t . The probability a patch is not visited by a digger from time t t to time t is given
by
( )
t
S(t) = exp m(y)dy . (B2)
tt
Hence,
n L(t, t) = S(t)n L(0, t t) = S(t)m(t (B3)
t).
Substituting equations (11) and (B3) into (B1) gives the following delay differential equation (DDE) for NL:
dNL
= m(t)(1 NL(t)) S(t)m(t (B4)
t).
dt
Differentiating equation (5) with respect to t gives
t
d nL
(NL(t)XL(t)) = x Lda,
dt t
0
t
( )
nL
= m(t)n L x Lda,
a
0
t t t
( )
nL dx L dx L
= xL nL da nL da n Lx Lda, (B5)
m(t)
a da da
0 0 0
t t
= (n Lx L )da n Lr(x L )da m(t)NL(t)XL(t),
a
0 0
= n L(0, t)x L(0, t) n L(t, t)x L(t, t) FL(t) m(t)NL(t)XL(t),
= m(t)x D S(t)m(t FL(t) m(t)NL(t)XL(t),
t)x G
where
t
FL(t) = n L(a, t)r(x L(a))da.
0
When resource renewal is described by the free-space model (eq. [33]),
FL(t) = r0(NL(t) NL(t)XL(t)/K ).
Similarly, we can differentiate equation (6) with respect to t, which, after a little algebra, gives
278 The American Naturalist
d
(N (t)XH(t)) = [S(t)m(t XL(t))]x G FH(t) (m(t) n(t))NH(t)XH(t), (B6)
t) n(t)(1
dt H
where
FH(t) = n H(a, t)r(x H(a))da.
0
For the case of free-space renewal,
FH(t) = r0(1 NL(t) NH(t)XH(t)/K ).
Differentiating equation (B2) with respect to t gives the final DDE that closes the system:
dS
= [m(t (B7)
t) m(t)]S(t).
dt
Consumer-resource dynamics can be generated by numerically approximating solutions to the following sets of coupled
equations: (B4)–(B7), (17), and (18). Subtleties related to initializing DDEs are discussed by Nisbet (1997).
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