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The foraging benets of information and the penalty of ignorance
The foraging benets of information and the penalty of ignorance
Ola Olsson and Joel S. Brown
Olsson, O. and Brown, J. S. 2006. The foraging benefits of information and the penalty
of ignorance. �
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Oikos 112: 260 �
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273.
Patch use theory and the marginal value theorem predict that a foraging patch should
be abandoned when the costs and benefits of foraging in the patch are equal. This has
generally been interpreted as all patches being abandoned when their instantaneous
intake rate equals the foraging costs. Bayesian foraging �
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patch departure is based on a
prior estimate of patch qualities and sampling information from the current patch �
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predicts that instantaneous quitting harvest rates sometimes are not constant across
patches but increase with search time in the patch. That is, correct Bayesian foraging
theory has appeared incompatible with the widely accepted cost �
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benefit theories of
foraging. In this paper we reconcile Bayesian foraging with cost �
/
benefit theories. The
general solution is that a patch should be left not when instantaneous quitting harvest
rate reaches a constant level, but when potential quitting harvest rate does. That is, the
forager should base its decision on the value now and in the future until the patch is
left. We define the difference between potential and instantaneous quitting harvest rates
as the foraging benefit of information, FBI. For clumped prey the FBI is positive, and
by including this additional benefit of patch harvest the forager is able to reduce its
penalty of ignorance.
O. Olsson and J. S. Brown, Dept of Biological Sciences, Univ. of Illinois at Chicago, 845
W. Taylor St, 60607 IL, USA. Present address for OO: Dept of Animal Ecology, Lund
Univ., Ecology Building, SE-223 62 Lund, Sweden (ola.olsson@zooekol.lu.se).
A foragers food may be distributed patchily. Spatial
variability in food availability poses opportunities and
challenges to the forager. As opportunity, variability
allows the forager to bias its searching efforts towards
rich patches and away from poor patches (Stephens
1989). Most models of patch use (Charnov 1976, Oaten
1977, Brown 1988) and habitat selection (Fretwell and
Lucas 1970, Rosenzweig 1981) assume that foragers
assess spatial heterogeneity in feeding or fitness oppor-
tunities and respond accordingly. As challenge, though,
the forager must be able to assess this heterogeneity
before it can benefit from a more efficient allocation of
effort. Furthermore, variability in food availability can
lead to variability in the individuals food intake rate.
Depending on the relationship between food consump-
tion and fitness (often an increasing and decelerating
curve) a foragers fitness may be influenced by temporal
variability in feeding rates (for instance, risk sensitive
foraging, Caraco 1980, Real 1980). Here we are inter-
ested in the assessment challenge posed by patchily
distributed foods.
Foragers may obtain information from a variety of
sources. Prior to selecting a food patch, a forager may
gain knowledge on the whereabouts of rich and poor
patches from visual, auditory or chemical cues that can
be detected at a distance. Such long distance cues include
patch appearance (e.g. floral color or abundance,
Sandlin 2000; the fruit scent or burdens of trees,
Sallabanks 1993) or observations on the foraging
successes of other individuals (Valone and Giraldeau
1993). This allows a forager to be periscopic (sensu
Mitchell 1989) by selectively visiting higher quality
patches and/or by optimizing the order within which
patches are visited (variations on the traveling salesman
problem, Gross et al. 1995). Upon encountering a
patch, a forager may gain information on its quality,
Accepted 6 September 2005
Copyright # OIKOS 2006
ISSN 0030-1299
OIKOS 112: 260
�
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273, 2006
260
OIKOS 112:2 (2006) usually represented as the abundance of resources. A
forager may be prescient (sensu Valone and Brown
1989). Upon entering the patch, such a forager uses
cues from the patch to make the most accurate assess-
ment of quality. Alternatively, the forager uses informa-
tion gained while exploiting the patch to make and
continuously update an estimate of patch quality (Oaten
1977, Green 1980, Iwasa et al. 1981, McNamara 1982,
Olsson and Holmgren 1998). Here we are interested in
this last information challenge. Specifically, how should
a forager use knowledge of cumulative search time and
cumulative harvest within a patch to effect an optimal
patch use decision?
Bayesian foraging has provided the conceptual frame-
work for how a forager should estimate patch quality
from three sources of information: 1) an a priori
knowledge of the distribution of patch qualities through-
out the environment, 2) time spent thus far in searching
for food items within the current patch, and 3) number
of food items thus far encountered and harvested. When
search within a patch is random (equal and constant
encounter probability on all food items within the
patch), authors agree on how a Bayesian forager can
estimate current patch quality.
However, authors disagree on how a fitness maximiz-
ing forager should use this information to decide how
thoroughly to use each food patch. Cost �
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benefit models
of patch use suggest that a forager should remain in the
patch until the expected instantaneous harvest rate no
longer exceeds foraging costs (Charnov 1976, Iwasa et al.
1981, Valone and Brown 1989). If foraging costs do not
vary among patches, then the forager should strive to
equalize quitting harvest rates among patches (Brown
and Mitchell 1989) and leave each patch when its
estimated quality falls to a threshold giving-up density
(GUD, Brown 1988).
While seductively straightforward, the balancing of
expected instantaneous harvest rate with foraging costs
can be wrong for a Bayesian forager (Green 1980, 1984).
That is, such a forager does not necessarily maximize its
fitness by leaving all patches at the same instantaneous
intake rate (Olsson and Holmgren 1998, 2000). Here
we are interested in reconciling the attractiveness of
cost �
/
benefit models of patch use with the following
conceptual facts about Bayesian foraging and patch
use.
The peculiarities of Bayesian patch use and its
incompatibility with traditional patch use models such
as the marginal value theorem (Charnov 1976), or H 0
/
C'
/
P'
/
MOC (Brown 1988) go back to Oaten (1977),
Green (1980) and McNamara (1982). But, it is a
technical report by Green (1988) that fully illuminates
the potential incompatibilities of a fixed quitting harvest
rate patch use rule with optimal foraging under Bayesian
patch use. Iwasa et al. (1981, and Valone and Brown
1989, Rodr�guez-Girone�s and Va�squez 1997) merely
assume a fixed quitting harvest rate strategy as optimal.
Green shows beautifully how this is true only when
resources are not clumped (e.g. binomial or Poisson
distributions).
When the distribution of resources among patches is
clumped (e.g. negative binomial �
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most patches have
few items and a few patches have many items), a
forager should use a potential value rule and only leave
the patch when the expected average harvest rate
within the patch for the remainder of the visit no
longer exceeds foraging costs. Olsson and Holmgren
(1998) showed how this subtle yet crucial distinction
leads to an interesting pattern of patch use. Instead of
a constant relationship between quitting harvest rate
and time spent within a patch, quitting harvest rate
actually rises with patch residence time. They show
how a forager may be willing to persevere in a
seemingly unsatisfactory patch in the knowledge that
finding a food item may provide the good news that
this patch is not so bad after all. Here, we show how
incorporating this good news into patch use models
can reconcile potential value rule of Green (1988)
and Olsson and Holmgren (1998) with the instanta-
neous rate rule of the patch use model of Brown (1988,
1992) and with the marginal value theorem (Charnov
1976).
Our specific goals include:
1)
Highlight the salient differences and consequences
of a Bayesian forager using a fixed quitting harvest
rate strategy versus a potential value rule. Under
clumped distributions of food, the fixed quitting
harvest rate strategy that seems so compatible with
cost �
/
benefit models of patch use is wrong. The
potential value rule, which is optimal, does not
seem compatible with traditional cost �
/
benefit
models.
2)
Introduce the concept of a foraging benefit of
information (FBI). With this concept we suggest
that the optimal patch use strategy of a Bayesian
forag