FW662 Lecture 10 – Predation 1
Lecture 10. Predation, Parasitism, and Herbivory.
Reading:
Gotelli, 2001, A Primer of Ecology, Chapter 6, pages 125-153.
Renshaw (1991) Chapter 6 Predator-prey processes, Pages 166-204.
Optional:
Gasaway, W. C., R. D. Boertje, D. V. Grangaard, D. G. Kelleyhouse, R. O. Stephenson,
and D. G. Larsen. 1992. The role of predation in limiting moose at low densities
in Alaska and Yukon and implications for conservation. Wildlife Monograph
120. 59 pp.
Boutin, S. 1992. Predation and moose population dynamics: a critique. Journal of
Wildlife Management 56:116-127.
Krebs, C. J. et al. 1992. What drives the snowshoe hare cycle in Canada's Yukon?
Pages 886-896 in D. R. McCullough and R. H. Barrett, eds. Wildlife 2001:
Populations. Elsevier Applied Science, New York, New York, USA.
The notion that one trophic level controls the dynamics of another trophic level is central to
ecology. Energy flows between trophic levels. The input of energy to a higher trophic
level population from a lower trophic level population can be controlled by the amount of
energy available in the lower level, or by the amount of energy the higher trophic level is
able to consume. Hence, the ideas of this chapter are as pertinent to herbivores and plants
as they are to predators and their prey. If the prey population limits the predator
population, the system is referred to as bottom-up control. Vice versa, if the predator
limits the prey population, then the system is referred to as top-down control.
The idea that predators control prey (Hairston et al. 1960), or top-down control, has been
pervasive in the literature for a long time. In a review of the impact of small-rodent
population dynamics, Hanski et al. (2001) conclude that predators are indeed causing the
regular multi-annual population oscillations of boreal and arctic small rodents (voles and
lemmings).
Alternatively, the notion that predators only take the weak and the sick (Errington 19??) suggests
that predators are not controlling prey populations. Rather, predators are only taking the
“doomed surplus”, and therefore, are not additive to the prey death rate, but are
compensatory.
Cause and Effect. Demonstration of predators having an impact on prey population must be
done via manipulative experiments. The following graph shows a correlation between
predator density and the proportion of prey consumed (killed by predators). However,
this correlation is not evidence that the predator population is in any way controlling the
prey population density. For example, reduction in predator numbers might result in
increased starvation of prey because of density dependence, and hence predation is
compensatory to starvation. Unfortunately, actual manipulations of predator and prey
FW662 Lecture 10 – Predation 2
populations are limited, and the few that do appear in the literature usually lack
replication and appropriate spatial and temporal controls. As a result, this area of
population dynamics is still highly controversial.
Most of the predator-prey models stem either from the original models of Lotka (1925) and
Volterra (1926) or from those of Nicholson (1933) and Nicholson and Bailey (1935).
These models provides an historical perspective and a foundation on which to build a
rigorous mathematical to base our discussion. Both types of model are based on simple
assumptions. Both are a closed system involving coupled interactions. Neither model
involves age structure for either the predator or prey. Predation is a linear function of
prey density, implying insatiable predators with not handling time. And, both models
assume that the prey eaten are direcytly converted into new predators. The chief
difference lies in the Lotka-Volterra models being differential equations, while the
Nicholson-Bailey models are based on difference equations giving discrete, non-
overlapping generations (Hassell and Anderson 1989). The following equations are the
Lotka-Volterra model:
dN1
' b1N1 & d1N1 ' b1N1 & *N1N2
dt
dN2
' b2N2 & d2N2 ' $N1N2 & d2N2
dt
The death rate (d1) of the prey (N1) is set proportional to the size of the predator
population (N2) as *N2. Likewise, the birth rate (b2) of the predator population is set
proportional to the prey population as $N1. Units of b1, b2, d1, and d2 are time-1, and units
for $ are predators born/predator/prey/time and for * prey eaten/prey/predator/time.
Thus, the equations generate units of prey/time or predators/time for the respecitve rates.
Some of the important characteristics of these equations are:
lack of density-dependence,
death rate of prey only depends on predator and prey,
FW662 Lecture 10 – Predation 3
death rate of predator only depends on predator, and
birth rate of predator depends on predator and prey.
Lotka-Volterra equations predict a stable cycle of predator and prey for a “pathological” set of
parameter values (Hassell and Anderson 1989), whereas the difference equation formulation
results in a expanding oscillations, as shown in the following graph.
Equilibrium can only be obtained via very carefully set parameter values. Yet, the
literature is full of theoretical results derived from these simple equations.
More appalling, these equations have been suggested as explaining some
of the classic predator-prey cycles observed in nature, e.g., the lynx cycle
based on fur returns for the Mackenzie River region of Canada, 1821-
1934, reported by Elton and Nicholson (1942). These equations are a
classic case of the mathematical tail wagging the biological dog. The
equations have so little reality that its hard to see how they have much to
do with real systems. Still, numerous hypotheses have been developed
from these equations. I will now explore some of the extensions, and the
consequences of these modifications to the predicted behavior of the
predator and prey populations.
Density Dependence. As a first attempt to incorporate some biological realism, density
dependence of the prey population might be included. As formulated above, the prey
population grows exponentially without predators. Density dependence might also be
incorporated into the predator population, giving the following equations:
dN1 2
' b1N1 & *N1N2 & c1N1
dt
dN2 2
' $N1N2 & d2N2 & c2N2
dt
Cycles no longer result with these equations.
FW662 Lecture 10 – Predation 4
Extensions commonly made to the Lotka-Volterra model involve the functional and
numerical responses. The functional response defines the changes in the per
capita predation rate as prey density increases. The numerical response defines
the changes in predator density as prey density increases.
Functional Response Curves. A predator’s functional response is its per capita feeding
rate on prey. Holling (1965, 1966) suggested that the predator should not be able
to consume an unlimited number of prey as the prey population increases. That is,
in the Lotka-Volterra equations, the number of prey consumed per predator is
unlimited as the prey population increases. The number of prey removed is
*N1N2, so that the number of prey eaten per predator is unlimited as N1 increases
to infinity. Holling proposed 3 models of the rate of prey capture per predator as a
function of prey population density: Types I, II, and III. Type I is the default case
already modeled in the Lotka-Volterra equations.
Type I: m1 = aTN1, where m1 is number of prey eaten over a time period T,
per predator, and a is a proportionality constant.
Type II: m1 = a T N1 / (1 + a TH N1), where TH is the time required to
catch and devour a prey (handling time). This equation is obtained
by substituting into the Type I and correcting for the handling time,
m1 = a(T - m1TH)N1, and solving for m1.
Type III: m1 = a T N10 / (1 + a TH N10), where 0 is generally set equal to 2.
0 generates a lag time in the learning curve of the predator, or a
"training effects". The idea of adding a power term is a common
mathematical trick, e.g., the Richard’s equation described as a
modification of the logistic population growth model. For the
following examples, the parameter values are I: a = 0.5, T = 1; II:
a = 0.4, T = 1, TH = 0.3, and III: a = 0.015, T = 1, TH = 0.3, and
0 = 2.
Functional Response Curves
Prey Eatern per Predator per Time
5
Type I
4
2
Type II Type III
1
0
0 5 10 15 20 25 30
Prey Population
FW662 Lecture 10 – Predation 5
Depensatory mortality is a term used to describe the decrease in the rate of prey
mortality as the prey population increases. That is, as all the predators become
satiated because of the large numbers of prey, the number of prey killed per
number of prey available declines, so that the prey survival rate will increase.
This phenomena will occur as the Type II and III curves reach asymptotic values
of the prey population. The Lotka-Volterra model is extended to incorporate Type
II and III curves, i.e., the rate at which predators eat prey is modified by the
functional responses shown above. The cyclic behavior can persist with Type II
and III functional responses incorporated into the Lotka-Volterra equations.
Holling (1965) justified the Type III functional response curve based on empirical
evidence. He buried sawfly cocoons in sand (the prey), and let mice (Peromyscus)
search for them. Until they learned how to find the cocoons, they were less
effective than after they had become proficient. Swenson (1977) (in Emlen
1984:108) constructed functional response curves for walleye and sauger, and
found that a Type II curve was adequate.
Skalski and Gilliam (2001) review the literature on functional response curves and
presented statistical evidence from 19 predator-prey systems that three predator-
dependent functional responses (Beddington-DeAngelis, Crowley-Martin, and
Hassell-Varley), i.e., models that are functions of both prey and predator
abundance because of predator interference, can provide better descriptions of
predator feeding over a range of predator-prey abundances. No single functional
response best described all of the data sets. The Beddington-DeAngelis
functional response curve (Beddington 1975, DeAngelis et al. 1975) is:
aN1
f(N1, N2) ' ,
1 % bN1 % c(N2 & 1)
where when the value of c becomes zero, this functional response curve becomes
identical to Hollings Type II curve. This model assumes that handling and
interfering are exclusive activities. The Crowley-Martin (Crowley and Martin
1989) allows for interference among predators regardless of whether a particular
individual is currently handling prey or searching for prey. The Crowley-Martin
model thus adds an additional term in the denominator:
aN1
f(N1, N2) ' ,
1 % bN1 % c(N2 & 1) % bcN1(N&2 & 1)
which can be simplified to
aN1
f(N1, N2) ' .
(1 % bN1)(1 % c(N2 & 1))
As with the Beddington-DeAngelis curve, making c equal to zero results in a Type
II curve. The third functional response curve considered by Skalski and Gilliam
(2001) was the Hassell-Varley (Hassell and Varley 1969) model:
FW662 Lecture 10 – Predation 6
aN1
f(N1, N2) ' ,
m
bN1 % N2
where when m becomes zero reduces to a Type II curve.
Vucetich et al. (2002) also consider a number of different functional response
curves to model the relationship between wolves and moose on Isle Royale.
Predator density explained more variation in kill rate than did prey density
( R 2 = 0.36 vs. R 2 = 0.17, respectively). The ratio-dependent model greatly
outperformed the prey-dependent model. Nevertheless, the ratio-dependent model
failed to explain most of the variation in kill rate ( R 2 = 0.34). The ratio-
dependent – prey-dependent controversy should disappear as investigators
recognize that both models are overly simplistic.
Numerical Response Curves. Numerical response is the response of predator populations
to prey populations. Predator birth rate may be a function of the food intake rate,
so that increased prey availability may result in an increased birth rate, up to some
asymptotic value. Another possibility is that predators immigrate to an area, or
aggregate in an area of high prey density. Type II and III curves are also useful for
modeling this process. This type of response results in a numerical response of
the predator to the prey, i.e., the number of predators increases in response to the
number of prey. The following graph illustrates an hypothetical example.
Maker (1970) (in Emlen 1984) demonstrated a Type III response of jaeger nest
density to lemming density. Johnson and Carpenter (1994) used functional and
numerical responses to develop a fish-angler interaction model.
Stochasticity. The cyclic behavior of these equations will not be retained in a model that
includes demographic stochasticity. Typically, both populations will go extinct.
FW662 Lecture 10 – Predation 7
What predator-prey system can you think of that does not have demographic
stochasticity? Environmental stochasticity might also reasonably be added to the
Lotka-Volterra model, i.e., the basic rate constants become functions of the
environment. For example, wolves pursuing moose benefit from snow conditions
that support the weight of a wolf, but not the weight of a moose. An example of a
Lotka-Volterra model which incorporates demographic stochasticity is shown in
the following graph.
Stochastic Predator-Prey Model
60
Prey
50
Population Size
40 Predator
30
20
10
0
0 1 2 3 4
Time (t)
Spatial stochasticity should also be added to provide a realistic model.
Examples of some of the hypotheses (some authors consider the following as
“conclusions”) (taken from Emlen 1984):
1. Predator-prey system is more likely stable if the predator is not highly efficient
at finding and capturing prey.
2. Predator-prey system is more likely stable if predator is not highly efficient in
handling prey.
3. Predator-prey system is more likely stable if predator is not highly efficient at
converting food to growth and reproduction.
4. Enrichment of a predator-prey system by addition of food for the prey or
alternative sustenance for the predator destabilizes the system.
5. Many (most) prey species have available to them some form of “refuge”
that prevents extinction by the predator. As a result, the predator-prey
system may show greater amplitude in the cycling.
6. Time lags destabilize a predator-prey system.
7. Density-dependence takes out cyclic behavior of a predator-prey system.
8. Stochasticity (both demographic and environmental) takes out cyclic behavior
of a predator-prey system.
FW662 Lecture 10 – Predation 8
9. Spatial stochasticity can be viewed as a refuge system, or as a metapopulation
system. Too low of prey dispersal makes the system unstable, just as too
high of prey dispersal does.
Coevolution. The predator-prey system is sensitive to the efficiency of the predator
taking prey, and ultimately determines the stability of the system. We can assume
that natural selection is constantly operating on the system, so that the predator is
improving its abilities to capture prey, whereas the prey is improving its abilities
to avoid capture. The predator and prey are co-evolving, like an evolutionary
arms race.
Experimental Studies
Huffaker experiments (Huffaker 1958). Prey was six-spotted mite (Eotetranychus
sexmaculatus), predator was a predatory mite, Typhlodromus occidentalis.
Oranges, rubber balls, and wax paper were used to construct experiments where
the prey had different habitat patches available to it, providing different
configurations of prey to the predator. “To ensure that his experiment was
sufficiently complex, Huffaker placed 40 oranges or balls in a 4 X 10 rectangular
array on each of a number of adjacent trays. Migration of predators across trays
was restricted by inserting vaseline barriers; whilst migration of prey over these
barriers was achieved by providing each tray with wooden posts from which the
prey could launch themselves on a silken thread, aided by currents from an
electric fan.” (Renshaw 1991:206). This is an example of biological modeling not
using equations. Huffaker did observe oscillations in the predator and prey
system.
Wolves and Moose in Alaska and Yukon (Gasaway et al. 1983, 1992; Boutin 1992).
High predation on moose calves by wolves was thought to be limiting the moose
population. After wolves were removed from the system, the moose populations
responded upward. Hence, moose at low densities were being controlled by
wolves. Gasaway et al. (1992) termed the original state of the system as a Low
Density Dynamic Equilibrium (LDDE). When wolves were removed temporarily
from the system, the moose population climbed towards the carrying capacity set
by browse, and escaped the wolf predation limitation because relatively more
moose calves escaped wolf predation.
Mathematics (if any) of LDDE (Messier 1994)
Is the reason that cycles are most often observed in Arctic systems because of the
simplicity of these systems, and the strong seasonality?
Wolves and Moose on Isle Royale (McLaren and Peterson 1994).
This article infers top down control because the only manipulation(?) was a
change in the wolf population. Are their inferences really valid, given the lack of
a true experiment?
Coyotes and Mule Deer in Northwestern Colorado (Bartmann et al. 1992). These authors
FW662 Lecture 10 – Predation 9
observed high overwinter fawn mortality from coyote predation.
Removal of coyotes (93, 78, and 47 in 1985-86, 86-87, and 87-88,
respectively) did not increase fawn survival compared to the previous 4
winters prior to coyote removal, and suggested an increase in starvation of
fawns. Hence coyotes are providing compensatory mortality in the fawn
population, not additive mortality as would be suggested by the large
numbers of fawns killed by coyotes each winter.
Predators only take weak and sick -- NOT!, but prudent predators do take the
easiest prey.
FW662 Lecture 10 – Predation 10
Mule deer at KCC.
Snowshoe hare cycles (Krebs et al. 1992). Krebs et al. (1992) present good evidence of
snowshoe hare (Lepus americanus) cycles in southwestern Yukon. Four
hypotheses have been proposed to explain the hare cycles:
1. Keith hypothesis: winter food shortage reduces reproduction at the population
peak and causes starvation losses which start the cyclic downturn, and
predation, which continues the downturn and reduces hares to low
numbers.
2. Plant chemistry hypothesis: qualitative (nutritional) changes in the hare’s food
plants.
3. Predation hypothesis: differs from Keith’s hypothesis by being a single-factor
model for the hare cycle.
4. Chitty hypothesis, or polymorphic behavior hypothesis: driving mechanism is
the spacing behavior of the hares themselves.
In their study at Kluane Lake during 1977-84, they have performed experiments
with winter food supplementation of hares. “The overall results of these studies
were rather inconsistent with the food hypothesis. Winter food shortage was not
necessary for the cyclic decline, and extra food would neither slow the rate of
decline of the hares nor delay in all populations the start of the decline. In these
experiments, we did not control for increased predation on the food-supplemented
areas...”. (Krebs et al. 1992:888). “Keith et al. (1984), studying the same cyclic
decline in central Alberta hares, also concluded that food shortage was not present
on all areas where hares were declining. If these results are accepted, neither the
original Keith hypothesis nor the plant chemistry hypothesis can be supported as
an explanation of cyclic events on a local scale.” (Krebs et al. 1992:888). Krebs
et al. (1992) eliminate the Chitty hypothesis because no evidence of social
mortality, either directly through infanticide or fighting, or indirectly through
territoriality and dispersal. Through the process of elimination, the only
hypothesis left is predation. From 1986-present, they have set up experiments that
provide controls, fertilization plots, food supplementation, predator-proof fences
(which do not eliminate avian predators), and food+predator fence. Results as of
1992 suggested food supplementation had increased April 1 populations, but they
had not yet observed the down-swing in the hare cycle. Presumably, eliminating
predators would keep the hare population high if predation was the mechanism
that causes the down turn.
Largemouth bass in a Michigan lake (Mittelbach et al. 1995).
This paper presents the results of a long-term study of changing predator densities
and cascading effects in a Michigan lake in which the top carnivore, the
largemouth bass (Micropterus salmoides), was eliminated in 1978 and then
reintroduced in 1986. The elimination of the bass was followed by a dramatic
increase in the density of planktivorous fish, the disappearance of large
zooplankton (e.g., two species of Daphnia that had historically dominated the
zooplankton community), and the appearance of a suite of small-bodied
FW662 Lecture 10 – Predation 11
cladoceran (zooplankton) species. The system remained in this state until bass
were reintroduced. As the bass population increased, the system showed a steady
and predictable return to its previous state; planktivore numbers declined by two
orders of magnitude, large-bodied Daphnia reappeared and again dominated the
zooplankton, and the suite of small-bodied cladocerans disappeared. Within each
cladoceran species there was a steady increase in mean adult body size as
planktivore numbers declined. Total zooplankton biomass increased _10-fold
following the return of large-bodied Daphnia, and water clarity increased
significantly with increases in Daphnia biomass and total cladoceran biomass.
These changes in community structure and trophic-level biomasses demonstrate
the strong impact of removing a single, keystone species, and the capacity of the
community to return to its previous state after the species is reintroduced.
Competition for shelter space causes density-dependent predation in damselfishes.
Holbrook and Schmitt (2002) demonstrated that competition for shelter spaces
caused more predation in damselfishes. These species shelter in branching corals
or anemones, and when refuge spaces to protect them were filled with their intra-
specific competitors, more mortality from predation was found.
Cycles. Post et al. (2002) suggest that wolves cause moose population on Isle Royale
(island in Lake Superior) to cycle. Although the data do suggest that moose
populations have peaked twice in the interval 1958-1998, claiming that wolf
populations are responsible is not defensible. Rather, there does appear to be a
correlation between these populations. However, many different variables,
particularly weather variables, could be found to correlate with these populations,
and most investigators would not claim a cause-and-effect relationship had been
found.
Literature Cited
Bartmann, R. M., G. C. White, and L. H. Carpenter. 1992. Compensatory mortality in a
Colorado mule deer population. Wildlife Monograph 121:1-39.
Beddington, J. R. 1975. Mutual interference between parasites or predators and its effect on
searching efficiency. Journal of Animal Ecology 51:331-340.
Boutin, S. 1992. Predation and moose population dynamics: a critique. Journal of Wildlife
Management 56:116-127.
Crowley, P. H., and E. K. Martin. 1989. Functional responses and interference within and
between year classes of a dragonfly population. Journal of the North American
Benthological Society 8:211-221.
DeAngelis, D. L., R. A. Goldstein, and R. V. O’Neill. 1975. A model for trophic interaction.
Ecology 56:881-892.
FW662 Lecture 10 – Predation 12
Elton, C. and M. Nicholson. 1942. The ten-year cycle in numbers of the lynx in Canada.
Journal of Animal Ecology 11:215-244.
Errington, P. 19??.
Gasaway, W. C., R. O. Stephenson, J. L. Davis, P. E. K. Shepherd, and O. E. Burris. 1983.
Interrelationships of wolves, prey, and man in interior Alaska. Wildlife Monograph 84.
50 pp.
Gasaway, W. C., R. D. Boertje, D. V. Grangaard, D. G. Kelleyhouse, R. O. Stephenson, and D.
G. Larsen. 1992. The role of predation in limiting moose at low densities in Alaska and
Yukon and implications for conservation. Wildlife Monograph 120. 59 pp.
Hairston, N. G., F. E. Smith, L. B. Slobodkin. 1960. American Naturalist 94:421.
Hanski, I., H. Henttonen, E. Korpimäki, L. Oksanen, and P. Turchin. 2001. Small-rodent
dynamics and predation. Ecology 82:1505-1520.
Hassell, M. P., and R. M. Anderson. 1989. Predator-prey and host-pathogen interactions. Pages
147-196 in J. M. Cherrett, ed. Ecological concepts. Blackwell scientific Publ., Oxford,
United Kingdom.
Hassell, M. P., and G. C. Varley. 1969. New inductive population model for insect parasites and
its bearing on biological control. Nature 223:1133-1136.
Holbrook, S. J., and R. J. Schmitt. 2002. Competition for shelter space causes density-
dependent predation mortality in damselfishes. Ecology 83:2855-2868.
Holling, C. S. 1965. The functional response of predators to prey density and its role in mimicry
and population regulation. Memorandum Entomological Society of Canada No. 45
Holling, C. S. 1966. The functional response of invertebrate prey to prey density. Memorandum
Entomological Society of Canada 48:1-48.
Huffaker, C. B. 1958, Experimental studies on predation: dispersion factors and predator-prey
interactions. Hilgardia 27:343-383.
Johnson, B. M., and S. R. Carpenter. 1994. Functional and numerical responses: a framework
for fish-angler interactions. Ecological Applications 4:808-821.
Keith, L. B., J. R. Cary, O. J. Rongstad, and M. C. Brittingham. 1984. Demography and ecology
of a declining snowshoe hare populations. Wildlife Monograph 90.
FW662 Lecture 10 – Predation 13
Krebs, C. J. et al. 1992. What drives the snowshoe hare cycle in Canada's Yukon? Pages 886-
896 in D. R. McCullough and R. H. Barrett, eds. Wildlife 2001: Populations. Elsevier
Applied Science, New York, New York, USA.
Lotka, A. J. 1925. Elements of Physical Biology. Williams and Wilkins, Baltimore (Reissued as
Elements of Mathematical Biology by Dover, 1956).
Maker, W. J. 1970. The pomarine jaeger as a brown lemming predator in northern Alaska.
Wilson Bull. 82:130-157.
McLaren, B. E., and R. O. Peterson. 1994. Wolves, moose, and tree rings on Isle Royale.
Science 266:1555-1558.
Messier, F. 1994. Ungulate population models with predation: a case study with the North
American moose. Ecology 75:478-488.
Mittelbach, G. G., A. M. Turner, D. J. Hall, J. E. Rettig, and C. W. Osenberg. 1995.
Perturbation and resilience: a long-term whole-lake study of predator extinction and
reintroduction. Ecology 76:2347-2360.
Nicholson, A. J. 1933. The balance of animal populations. Journal of Animal Ecology 2:131-
178.
Nicholson, A. J., and V. A. Bailey. 1935. The balance of animal populations. Part 1.
Proceedings of the Zoological Society of London 1935:551-598.
Post, E., N. C. Stenseth, R. O. Peterson, J. A. Vucetich, and A. M. Ellis. 2002. Phase
dependence and population cycles in a large-mammal predator-prey system. Ecology
83:2997-3002.
Skalski, G. T., and J. F. Gilliam. 2001. Functional responses with predator interference: viable
alternatives to the Holling Type II model. Ecology 82:3083-3092.
Swenson, W. A. 1977. Food consumption of walleye (Stizostedion vitreum vitreum) and sauger
(S. canadense) in relation to food availability and physical conditions in Lake of the
Woods, Minnesota, Shagawa Lake and western Lake Superior. Journal Fisheries
Research Board of Canada 34:1643-1654.
Volterra, V. 1926. Variazioni e fluttuazioni del numero d’individui in specie animali
conviventi. Memorie della R. Accademia Nazionale dei Lincei 2:31-113. (Translation in
Chapman, R. N. 1931. Animal ecology, pages 409-448. McGraw-Hill, New York, New
York, USA.)
FW662 Lecture 10 – Predation 14
Vucetich, J. A., R. O. Peterson, and C. L. Schaefer. 2002. The effect of prey and predator
densities on wolf predation. Ecology 83:3003-3013.
