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Chapter 18: Dynamics of Predation
Robert E. Ricklefs
The Economy of Nature, Fifth Edition
(c) 2001 by W. H. Freeman and
Company
Population Cycles of
Canadian Hare and Lynx
Charles Elton’s seminal paper focused on
fluctuations of mammals in the Canadian boreal
forests.
Elton’s analyses were based on trapping records
maintained by the Hudson’s Bay Company
of special interest in these records are the regular
and closely linked fluctuations in populations of the
lynx and its principal prey, the snowshoe hare
What causes these cycles?
(c) 2001 by W. H. Freeman and
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Some Fundamental
Questions
The basic question of population biology
is:
what factors influence the size and stability
of populations?
Because most species are both consumers
and resources for other consumers, this
basic question may be refocused:
are populations limited primarily by what
they eat or by(c) 2001
what
eats them?
by W. H. Freeman and
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More Questions
Do predators reduce the size of prey
populations substantially below the carrying
capacity set by resources for the prey?
this question is prompted by interests in
management of crop pests, game populations, and
endangered species
Do the dynamics of predator-prey interactions
cause populations to oscillate?
this question is prompted by observations of
predator-prey cycles in nature, such as Elton’s lynx
(c) 2001 by W. H. Freeman and
and hare
Company
Consumers can limit
resource populations.
 An example: populations of cyclamen mites, a pest of
strawberry crops in California, can be regulated by a
predatory mite:
cyclamen mites typically invade strawberry crops soon after
planting and build to damaging levels in the second year
predatory mites invade these fields in the second year and keep
cyclamen mites in check
 Experimental plots in which predatory mites were
controlled by pesticide had cyclamen mite populations
25 times larger than untreated plots.
(c) 2001 by W. H. Freeman and
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What makes an effective
predator?
Predatory mites control populations of
cyclamen mites in strawberry plantings
because, like other effective predators:
they have a high reproductive capacity
relative to that of their prey
they have excellent dispersal powers
they can switch to alternate food resources
when their primary prey are unavailable
(c) 2001 by W. H. Freeman and
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Consumer Control in
Aquatic Ecosystems
An example: sea urchins exert strong
control on populations of algae in rocky
shore communities:
in urchin removal experiments, the biomass
of algae quickly increases:
in the absence of predation, the composition of
the algal community also shifts:
• large brown algae replace coralline and small green
algae that can persist in the presence of predation
(c) 2001 by W. H. Freeman and
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Predator and prey
populations often cycle.
Population cycles observed in Canada are
present in many species:
large herbivores (snowshoe hares, muskrat, ruffed
grouse, ptarmigan) have cycles of 9-10 years:
predators of these species (red foxes, lynx, marten, mink,
goshawks, owls) have similar cycles
small herbivores (voles and lemmings) have cycles of
4 years:
predators of these species (arctic foxes, rough-legged
hawks, snowy owls) also have similar cycles
cycles are longer in forest, shorter in tundra
(c) 2001 by W. H. Freeman and
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Predator-Prey Cycles: A
Simple Explanation
Population cycles of predators lag slightly
behind population cycles of their prey:
predators eat prey and reduce their numbers
predators go hungry and their numbers drop
with fewer predators, the remaining prey
survive better and prey numbers build
with increasing numbers of prey, the
predator populations also build, completing
the cycle
(c) 2001 by W. H. Freeman and
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Time Lags in Predator-Prey
Systems
Delays in responses of births and deaths to an
environmental change produce population
cycles:
predator-prey interactions have time lags associated
with the time required to produce offspring
4-year and 9- or 10-year cycles in Canadian tundra
or forests suggest that time lags should be 1 or 2
years, respectively:
these could be typical lengths of time between birth and
sexual maturity
the influence of conditions in one year might not be felt until
young born in that
year
old enough
to reproduce
(c) 2001
by W.are
H. Freeman
and
Company
Time Lags in PathogenHost Systems
Immune responses can create cycles of infection
in certain diseases:
measles produced epidemics with a 2-year cycle in
pre-vaccine human populations:
two years were required for a sufficiently large population of
newly susceptible infants to accumulate
(c) 2001 by W. H. Freeman and
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Time Lags in PathogenHost Systems
 other pathogens cycle because they kill sufficient hosts
to reduce host density below the level where the
pathogens can spread in the population:
such cycling is evident in polyhedrosis virus in tent caterpillars
In many regions, tent caterpillar infestations last about 2 years
before the virus brings its host population under control
In other regions, infestations may last up to 9 years
Forest fragmentation – which creates abundant forest edge –
tends to prolong outbreaks of the tent caterpillar
Why?
Increased forest edge exposes caterpillars to more intense sunlight 
inactivates the virus  thus, habitat manipulation here has secondary
effects
(c) 2001 by W. H. Freeman and
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Laboratory Investigations
of Predators and Prey
G.F. Gause conducted simple test-tube
experiments with Paramecium (prey) and
Didinium (predator):
in plain test tubes containing nutritive medium, the
predator devoured all prey, then went extinct itself
in tubes with a glass wool refuge, some prey
escaped predation, and the prey population
reexpanded after the predator went extinct
Gause could maintain predator-prey cycles in such tubes by
periodically adding more predators
(c) 2001 by W. H. Freeman and
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Predator-prey interactions can be
modeled by simple equations.
 Lotka and Volterra independently developed
models of predator-prey interactions in the
1920s:
dR/dt = rR - cRP
describes the rate of increase of the prey
population, where:
R is the number of prey
P is the number of predators
r is the prey’s per capita exponential growth rate
c is a constant(c)expressing
efficiency
of predation
2001 by W. H. Freeman
and
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Lotka-Volterra PredatorPrey Equations
A second equation:
dP/dt = acRP - dP
describes the rate of increase of the predator
population, where:
P is the number of predators
R is the number of prey
a is the efficiency of conversion of food to growth
c is a constant expressing efficiency of predation
d is a constant related to the death rate of predators
(c) 2001 by W. H. Freeman and
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Predictions of LotkaVolterra Models
Predators and prey both have equilibrium
conditions (equilibrium isoclines or zero
growth isoclines):
P = r/c for the predator
R = d/ac for the prey
when these values are graphed, there is a single
joint equilibrium point where population sizes of
predator and prey are stable:
when populations stray from joint equilibrium, they cycle
with period T = 2 / rd
(c) 2001 by W. H. Freeman and
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Cycling in Lotka-Volterra
Equations
A graph with axes representing sizes of
the predator and prey populations
illustrates the cyclic predictions of LotkaVolterra predator-prey equations:
a population trajectory describes the joint
cyclic changes of P and R counterclockwise
through the P versus R graph
(c) 2001 by W. H. Freeman and
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Factors Changing
Equilibrium Isoclines
The prey isocline increases if:
r increases or c decreases, or both:
the prey population would be able to support the
burden of a larger predator population
The predator isocline increases if:
d increases and either a or c decreases:
more prey would be required to support the
predator population
(c) 2001 by W. H. Freeman and
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Other Lotka-Volterra
Predictions
Increasing the predation efficiency (c) alone in
the model reduces isoclines for predators and
prey:
fewer prey would be needed to sustain a given
capture rate
the prey population would be less able to support the
more efficient predator
Increasing the birth rate of the prey (r) should
lead to an increase in the population of
predators but not the prey themselves.
(c) 2001 by W. H. Freeman and
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Modification of Lotka-Volterra
Models for Predators and Prey
There are various concerns with the
Lotka-Volterra equations:
the lack of any forces tending to restore the
populations to the joint equilibrium:
this condition is referred to as a neutral
equilibrium
the lack of any satiation of predators:
each predator consumes a constant proportion of
the prey population regardless of its density
(c) 2001 by W. H. Freeman and
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The Functional Response
A more realistic description of predator behavior
incorporates alternative functional responses,
proposed by C.S. Holling:
type I response: rate of consumption per predator
is proportional to prey density (no satiation)
type II response: number of prey consumed per
predator increases rapidly, then plateaus with
increasing prey density
type III response: like type II, except predator
response to prey is depressed at low prey density
(c) 2001 by W. H. Freeman and
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The Holling Type III
Response
What would cause the type III functional
response?
heterogeneous habitat, which provides a
limited number of safe hiding places for prey
lack of reinforcement of learned searching
behavior due to a low rate of prey encounter
switching by predator to alternative food
sources when prey density is low
(c) 2001 by W. H. Freeman and
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The Numerical Response
If individual predators exhibit satiation
(type II or III functional responses),
continued predator response to prey must
come from:
increase in predator population through local
population growth or immigration from
elsewhere
this increase is referred to as a numerical
response
(c) 2001 by W. H. Freeman and
Company
Several factors reduce
predator-prey oscillations.
All of the following tend to stabilize predator
and prey numbers (in the sense of maintaining
nonvarying equilibrium population sizes):
predator inefficiency
density-dependent limitation of either predator or
prey by external factors
alternative food sources for the predator
refuges from predation at low prey densities
reduced time delays in predator responses to
changes in prey abundance
(c) 2001 by W. H. Freeman and
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Destabilizing Influences
The presence of predator-prey cycles indicates
destabilizing influences:
such influences are typically time delays in predatorprey interactions:
developmental period
time required for numerical responses by predators
time course for immune responses in animals and induced
defenses in plants
when destabilizing influences outweigh stabilizing
ones, population cycles may arise
(c) 2001 by W. H. Freeman and
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Predator-prey systems can have
more than one stable state.
Prey are limited both by their food supply
and the effects of predators:
some populations may have two or more
stable equilibrium points, or multiple stable
states:
such a situation arises when:
• prey exhibits a typical pattern of density-dependence
(reduced growth as carrying capacity is reached)
• predator exhibits a type III functional response
(c) 2001 by W. H. Freeman and
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Three Equilibria
The model of predator and prey responses to prey
density results in three stable or equilibrium
states:
a stable point A (low prey density) where:
any increase in prey population is more than offset by
increasingly efficient prey capture by predator
an unstable point B (intermediate prey density) where:
control of prey shifts from predation to resource limitation
a stable point C where:
prey escapes control by predator and is regulated near its
carrying capacity by resource scarcity
(c) 2001 by W. H. Freeman and
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Implications of Multiple
Stable States
Predators may control prey at a low level (point
A in model), but can lose the potential to
regulate prey at this level if prey density
increases above point B in the model:
a predator controlling an agricultural pest can lose
control of that pest if the predator is suppressed by
another factors for a time:
once the pest population exceeds point B, it will increase to a
high level at point C, regardless of predator activity
at this point, pest population will remain high until some
other factor reduces the pest population below point B in the
(c) 2001 by W. H. Freeman and
model
Company
Effects of Different Levels
of Predation
Inefficient predators cannot maintain prey at
low levels (prey primarily limited by resources).
Increased predator efficiency adds a second
stable point at low prey density.
Further increases in predator functional and
numerical responses may eliminate a stable
point at high prey density
Intense predation at all prey levels can drive the
prey to extinction
(c) 2001 by W. H. Freeman and
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When can predators drive
prey to extinction?
It is clearly possible for predators to drive
their prey to extinction when:
predators and prey are maintained in simple
laboratory systems
predators are maintained at high density by
availability of alternative, less preferred prey:
biological control may be enhanced by providing
alternative prey to parasites and predators
(c) 2001 by W. H. Freeman and
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What equilibria are likely?
Models of predator and prey suggest that:
prey are more likely to be held at relatively
low or relatively high equilibria (or perhaps
both)
equilibria at intermediate prey densities are
highly unlikely
(c) 2001 by W. H. Freeman and
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Summary 1
Predators can, in some cases, reduce prey
populations far below their carrying capacities.
Predators and prey often exhibit regular cycles,
typically with cycle lengths of 4 years or 9-10
years.
Lotka and Volterra proposed simple
mathematical models of predator and prey that
predicted population cycles.
(c) 2001 by W. H. Freeman and
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Summary 2
Increased productivity of the prey should increase
the predator’s population but not the prey’s.
Functional responses describe the relationship
between the rate at which an individual predator
consumes prey and the density of prey.
The Lotka-Volterra models incorporate a type I
functional response, which is inherently unstable.
Type III functional responses can result in stable
regulation of prey populations at low densities.
(c) 2001 by W. H. Freeman and
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Summary 3
Type III functional responses can result from
switching.
Numerical responses describe responses of
predators to prey density through local
population growth and immigration.
Several factors tend to stabilize predator-prey
interactions, but time lags tend to destabilize
them.
Predator-prey systems may have multiple stable
points.
(c) 2001 by W. H. Freeman and
Company