Download Week 8 2010

Improved Predator-Prey models
• Self limitation of prey and predators
• Asymptotic prey consumption by
predators
• Spatial refuges for prey
• graphical approach
– Rosezweig & MacArthur (1963)
• mathematical approach
– Williams (1980) Grover (1997)
– Gilpin & Ayala (1973) Populus 5.4
Rosenzweig & MacArthur
predator-prey isoclines
PREY ISOCLINE
PREDATOR ISOCLINE
dP / dt < 0
dN / dt < 0
dP / dt = 0
dN / dt = 0
dN / dt > 0
Prey (N)
K
dP / dt > 0
Prey (N)
Predictions of R. & MacA.
• 1. Inefficient predator
1
– isoclines don’t cross
– predicts predator extinction
Prey (N)
2

2. Intermediate predator
efficiency #1
– isoclines cross to right of hump
– predicts stable coexistence
with damped oscillations
Prey (N)
Predictions of R. & MacA.
• 3. Intermediate predator
efficiency #2
3
– isoclines cross near hump
– predicts stable oscillations
Prey (N)
4

4. Highly efficient predator
– isoclines cross to left of hump
– predicts expanding oscillations
& extinction
Prey (N)
2
Inefficient Predator:
Predator Extinction
Density (N or P)
1
Density (N or P)
Predictions of R. & MacA.
Intermediate efficiency #2:
Stable Oscillations
Time (t )
Time (t )
4
Density (N or P)
3
Density (N or P)
Time (t )
Intermediate efficiency #1:
Damped Oscillations
Efficient predator:
Expanding
Oscillations
& Extinction
Time (t )
Further improvements:
A refuge for prey
• If prey have a refuge, then a certain
proportion can escape predation
• Prey population in the refuge tolerates
infinitely large predator population
• Makes stable coexistence more likely
REFUGE
A prey refuge stabilizes the
system
Prey (N)
Implications of graphical
predator-prey models
• Many different patterns of dynamics are
possible
• Stable oscillations are only one special case
– and not a very likely one
• Prey may be exterminated (efficient predators)
• Prey may be reduced to stable populations
below K
 logistic model: prey dynamics
•
•
•
•
dN / dt = r N [ 1 - (N / K) ] - f P
r = prey intrinsic rate of increase
K = prey carrying capacity
 quantifies form of density
dependence
– =1
Fmax
yields ordinary logistic
• f = functional response
– function relating number eaten per
predator to N
K1/2
Prey
 logistic model: predator dynamics
• dP / dt = sP( f - D)
• f = the number of prey eaten per predator
(functional response)
• D = minimum feeding required for dP / dt >0
(predator efficiency)
• s = constant relating predator rate of
increase to amount eaten
Resource models of
predator-prey interactions
• Prey consume resources and prey
population grows
• Predator eats prey and population grows
• Chemostat system
Chemostat
C0
k0
C
k0
Resource based model
(without predator)
• Prey consumes nutrient C
FmaxC / ( K1/2 + C ) [“saturation kinetics”]
– Fmax = maximal feeding rate, K1/2 = 1/2 saturation
• Prey growth rate
dN / dt = N a [ Fmax C / ( K1/2 + C )] - k0 N
– N = prey density
– a = conversion of feeding to growth
• Resource dynamics
dC / dt = k0 C0 - k0 C - N [FmaxC / ( K1/2 + C )]
Resource based model
(with predator)
• Add predator ( P )
– Predator consumes prey (saturation kinetics)
dP / dt = P b [ FP N / ( KP + N )] - k0 P
– FP = maximum feeding rate, KP = 1/2 saturation
– b = conversion of prey eaten to predator
• Prey with the predator
dN/dt =
Na[FmaxC/(K1/2 + C)] - k0N - P[FP N/(KP + N)]
What do isoclines look like?
Prey (N)
Prey (N)
Positions depend heavily on:
k0 (turnover rate, mortality)
C0 (nutrient input)
KP (predator 1/2 saturation)
Simplifying Assumptions
• Simplifying Environmental
– Constant in time (except resources)
– Uniform or random in space
• Simplifying Biological
– Individuals are identical & constant in time
– Prey limited only by resource and
predation
– Predator growth dependent only on
predation
Explanatory Assumptions
• Predator growth is a saturation kinetics
function of prey density
• Prey growth rate is a saturation kinetics
function of resource density
Predictions
• See graphical models
– expanding oscillations, stable oscillations,
damped oscillations
– dependent on positions of isoclines
• Models specify environmental variable
that may modify outcome
– Turnover rate k0
– Nutrient input C0
Predictions for increasing k0
raise k0
lower k0
Prey (N)
Predictions for increasing k0
lower k0
raise k0
Prey (N)
Predictions for k0
• Increasing k0 yields outcomes:
– expanding oscillations & extinction
– stable oscillations
– damped oscillations
– predator extinction
Predictions for increasing C0
raise C0
Prey (N)
Predictions for C0
• Increasing C0 for a stable system yields:
– destabilization
– expanding oscillations and extinction
Experimental tests
• Varying k0
– Predictions generally confirmed
– Details of sequence of changes may vary
– Williams 1980
• Varying C0
– largely untested
• Testing these predictions in a real
system chemostat settting is difficult
Exam
• Mean (SD):
83 (8.3)
– 2007: 84
Predator Isoclines:
Resource-based models
• Chase & Leibold
• zero growth conditions
• impact vector
– generalization of consumption vector
– impact of consumer (N) on resource (R) is
depletion
– impact of prey (N) on predator (P) is predator
population growth
Isocline
IMPACT ON PREDATOR
(=PREDATOR
POPULATION GROWTH)
P
dN/dt < 0
dN/dt > 0
R*
IMPACT ON
RESOURCE
(=CONSUMPTION)
S
R
Community effects of predation?
• Basic effect - reduce prey abundance
– Increase likelihood of prey elimination?
– Reduce diversity?
• Single predator - single prey
–
–
–
–
–
Smith 1983
Pseudacris tadpoles
Anax dragonfly nymphs
Anax exterminates Pseudacris within a pond
Pseudacris only in ephemeral ponds
Other effects of predation
• Effect: reduce diversity
• Keystone predator: A predator whose
removal from a community results in
reduced species diversity in that
community
– therefore keystone predators increase
community diversity
• Keystone predator effect requires both
interspecific competition and predation
Isocline
P
sp. 2
sp. 1


R*
R*
&
R
Isocline
P
sp. 2
sp. 1


R*
R*
 or 
R
Models of keystone predation
• Leibold 1996
• What environmental conditions promote
keystone effects?
• 3 tropic levels
– resource …R
– prey … N (consumes resource)
– predator … P (consumes prey)
Keystone predation isocline
P
consumer
impact
why is this part horizontal?
system
trophic
balance
R*
S
R
Keystone predation isocline
P
B
A
consumer
impacts
system
trophic
balance
R
R*
R*
2 sp [PA]
Productivity
3 sp [PAB]
2 sp [PB]
NB
B excludes A
Predator & Resource
isoclines
Resource zero growth
isoclines for different
supply points:
S1 = low; S4 = high
S4
S1
S2
S3
NA
A excludes B
Keystone predator effect
• Simple models – linear increase of predator
feeding with prey density (and of prey feeding
with resource density)
• Keystone effect most likely at intermediate
levels of productivity
– high productivity favors predator resistant sp.
– low productivity favors best competitor
• Predicts unimodal diversity - productivity
relationship
Keystone predator & >2 prey
• For any given productivity (S ), there is a
stable equilibrium with up to 2 prey spp.
• Across a gradient of productivity, prey
species replace each other
– low productivity … best competitors
– high productivity … least vulnerable to
predator
• May create large-scale unimodal diversityproductivity relationship
Keystone predator
& spatial heterogeneity
• With spatial heterogeneity in productivity,
>2 species of prey can coexist locally
• Strong unimodal diversity-productivity
relationship
– local patches of different productivities have 2
prey
– regionally >2 species coexist at intermediate
average productivities
Prediction
• High productivity: Predator resistant
species dominate
• Low productivity: Competitive species
dominate
• Intermediate productivity: Keystone
predator mediated coexistence
• Assumes: Trade-off between competitive
ability and resistance to predation
Experimental test
(Bohannon & Lenski 2000)
• Chemostat
• T2 bacteriophage
virus feeding on
• 2 strains of
Escherichia coli
– more resistant to T2
– more vulnerable to T2
– Trade-off w/glucose
exploitation
Design
• Treatment chemostats
– T2 + E. coliResistant + E. coliVulnerable
• Control chemostats
– T2 + E. coliResistant
– T2 + E. coliVulnerable
– E. coliResistant + E. coliVulnerable
Productivity
• Productivity manipulation: input glucose
– 0.1 mg/ml, 0.5 mg/ml
– 0.09 mg/ml, 0.5 mg/ml
– 0.07 mg/ml, 0.08 mg/ml, 0.09 mg/ml,
0.10 mg/ml, 0.11 mg/ml, 0.12 mg/ml
Some complicating details
• Bacterial evolve
resistance (assayed)
• Phage feeding rate
is a linear function of
bacterial density in
the range of
densities used (not
saturation kinetics)
Curvilinear isoclines
Narrow range for coexistence
Results
low productivity
phage
high productivity
more
phage
less
more
less
Results
• Without T2: Less vulnerable E. coli
declines to nearly 0 (due to competition)
• Low productivity: Decline, but not
elimination of less vulnerable E. coli
– not explained by evolution of resistance
• High productivity: Decline, but not
elimination of more vulnerable E. coli
– decline stops when invulnerable mutants
begin to increase
Conclusions
• Models correctly predict how productivity
is related to the relative importance of
competition vs. predation
• Extinctions not observed
– Wall growth?
– Evolution of the trade-off?
Keystone predation in the
Rocky intertidal zone
• Predator
– Pisaster sea star
– Nucella snails
• Grazers
– limpets snails
– chitons snails
Keystone predation in
the Rocky intertidal
zone
• Sessile species
– Mytilus Mussels
– Pollicipes Gooseneck barnacle
– Chthamalus,
Balanus acorn
barnacles
Pacific Northwest Intertidal
(Paine 1966)
•
•
•
•
•
•
•
Competition for space
Mytilus the competitive dominant species
Pisaster preys on all speces, prefers Mytilus
Natural intertidal community: 15 species
Exclude Pisaster with cages
1 to 2 years: 8 species
Without Pisaster, Mytilus dominates
The Keystone effect
Predator
(Pisaster)
Competitor 1
(Mytilus)
Competitor 2
(other species)
Pisaster is a keystone predator
• Keeps competitive dominant (Mytilus) from
eliminating other species
• Other predators do not have this effect (e.g.,
Nucella)
• Disturbance (e.g., storms, wave action,
scouring) can have a similar keystone effect
• Create open space, allow poorer competitors
to survive
Number of Species
or diversity
Related concept:
Intermediate disturbance hypothesis
Low
Disturbance or Predation
High
Intermediate Disturbance
or
Intermediate predation
• Disturbance … disruption of community
progress toward competitive equilibrium
• Predation or physical disturbance
• Diversity maximal at intermediate
disturbance
• Keystone effect may be a special case of
intermediate predation
Intermediate Disturbance
• Low disturbance (frequency, intensity)
– Competitive dominant excludes other spp.
– low diversity, low S
• High disturbance (frequency, intensity)
– few species can endure disturbances
– low diversity, low S
• Intermediate disturbance (frequency, intensity)
– disturbance doesn’t elimnate species
– reduces or eliminates competition among prey
– maximal diversity, maximal S
Intermediate predation:
Temporary pond amphibians
•
•
•
•
•
Woodland ponds, SE United States
Fill with spring rains; later dry up
Up to 17 spp. amphibian larvae in one pond
Up to 25 spp. present locally
Morin 1983, 1981; Wilbur 1983; and many
more recent papers
Temporary pond amphibians
• Predators - Newts (Notophthalmus) adults and larvae
• Prey on larvae of anurans (frogs & toads)
photo © Michael Righi on Flickr -
Temporary pond
amphibians
• Common anurans
– Spadefoot toad
• (Scaphiopus holbrooki)
– Leopard frog
• (Rana sphenocephala)
– Southern toad
• (Bufo terrestris)
• All filter feeders & scrapers
Temporary pond
amphibians
• Other common anurans
– Spring peeper
• (Hyla crucifer)
– Barking tree frog
• (Hyla gratiosa)
– Grey tree frog
• (Hyla crhysocselis)
• Also filter feeders &
scrapers
Experiment 1: Artificial ponds
• Cattle tanks
• Stock with leaf litter, plants, invertebrates
• 1200 newly hatched larvae of a mix of the 6
anuran species (150 to 300 each species)
• Predators: 0, 2, 4, 8 adult newts
• 0 newts
– Scaphiopus dominates,
Hyla rare
• 2 newts
– Scaphiopus dominates,
Hyla crucifer increases
– Maximal mass of anuran
adults; Maximal evenness
• 4 newts
– Hyla crucifer & Scaphiopus
equally abundant
• 8 newts
– 60% Hyla crucifer, all others
rare
Effect of
newt
predation
Supporting data
• Scaphiopus most vulnerable to newt predation
– Most active, moves, forages most
– Best competitor
• Hyla crucifer poorest competitor
– Moves very little
• General tradeoff -- high vs. low activity
– High activity, effective foraging, good competitor,
vulnerable to predation
– Low activity, lower foraging success, poor competitor,
less subject to predation
Temporary pond amphibians
• Newt predation concentrated on competitive
dominant species
• Intermediate predation yields maximal
diversity
• Both competition and predation are
necessary for the keystone predator or
intermediate predation effect
• Predators … salamanders
Temporary pond
amphibians
– Tiger salamanders
(Ambystoma)
• larvae
• Prey on larvae of anurans
– (frogs & toads)
Experiment 2: Artificial ponds
• Ambystoma as a predator, same prey
species
• With any Ambystoma present, anuran
larvae are exterminated
• No intermediate predation effect
• No keystone effect
• Effect on diversity specific to the
predator prey combination
Beyond keystone predation
•
•
•
•
•
Predation is a pairwise interaction
Interference competition is a pairwise interaction
Effects on the two species involved
There can be effects beyond the pair of species
Indirect effect: An effect of one species on
another that occurs via an effect on a third
species