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 Predator (P) Prey (N) dN / dt < 0 dN / dt > 0 dN / dt = 0 PREY ISOCLINE K Predator (P) Prey (N) dP / dt < 0 dP / dt > 0 dP / dt = 0 PREDATOR ISOCLINE

Predictions of R. & MacA. 1. Inefficient predator –isoclines don’t cross –predicts predator extinction u 2. Intermediate predator efficiency #1 –isoclines cross to right of hump –predicts stable coexistence with damped oscillations Predator (P) Prey (N) 1 Predator (P) Prey (N) 2

Predictions of R. & MacA. 3. Intermediate predator efficiency #2 –isoclines cross near hump –predicts stable oscillations u 4. Highly efficient predator –isoclines cross to left of hump –predicts expanding oscillations & extinction Predator (P) Prey (N) 3 Predator (P) Prey (N) 4

Predictions of R. & MacA. Time (t ) Density (N or P) 1 Inefficient Predator: Predator Extinction Time (t ) Density (N or P) 3 Intermediate efficiency #2: Stable Oscillations Time (t ) Density (N or P) 4 Efficient predator: Expanding Oscillations & Extinction Time (t ) Density (N or P) 2 Intermediate efficiency #1: Damped Oscillations

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

A prey refuge stabilizes the system dN / dt > 0 Predator (P) Prey (N) dN / dt < 0 REFUGE

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 yields ordinary logistic f = functional response –function relating number eaten per predator to N Prey Prey eaten F max K 1/2

 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 C0C0 k0k0 C k0k0

Resource based model (without predator) Prey consumes nutrient C F max C / ( K 1/2 + C ) [“saturation kinetics”] –F max = maximal feeding rate, K 1/2 = 1/2 saturation Prey growth rate dN / dt = N a [ F max C / ( K 1/2 + C )] - k 0 N –N = prey density –a = conversion of feeding to growth Resource dynamics dC / dt = k 0 C 0 - k 0 C - N [F max C / ( K 1/2 + C )]

Resource based model (with predator) Add predator ( P ) –Predator consumes prey (saturation kinetics) dP / dt = P b [ F P N / ( K P + N )] - k 0 P –F P = maximum feeding rate, K P = 1/2 saturation –b = conversion of prey eaten to predator Prey with the predator dN/dt = Na[F max C/(K 1/2 + C)] - k 0 N - P[F P N/(K P + N)]

What do isoclines look like? Predator (P) Prey (N) Predator (P) Prey (N) Positions depend heavily on: k 0 (turnover rate, mortality) C 0 (nutrient input) K P (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 k 0 –Nutrient input C 0

Predictions for increasing k 0 Predator (P) Prey (N) raise k 0 lower k 0

Predictions for increasing k 0 Predator (P) Prey (N) raise k 0 lower k 0

Predictions for k 0 Increasing k 0 yields outcomes: –expanding oscillations & extinction –stable oscillations –damped oscillations –predator extinction

Predictions for increasing C 0 Predator (P) Prey (N) raise C 0

Predictions for C 0 Increasing C 0 for a stable system yields: –destabilization –expanding oscillations and extinction

Experimental tests Varying k 0 –Predictions generally confirmed –Details of sequence of changes may vary –Williams 1980 Varying C 0 –largely untested Testing these predictions in a real system chemostat settting is difficult

Exam Mean (SD): 83 (8.3) –2007: 84 (4.2) –2005: 82 (9.8)

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 R P IMPACT ON PREDATOR (=PREDATOR POPULATION GROWTH) IMPACT ON RESOURCE (=CONSUMPTION) S R*R* dN/dt > 0 dN/dt < 0

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 R P R*R* R*R* sp. 1 sp. 2   &&

Isocline R P R*R* R*R* sp. 1 sp. 2    or 

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 R P S consumer impact system trophic balance R* why is this part horizontal?

Keystone predation isocline system trophic balance consumer impacts R* A B Productivity P R 2 sp [PA]3 sp [PAB]2 sp [PB]

Predator & Resource isoclines NANA NBNB A excludes B B excludes A S1S1 S2S2 S3S3 S4S4 Resource zero growth isoclines for different supply points: S 1 = low; S 4 = high

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 diversity- productivity 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. coli Resistant + E. coli Vulnerable Control chemostats –T2 + E. coli Resistant –T2 + E. coli Vulnerable –E. coli Resistant + E. coli Vulnerable

Productivity Productivity manipulation: input glucose –0.1  g/ml, 0.5  g/ml –0.09  g/ml, 0.5  g/ml –0.07  g/ml, 0.08  g/ml, 0.09  g/ml, 0.10  g/ml, 0.11  g/ml, 0.12  g/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 productivityhigh productivity morephage 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 Goose- neck 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

Related concept: Intermediate disturbance hypothesis Disturbance or Predation LowHigh Number of Species or diversity

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 -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

Effect of newt predation 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

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

Temporary pond amphibians Predators … salamanders –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