Lecture 6: Dynamics of Consumer-Resource Interactions
Lecture 6: Dynamics of Consumer-Resource Interactions
Huang He
Phone: 18972127775 QQ:105367750 E-mail: [email protected]
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Population cycles of predators and their prey
Data from records of purchase by Hudson’s Bay Company, Canada
MacLuich 1937
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Topics
6..1 Consumers can limit resource populations6.2 Many predator and prey populations increase and
decrease in regular cycles6.3 Mathematic models for predator-prey interaction6.4 Pathogen-host dynamics can be described by the S-I-R
model6.5 Lotka-Volterra model can be stabilized by predator
satiation6.6 Factors can reduce oscillation of predator-prey models6.7 Consumer-Resource system can have more than one
stable state
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6.1 Consumers can limit resource populations
Populations of consumers are self-regulatedbecause of their effects on their resources
Consumers contribute to the regulation of resource population.
Thus, populations are regulated from above and below.
Questions: how large is the rule of consumers?
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Predation on cyclamen 【植】仙客来 ( 属 ) mites
Cyclamen mite is a pest of strawberry in CA
Typhlodromus 盲走螨属 mite is a predatory mite
Greenhouse Experiment
One with predatory mite and one without by applying parathion
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Herbivores and Plant Populations
Herbivores can control plant populations
Klamath 克拉马斯人 weed, or St. John’s wort麦芽汁 , 植物 , became a widely spread pest following its introduction.
When Chrysolina beetle 甲虫 was introduced, the Klamath weed was finally under control.
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Effects of herbivores on plant production can be measured using exclosure experiments
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6.2 Many Predator and Prey populations increase and decrease in regular cycles
Cycles of predator and prey populations are common The periods of cycles vary from species to species
Large herbivores (snowshoe hare, muskrat[ 动 ]麝鼠 , ruffed grouse 环毛松鸡 ) 9-10 year
Small ones (vole, mice, lemming [ 动 ] 旅鼠 ) 4 years cycle Predators feed on large prey have long cycle (red foxes, lynx 猞猁 ,
marten 貂鼠 , mink [ 动 ] 貂 (尤指水貂 ), 貂皮衣 ) Predators feed on small prey have short cycle (Artic fox,
hawks, snowy owls) Cycles (oscillations 摆动 , 振动 ) are caused by predator and
prey interaction (predator – prey).
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Time delays and population cyclesTime delays and population cycles
Time delays in birth and death caused oscillation in population
Time delays also occur in predation
Period of population cycle should be 4 ~ 5 times the time delay
Hare populations fluctuated less on an island with few predators than on the surrounding mainland.
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Physical conditions may change the period of cyclesPhysical conditions may change the period of cycles
4-year cycle in northern Scandinavia 斯堪的纳维亚 ( 半岛 )( 瑞典、挪威、丹麦、冰岛的泛称 ), but annually in southern Sweden. Winter delay in north maintain a long cycle. In the south, owls hunt voles whole year, create a short cycle. Climate warming may cause the shift from 4-yr to annual shown in this figure (or multiple prey). 10
Development of host immunityDevelopment of host immunity 免疫性免疫性 influences host populationsinfluences host populations
Cases of measles [ 医 ] 麻疹reported in London (before vaccine 疫苗 had been developed)
Peaked about every two years
Periodicity in pathogen-host relationshipsPeriodicity in pathogen-host relationships
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Habitat structure can affect population cycles
Disease outbreak is density dependent
Forest tent caterpillar 毛虫 as host
Nuclear 细胞核的 polyhedrosis [ 昆 ]( 幼虫的 ) 多面体病 virus as pathogen
In many regions, tent caterpillars infestations 横行 last about 2 years before the virus brings its host population under control. In other regions, it may last 9 yearsForest fragmentation plays a role.Forest edges with more light, inactivate the virus.Habitats has second effects on population
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Creating predator-prey cycles in the laboratoryCreating predator-prey cycles in the laboratory
Modeling and lab experiments Studies by GF Gause on protists [ 生 ]原生生物
Predator: Ciliated有纤毛的 protist, Didnium Prey: Protist, Paramecium [ 动 ]草履虫 Culture medium: test tube
Difficult to demonstrate the oscillations Predators eat all prey, then die Add refuge, predators would die and left some prey to survive Add small number of predators periodically oscillations.
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Huffaker’s mite experimentHuffaker’s mite experimentC.B. Huffaker, UC Berkeley (1958)
Predator: mite螨 ,Typhlodromus. 盲走螨 Prey: six-potted mite (Eotetranychus 叶螨 ), pest of citrus fruits 柑橘类的水果Reproduction: parthenogenesis单性生殖 , 孤雌生殖
Control food resources: number and dispersion
First study: 40 positions, 4 fruits, 20 prey, after 11 days, add 2 predators 14
A spatial mosaic 【生】嵌合体 of habitats allows predators and prey to coexist
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6.3 Mathematical model for predation
Lotka and Volterra equation for predation Prey
Where cNpredNprey is mortality of prey due to predator. c is per capita capture rate, and Npred, Nprey are the number of predators and prey, respectively.
Predator
Where b is efficiency of conversion of prey consumed (cNpredNprey) and d is death rate of predators
predpreypreyprey NcNrNdt
dN
predpredpreypred dNNcNbdt
dN )(
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Solving the equations
For prey growth (dN_Prey/dt=0) Npred = r/c
Growth rate of prey population is zero when density of predators equals per capita growth rate of prey divided by per capita capture rate of predators.
Any increase in predator density will result in negative growth in prey population
For predator growth (dN_Pred/dt=0) Nprey = d/bc
Growth rate of predator population is zero when rate of increase of prey is equal to rate of mortality divided by the product乘积 of b and c.
Thus the two equations interact and this can be done graphically
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There is a cyclical rise and fall in both the predator and prey populations with time
Density of predators lags behind density of prey
Feast and Famine scenario 盛宴和饥荒的情况Prey and predators are never quite driven to extinction
Mutual population regulation
Pred
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Trajectories[ 物 ]( 射线的 ) 轨道 , 弹道 , 轨线 of predator and prey populations and their joint
equilibrium point
dP/dt=0 or dv/dt=0
Equilibrium isocline or more common, zero growth isocline
The change in predator and prey populations together follows a closed cycle that combines the individual changes in the predator and prey population, called joint population trajectory.
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Another chart to show that Lotka-Volterra model predicts a regular cycling of predator and prey populations
Joint equilibrium point
This point is not stable, or neutral stable (exhibits neutral stability), as slightly change in either population will move to next cycle, rather than return
Period of oscillation:T=2Pi/sqrt(rd)
If r=2 (200%) and d=0.5 per year, then T=6.3
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Influence of growth rate on predator and prey populations
Nprey or V=d/ac is the minimum requirement to sustain the growth of predator populations
Npredator or P=r/c is the largest number of predators that the prey population can sustain.
A surprising prediction of the model is that increase in r of prey growth leads to an increase in predator population, not the prey
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An increase in the birth rate of prey increases the predator population, but no the prey population
Bohannan and Lenski, Michigan State University
Prey: E. coliPredator: bacteriphage T4
Prey food source: limited by glucose
Tow levels: 0.1 or 0.5 mg per litter
Add food supply only increased predator population.
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6.4 Pathogen-host dynamics can 6.4 Pathogen-host dynamics can be described by the S-I-R modelbe described by the S-I-R model
Parasites do not remove host from population, but can develop time delays that lead to population cycling
Course of epidemic depends on Rate of transmission (b) and rate of recovery (g): Reproduction ratio: number of secondary cases produced by a primary case during its period of infectiousness, R0=(b/g)SR0>1, an epidemic will occur, each infected individual will infect more than one before it recoversR0<1, fails to take hold in the populationR0: 5-18 for measles, chicken pox etc. HIV: 2-5; malaria: >100.
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The S-I-R model can predict the spread on epidemic through a host population
Total =100b=1, g=0.2, duration of infectiousness 1/g=5Beginning, S=1, R0=b/g*S=5
Assume no births of S, and no loss of resistance among previously infected individuals.
Influenza virusVaccination: remove individuals from S, reduce R0.
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Case study: The chytrid fungus and the global decline of amphibians
Pathogenic fungus: Batrachochytrium dendrobatisdisIt kills hosts and persists by infecting alternative species.
Karan Lips, Southern Illinois University, 2006El Cope: first found in July 2004, rapid spread and caused abrupt drop. 26
The Lotka–Volterra model is criticized for overemphasizing the mutual regulation of predator and prey populations Differential equations, no time delay(Difference equations, add time delay) No internal forces act to restore the populations to the
joint equilibrium point, random perturbations could increase oscillations to a point that V=0 or P=0
cNpreyNpredator: at a given Npred, the rate at wich prey are captured increases with Nprey. This is not true. There is predator satiation.
6.5 Lokta-Volterra model can be stabilized by predator satiation
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Functional and numerical responses
cN_preyN_pred (cVP): For prey population, this term serves to regulate
population growth through mortality For predator population, it serves to regulate population
growth through two distinct responses:
Predator’s Functional responses: the great the number of prey, the more the predator eats. The relationship between per capita rate of consumption and the number of prey (cNpreyNpred).
Predator’s Numerical response: an increase in consumption of prey results in an increase in predator reproduction (b(cNpreyNpred).
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The functional response is the relationship between the per capita predation rate (number of prey consumed per unit time) and prey population size This idea was introduced by M.E. Solomon in 1949
Three types of functional response (I, II, and III) Developed by C.S. Holling
Functional Responses Relate Prey Consumed to Prey Density
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Functional responseNe: per capita rate of predation, i.e., # of prey eaten
during a given period of search time.
Type I functional response Ne=c Nprey Passive predator such as spider or the prey is less
sufficiently abundant (e.g., kestrels and voles) All time allocated to feeding is searching.
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Type II response Ne increase with Nprey rapidly, but level off at high prey
density.
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Type III functional response
Sigmoid (S-shaped) response At high prey density, the response is the same as type II
response; however, the rate of prey consumed is low when the prey density is low at first, increasing in a S-shaped fashion.
Factors caused the S-shape response 1. availability of cover to escape the predators 2. predator’s search image 3. Prey switching. Switch to other preys (more abundant)
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Functional response As prey increases, predators
take more prey But how
Linear– Rate of predation is constant
Decreasing rate to maximum– Rate of predation decline
Sigmoidal– Reaches maximum then declines
Functional responses related prey consumed to prey density
(Right panel is expressed as proportion of prey density, # prey consumed divided by prey density)
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Linear Type 1 (European kestrel to vole)Mortality of prey simply density dependentNo limits on system
Decreasing Type 2 (weasel on rodent)Predators can only eat so much – satiationTime needed to kill and eat prey becomes
limiting Sigmoid Type 3 (warbler on budworm larvae)
Capture rate is density dependentAvailability of coverAlternative prey when preferred is rare
(prey switching)Prey not part of predators search image,
not a desirable food source
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Prey switching Palatable versus less palatable Better return per kill Less energy needed to find and kill
an abundant prey
Model of prey switching
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Numerical response Predators reproduce more
However reproduction usually slower than prey
Movement into high prey density areas
This aggregative response is very important as it rapidly increases predator density
Predators respond numerically to changing prey density
Aggregative response in the redshank
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Other numerical response as increased reproductive effort Weasels as predators Rodents as prey Predators followed prey in reproduction Increase of rodent was due to good harvest in 1990
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Predator population exhibits a numerical response to change in prey density
Most of the increase was due to local population growth rather than immigration from else where
After hare density fell to a low level, red squirrels and other small mammals were eaten by lynx.
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Numerical response of a predator population lags behind changes in prey density following counterclockwise joint population trajectory
predicted by the Lotka-Volterra model
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Stability: achievement of an unvarying equilibrium size, often the carrying capacity
Predator-prey: oscillations, but several factors could stabilizing: Predator inefficiency Density-dependent limitations of prey or predator by
other external factors Alternative food resource for predator Refuges for prey at low prey density Reduced time delays in predator responses to changes in
prey abundance
6.6 Factors that reduce oscillations in predator-prey models
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Population size is determined by: abundance of its resources and of its consumers
One Extreme: resource population is only limited by its own food supply
Another: resource population is depressed below its carrying capacity
Balance between these factors create multiple equilibrium points: alternative state states
6.7 Consumer-resource system can have more than one stable state
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Consumer-imposed equilibrium:At low density, prey can seek refuge, avoid predatorsAt low density, prey grows faster than predators
Low stable equilibrium point well below its carrying capacityResource-imposed equilibrium
in some cases, prey population can move up from the consumer-imposed equilibrium, due to the limited number of predators, predator satiation, or other factors that keep predators in check (nest limitation), reach equilibrium set by its carrying capacity.
Population could have two stable states and sometime move between these two (crop and forest pests, diseases).
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