gause ’ s and park ’ s competition experiments lotka-volterra competition equations
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Gause ’ s and Park ’ s competition experiments Lotka-Volterra Competition equations dN i /dt = r i N i ({K i – N i – S a ij N j }/K i ) Summation is over j from 1 to n , excluding i N i * = K i – S a ij N j [Diffuse competition] - PowerPoint PPT PresentationTRANSCRIPT
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Mutualism Equations (pp. 234-235, Chapter 11)
dN1 /dt = r1 N1 ({X1 – N1 + 12 N2 }/X1)
dN2 /dt = r2 N2 ({X2 – N2 + 21 N1 }/X2)
(X1 – N1 + 12 N2 )/X1 = 0 when N1 = X1 + 12 N2
(X2 – N2 + 21 N1 )/X2 = 0 when N2 = X2 + 21 N1
If X1 and X2 are positive and 12 and 21 are chosen so that isoclines cross,
a stable joint equilibrium exists.
Intraspecific self damping must be stronger than interspecific positive mutualistic
effects.
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N1*
N2*
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The ecological niche, function of a species in the community
Resource utilization functions (RUFs)
Niche breadth and overlap
Competitive communities in equilibrium with their resources
Hutchinson’s n-dimensional hypervolume concept
Niche dimensionality and diffuse competition
Euclidean distances in n- space (Greek mathematician, 300 BC)
Fundamental versus Realized Niches
Niche dynamics
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Resource matrices of utilization coefficients
Niche dynamics
Niche dimensionality and diffuse competition
Complementarity of niche dimensions
Niche Breadth: Specialization versus generalization.Similar resources favor specialists, different resources favor generalists
Periodic table of lizard niches (many dimensions)Thermoregulatory axis: thermoconformers —> thermoregulators
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Experimental EcologyControls
Manipulation
Replicates
Pseudoreplication
Rocky Intertidal Space Limited System
Paine’s Pisaster removal experiment
Connell: Balanus and Chthamalus
Menge’s Leptasterias and Pisaster experiment
Dunham’s Big Bend saxicolous lizards
Brown’s Seed Predation experiments
Simberloff-Wilson’s defaunation experiment
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Defaunation Experiments in the Florida Keys
Islands of mangrove trees weresurveyed and numbers ofarthropod species recorded
Islands then covered in plastic tentsand fumigated with methyl bromide
Islands then resurveyed at intervalsto document recolonization
Simberloff and Wilson 1970
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Simberloff and Wilson 1970
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Evidence for Stability of Trophic Structure? First number is the
number of species before defaunation, second in parentheses is the number after _______________________________________________________________________________________
Trophic Classes______________________________________________________________________________
Island H S D W A C P ? Total_______________________________________________________________________________________
E1 9 (7) 1 (0) 3 (2) 0 (0) 3 (0) 2 (1) 2 (1) 0 (0) 20 (11)E2 11 (15) 2 (2) 2 (1) 2 (2) 7 (4) 9 (4) 3 (0) 0 (1) 36 (29)E3 7 (10) 1 (2) 3 (2) 2 (0) 5 (6) 3 (4) 2 (2) 0 (0) 23 (26)ST2 7 (6) 1 (1) 2 (1) 1 (0) 6 (5) 5 (4) 2 (1) 1 (0) 25 (18)E7 9 (10) 1 (0) 2 (1) 1 (2) 5 (3) 4 (8) 1 (2) 0 (1) 23 (27)E9 12 (7) 1 (0) 1 (1) 2 (2) 6 (5) 13 (10) 2 (3) 0 (1) 37 (29)Totals 55 (55) 7 (5) 13 (8) 8 (6) 32 (23) 36 (31) 12 (9) 1 (3) 164 (140) _______________________________________________________________________________________H = herbivoreS = scavengerD = detritus feederW = wood borerA = antC = carnivorous predatorP = parasite? = undetermined
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Wilson 1969
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Predation and Parasitism
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Predator-Prey Experiments
Georgii F. Gause
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Predator-Prey Experiments
Georgii F. Gause
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Predator-Prey Experiments
Georgii F. Gause
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Lotka-Volterra
Predation Equations
coefficients of predation, p1 and p2
dN1 /dt = r1 N1 – p1 N1 N2
dN2 /dt = p2 N1 N2 – d2 N2
No self damping (no density dependence)
dN1 /dt = 0 when r1 = p1 N2 or N2 = r1 / p1 dN2 /dt = 0 when p2 N1 = d2 or N1 = d2 / p2
Alfred J. Lotka Vito Volterra
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Neutral Stability(Vectors spiralin closed loops)
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Vectors spiral inwards (Damped Oscillations)
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DampedOscillations
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Vectors spiral inwards (Damped Oscillations)
Prey self damping
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Mike Rosenzweig Robert MacArthur
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Mike Rosenzweig Robert MacArthur
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Moderately efficient predatorNeutral stability — Vectors form a closed ellipse. Amplitude of oscillations remains constant.
<—Mike Rosenzweig Robert MacArthur —>
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Unstable — extremely efficient predatorVectors spiral outwards until a Limit Cycle is reached
Robert MacArthur —><—Mike Rosenzweig
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Damped Oscillations — inefficient predatorVectors spiral inwards to stable equilibrium point
Robert MacArthur —><—Mike Rosenzweig
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Functional response = rate at which Individual predators capture and eat more prey per unit time as prey density increases C. S. Holling
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Numerical response = increased prey density raises the predator’spopulation size and a greater number of predators consume An increased number of prey
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Gause’s Didinium ExperimentsLotka-Volterra Predation Equations: N1 N2 = Contacts
coefficients of predation, p1 and p2
dN1 /dt = r1 N1 – p1 N1 N2
dN2 /dt = p2 N1 N2 – d2 N2
No self damping (no density dependence)
dN1 /dt = 0 when r1 = p1 N2 or N2 = r1 / p1
dN2 /dt = 0 when p2 N1 = d2 or N1 = d2 / p2
Neutral StabilityPrey RefugesFunctional and Numerical Responses
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Adding Prey self-damping stabilizes
Prey-Predator isocline analyses
Predator efficiency, Prey escape ability
Prey refuges, coevolutionary race
Predators usually destabilizing
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Prey Isocline Hump
Efficient Predator —> unstable
Inefficient Predator —> stable
Predator Switching, frequency dependence, stabilizes
“Prudent” Predation and Optimal Yield
Feeding territories
Consequence of senescence
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Prey Isocline Hump
Inefficient Predator —> stable
Damped Oscillations
Efficient Predator —> unstable
Increasing Oscillations —> Limit Cycle
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Limit Cycle
Prey Population Density
Pre
dato
r P
opul
atio
n D
ensi
ty
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Predator Escape TacticsAspect Diversity
Cryptic coloration (countershading)
Disruptive coloration
Flash coloration
Eyespots, head mimicry
Warning (aposematic) coloration
Alarm signals
Hawk alarm calls
Selfish callers
Plant secondary chemicals
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Aspect Diversity in Tide Pools
Cottid Fish <—
Shrimp —>
Secondary Chemical Defenses of Plants
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Head Mimicry Papilio caterpillar Pit Viper caterpillar DeVries Snake head
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Monarch(Model)
Viceroy(Mimic)
Batesian Mimicry
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Mullerian Mimicry
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Batesian Mimicry
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Parasitism > Commensalism > Mutualism
(+, –) < (+, 0) < (+, +)
Host-Altered Behavior
Evolution of Virulence
Biological Control