Transcript 15-Competition
Exploitation vs. interference competition Lotka-Volterra Competition equations Assumptions: linear response to crowding both within and between species, no lag in response to change in density,
r
,
K
, a Competition coefficients a
ij
,
i
having the effect is species affected and
j
constant is the species Solving for zero isoclines, resultant vector analyses Point attractors, saddle points, stable and unstable equilibria Four cases, depending on
K
/ a ’ s compared to
K
’ s Sp. 1 wins, sp. 2 wins, either/or, or coexistence Gause ’ s and Park ’ s competition experiments Mutualism equations, conditions for stability: Intraspecific self damping must be stronger than interspecific positive mutualistic effects.
Diffuse competition:
N i * = K i
Alpha matrices,
N
and
K
vectors
–
Matrix Algebra Notation: S a
ij
N = K – AN
N j
Partial derivatives, ∂
N i
/∂
N j
sensitivity of species
i
to changes in Jacobian matrix (community matrices), Lyapunov stability
j
Evidence for competition in nature Resource partitioning among sympatric congeneric pairs Resource Matrices, food, place, time niche dimensions Complementarity of niche dimensions Galapagos finches, beak depth, seed size Character displacement
Hydrobia
mud snails Hutchinsonian ratios Corixids, musical instruments, knives, pots, trikes, bikes Accipter hawks, monitor lizards
Evidence of Competition in Nature often circumstantial
1. Resource partitioning among closely-related sympatric congeneric species (food, place, and time niches) Complementarity of niche dimensions 2. Character displacement 3. Incomplete biotas: niche shifts 4. Taxonomic composition of communities
Complementarity of Niche Dimensions, page 276 Thomas Schoener
Prey size versus predator size
Prey size versus predator size
Ctenotus
skinks Hawks
Character Displacement, Galápagos finches Peter R. Grant David Lack
Character Displacement in
Hydrobia
mud snails in Denmark
Snail shell length, mm
Corixid Water Boatman G. E. Hutchinson
Hutchinsonian Ratios
Henry S. Horn
Hutchinsonian Ratios
Bob May
Henry S. Horn
Hutchinsonian Ratios Limiting Similarity
Bob May
Henry S. Horn
Hutchinsonian Ratios Limiting Similarity
Bob May Recorders
Wind Instruments
Kitchen Knives
Kitchen Pots
Tricycles
Bikes
Hutchinsonian ratios among short wing Accipiter hawks
Thomas W. Schoener
Nicole hugs A komodo monitor
Hutchinsonian ratios among Australian
Varanus
lizards 25 Expected Observed (R) Observed (L) 20 15 10 5 0 0 1 2 3 4 5 Hutchinsonian Ratio 6 7 8 9
The ecological niche, function of a species in the community Resource utilization functions (RUFs) Competitive communities in equilibrium with their resources Hutchinson ’ s
n
-dimensional hypervolume concept Fundamental and Realized Niches Resource matrices Niche Breadth (vector) Niche Overlap (matrix)
Ecological Niche = sum total of adaptations of an organismic unit How does the organism conform to its particular environment?
Resource Utilization Functions = RUFs
Within-phenotype versus between-phenotype components of niche width
Within Phenotype Between Phenotype Individuals are generalists More specialized individuals
n
-Dimensional Hypervolume Model
Fitness density Hutchinson ’ s Fundamental and Realized Niches G. E. Hutchinson
Euclid
Euclidean distance
d jk
= sqrt [
S
(
p ij
-
p ik
) 2 ] where
j
and
k
represent species
j
and species
k,
the
p ij
and
p ik
’
s represent the proportional utilization or electivities of resource state
i
used by species
j
and species
k
, respectively and the summation is from
i
to
n.
n
is the number of resource dimensions
Robert H. MacArthur
Geographical Ecology
Range of Available Resources Average Niche Breadth Niche Overlap
MacArthur, R. H. 1970. Species packing and competitive equilibrium for many species. Theoret. Population Biol. 1: 1-11. Species Packing, one dimension Resource Utilization Functions = RUFs
Species Packing , one dimension, two neighbors in niche space Three generalized abundant species with broad niche breadths Nine specialized less abundant species with with narrow niche breadths
Niche Breadth Jack of all trades is a master of none Robert H. MacArthur MacArthur & Levin ’ s Theory of Limiting Similarity Richard Levins Specialists are favored when resources are very different
Robert H. MacArthur Niche Breadth Jack of all trades is a master of none MacArthur & Levin ’ s Theory of Limiting Similarity Richard Levins Generalists are favored when resources are more similar
Niche Dimensionality
1 D = ~ 2 Neighbors 2 D = ~ 6 Neighbors 3 D = ~ 12 Neighbors 4 D = ~ 20 Neighbors NN = D + D 2
Diffuse Competition
dN i /dt = r i N i
(
K i dN i /dt =
0 when -
N i -
Sa
ij N j
)
N i = K i -
Sa
ij N j