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 constant
Competition coefficients a ij
, i is species affected and j is the species having the effect
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

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

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

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
N i
-
Sa ij
N j
) dN i
/dt = 0 when N i
= K i
-
Sa ij
N j