Exploitation vs. interference competition Lotka-Volterra Competition equations

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, constant

Competition coefficients 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/’s compared to K’sSp. 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: Ni* = Ki – ij Nj

Alpha matrices, N and K vectors

Matrix Algebra Notation: N = K – ANPartial derivatives, ∂Ni/∂Nj sensitivity of species i to changes in j

Jacobian matrix (community matrices), Lyapunov stability

Evidence for competition in nature Resource partitioning among sympatric congeneric pairsResource Matrices, food, place, time niche dimensionsComplementarity of niche dimensionsGalapagos finches, beak depth, seed sizeCharacter displacementHydrobia mud snailsHutchinsonian ratiosCorixids, musical instruments, knives, pots, trikes, bikesAccipter hawks, monitor lizards

Evidence of Competition in Natureoften 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

Peter R. Grant

David Lack

Character Displacement, Galápagos finches

Character Displacement in Hydrobia mud snails in Denmark

Snail shell length, mm

Corixid Water BoatmanG. E. Hutchinson

Hutchinsonian Ratios

Henry S. Horn Bob May

Hutchinsonian Ratios

Henry S. Horn Bob May

Hutchinsonian Ratios

Limiting Similarity

Henry S. Horn Bob May

Hutchinsonian Ratios

Limiting Similarity

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

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 componentsof niche width

Within Phenotype Between Phenotype

Individuals are generalists More specialized individuals

Fitness density

Hutchinson’s Fundamental and Realized Niches

n-Dimensional Hypervolume Model

G. E. Hutchinson

Euclidean distance

djk = sqrt [ (pij - pik)2]

where j and k represent species j and species k, the pij and pik

’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

Euclid

Robert H. MacArthur

Geographical Ecology

Range of Available Resources

Average Niche Breadth

Niche Overlap

Resource Utilization Functions = RUFs

Rat

e of

Res

ourc

eMacArthur, R. H. 1970. Species packing and competitive

equilibrium for many species. Theoret. Population Biol. 1: 1-11.

Species Packing, one dimension

Three generalized abundant

species with broad niche breadths

Nine specialized less abundant

species with with narrow niche

breadths

Species Packing , one dimension, two neighbors in niche space

Niche Breadth Jack of all trades is a master of none

MacArthur & Levin’s Theory of Limiting Similarity

Specialists are favored when resources are very different

Robert H. MacArthur Richard Levins

Generalists are favored when resources are more similar

MacArthur & Levin’s Theory of Limiting Similarity Robert H. MacArthur Richard Levins

Niche Breadth Jack of all trades is a master of none

Niche Dimensionality

1 D = ~ 2 Neighbors

2 D = ~ 6 Neighbors

3 D = ~ 12 Neighbors

4 D = ~ 20 Neighbors

NN = D + D2

Diffuse Competition

dNi/dt = riNi(Ki -Ni -ij Nj)

dNi/dt = 0 when Ni = Ki -ij Nj

Niche Overlap Hypothesis