#### 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 constant Competition coefficients aij, 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: Ni* = Ki – S aij Nj Alpha matrices, N and K vectors Matrix Algebra Notation: N = K – AN Partial 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 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 Hutchinsonian Ratios Henry S. Horn Bob May Hutchinsonian Ratios Limiting Similarity Henry S. Horn Bob May Hutchinsonian Ratios Limiting Similarity Henry S. Horn 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) Frequency 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 Individuals are generalists Between Phenotype More specialized individuals n-Dimensional Hypervolume Model Fitness density Hutchinson’s Fundamental and Realized Niches G. E. Hutchinson Euclid Euclidean distance djk = sqrt [S (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 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 Niche Breadth Jack of all trades is a master of none Robert H. MacArthur 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 + D2 Diffuse Competition dNi/dt = riNi(Ki -Ni -Saij Nj) dNi/dt = 0 when Ni = Ki -Saij Nj Niche Overlap Hypothesis