Exploitation vs. interference competition Lotka-Volterra Competition equations

1. 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.

2. 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

3. 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

4. Complementarity of Niche Dimensions, page 276 Thomas Schoener

5. Prey size versus predator size

6. Prey size versus predator size Ctenotus skinks Hawks

7. Character Displacement, Galápagosfinches Peter R. Grant David Lack

8. Character Displacement in Hydrobia mud snails in Denmark Snail shell length, mm

9. Corixid Water Boatman G. E. Hutchinson

10. Hutchinsonian Ratios

11. Hutchinsonian Ratios Henry S. Horn Bob May

12. Hutchinsonian Ratios Limiting Similarity Henry S. Horn Bob May

13. Hutchinsonian Ratios Limiting Similarity Henry S. Horn Bob May Recorders

14. Wind Instruments

16. Kitchen Knives

17. Kitchen Pots

18. Tricycles

19. Bikes

20. Hutchinsonian ratios among short wing Accipiter hawks Thomas W. Schoener

26. Nicole hugs A komodo monitor

30. Hutchinsonian ratios among Australian Varanus lizards

31. 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)

32. Ecological Niche = sum total of adaptations of an organismic unit How does the organism conform to its particular environment? Resource Utilization Functions = RUFs

33. Within-phenotype versus between-phenotype components of niche width Within Phenotype Between Phenotype Individuals are generalists More specialized individuals

34. n-Dimensional Hypervolume Model Fitness density Hutchinson’s Fundamental and Realized Niches G. E. Hutchinson

36. 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

39. Robert H. MacArthur Geographical Ecology Range of Available Resources Average Niche Breadth Niche Overlap

40. MacArthur, R. H. 1970. Species packing and competitive equilibrium for many species. Theoret. Population Biol. 1: 1-11. Species Packing, one dimension Rate of Resource Resource Utilization Functions = RUFs

41. 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

42. Niche Breadth Jack of all trades is a master of noneMacArthur & Levin’s Theory of Limiting Similarity Robert H. MacArthur Richard Levins Specialists are favored when resources are very different

43. Niche Breadth Jack of all trades is a master of none MacArthur & Levin’s Theory of Limiting Similarity Robert H. MacArthur Richard Levins Generalists are favored when resources are more similar

44. Niche Dimensionality 1 D = ~ 2 Neighbors 2 D = ~ 6 Neighbors 3 D = ~ 12 Neighbors 4 D = ~ 20 Neighbors NN = D + D2Diffuse CompetitiondNi/dt = riNi(Ki -Ni-ij Nj)dNi/dt = 0 when Ni =Ki-ij Nj

45. Niche Overlap Hypothesis