1. Gause ’ s Paramecium competition lab experiments Park’s Tribolium competition experiments Lotka-Volterra competition equations Zero isoclines dN 1 /dt = 0 when N 1 = K 1 –  12 N 2 Zero isoclines dN 2 /dt = 0 when N 2 = K 2 –  21 N 1 When competitor N 2 ’s density N 2 = K 1 /  12, dN 1 /dt = 0 When competitor N 1 ’s density N 1 = K 2 /  21, dN 2 /dt = 0 Possible Cases of Competition Competitive Exclusion Stable coexistence (Point Attractor) Either/or (Saddle Point) Lecture # 21 November 5th, 2015

2. Lotka-Volterra Competition Equations for n species dN i /dt = r i N i ({K i – N i –  ij N j }/K i ) N i * = K i –  ij N j where the summation is over j from 1 to n, excluding i Diffuse Competition  ij N j

3. Evidence of Competition in Nature (often circumstantial) Resource partitioning among closely-related sympatric congeneric species (food, place, and time niches) Complementarity of niche dimensions Resource matrix: niche breadth and overlap Character displacement: Galápagos finches Limiting Similarity Hutchinsonian ratios: Corixid water boatman Horn and May’s irreverent man-made items Short-winged accipiter hawks (Schoener) Australian monitor lizards

4. Gause ’ s and Park ’ s competition experiments Lotka-Volterra Competition equations dN i /dt = r i N i ({K i – N i –  ij N j }/K i ) Summation is over j from 1 to n, excluding i N i * = K i –  ij N j [Diffuse competition] 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, set dN/dt = 0 Resultant vector analyses Four cases, depending on K/  ’ s compared to K ’ s Sp. 1 wins, sp. 2 wins, either/or, or coexistence

5. Four Possible Cases of Competition Under the Lotka–Volterra Competition Equations _____________________________________________________________________ Species 1 can contain Species 1 cannot contain Species 2 (K 2 /  21 K 1 ) ______________________________________________________________________ Species 2 can contain Case 3: Either species Case 2: Species 2 Species 1 (K 1 /  12 K 2 ) always wins can contain the other; stable coexistence ______________________________________________________________________ Alfred Lotka Vito Volterra

6. Resultant Vectors

7. Saddle Point Point Attractor

8. Lotka-Volterra Competition Equations for 3 species: dN 1 /dt = r 1 N 1 ({K 1 – N 1 –  12 N 2 –  13 N 3 }/K 1 ) dN 2 /dt = r 2 N 2 ({K 2 – N 2 –  21 N 1 –  23 N 3 }/K 2 ) dN 3 /dt = r 3 N 3 ({K 3 – N 3 –  31 N 1 –  32 N 2 }/K 3 ) Isoclines: (K 1 – N 1 –  12 N 2 –  13 N 3 ) = 0 when N 1 = K 1 –  12 N 2 –  13 N 3 (K 2 – N 2 –  21 N 1 –  23 N 3 ) = 0 when N 2 = K 2 –  21 N 1 –  23 N 3 (K 3 – N 3 –  31 N 1 –  32 N 2 ) = 0 when N 3 = K 3 –  31 N 1 –  32 N 2 Lotka-Volterra Competition Equations for n species (i = 1, n): dN i /dt = r i N i ({K i – N i –  ij N j }/K i ) N i * = K i –  ij N j where the summation is over j from 1 to n, excluding i Diffuse Competition  ij N j

9. Mutualism Equations (pp. 234-235, Chapter 11) dN 1 /dt = r 1 N 1 ({X 1 – N 1 +  12 N 2 }/X 1 ) dN 2 /dt = r 2 N 2 ({X 2 – N 2 +  21 N 1 }/X 2 ) (X 1 – N 1 +  12 N 2 )/X 1 = 0 when N 1 = X 1 +  12 N 2 (X 2 – N 2 +  21 N 1 )/X 2 = 0 when N 2 = X 2 +  21 N 1 If X 1 and X 2 are positive and  12 and  21 are chosen so that isoclines cross, a stable joint equilibrium exists. Intraspecific self damping must be stronger than interspecific positive mutualistic effects.

12. 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 Euclidean distances in n- space (Greek mathematician, 300 BC) Fundamental versus Realized Niches

13. Resource matrices of utilization coefficients Niche dynamics Niche dimensionality and diffuse competition Complementarity of niche dimensions Niche Breadth: Specialization versus generalization. Similar resources favor specialists, different resources favor generalists Periodic table of lizard niches (many dimensions) Thermoregulatory axis: thermoconformers —> thermoregulators

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

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

17. Euclid One Dimension: Distance between two points along a line: simply subtract smaller value from larger one x 2 - x 1 = d x 1 dx 2 Two Dimensions: Score position of each point on the first and second dimensions. Subtract smaller from larger on both dimensions. d 1 = x 2 - x 1 d 2 = y 2 - y 1 Square these differences, sum them and take the square root. This is the distance between the points in 2D: sqrt (d 1 2 + d 2 2 ) = d Three Dimensions —> n-dimensions: follow this same protocol summing over all dimensions i = 1, n: sqrt  d i 2 = d ________________________ | ||

18. Euclidean distance between two species in n-space n-dimensional hypervolume d jk = sqrt [  (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 = 1 to n. n is the number of resource dimensions Euclid i = 1 n

20. Niche Dimensionality 1 D = ~ 2 Neighbors 2 D = ~ 6 Neighbors 3 D = ~ 12 Neighbors 4 D = ~ 16 Neighbors 5 D = ~ 30 Neighbors NN = 2D + (D 2 - D) Diffuse Competition dN i /dt = r i N i (K i -N i -  ij N j ) dN i /dt = 0 when N i = K i -  ij N j

22. 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

23. 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

24. Within-phenotype versus between-phenotype components of niche width

25. Compression Hypothesis: habitats contract, diets do not

26. Complementarity of Niche Dimensions, page 276 Thomas W. Schoener Anolis

30. Moloch Phrynosoma Ctenotus piankai Ctenotus leae Slope

31. Experimental Ecology Controls Manipulation Replicates Pseudoreplication Rocky Intertidal Space Limited System Paine ’ s Pisaster removal experiment Connell: Balanus and Chthamalus Menge ’ s Leptasterias and Pisaster experiment Dunham ’ s Big Bend saxicolous lizards Brown ’ s Seed Predation experiments Simberloff-Wilson ’ s defaunation experiment

32. R. T. Paine (1966) Food Web Complexity and Species Diversity

33. Joseph Connell (1961)

35. Bruce Menge (1972)

36. Menge 1972 Bruce Menge

37. Grapevine Hills, Big Bend National Park Sceloporus merriami and Urosaurus ornatus Six rocky outcrops: 2 controls, 2 Sceloporus removal plots and 2 Urosaurus removal areas. ======================================================== 4 year study: 2 wet and 2 dry: insect abundances Monitored density, feeding success, growth rates, body weights, survival, lipid levels Urosaurus removal did not effect Sceloporus density No effects during wet years (insect food plentiful) Insects scarce during dry years: Urosaurus growth and survival was higher on Sceloporus removal plots Arthur Dunham

38. James Brown Dipodomys kangaroo rats Pogonomyrmex harvester ants

39. Experimental Design of Seed Predation in the Chihuahuan Desert ___________________________________________________ PlotsTreatments ___________________________________________________ 11,14Controls 6,13Seed addition, large seeds, constant rate 2,22Seed addition, small seeds, constant rate 9,20Seed addition, mixed seeds, constant rate 1,18Seed addition, mixed seeds, temporal pulse 5,24Rodent removal, Dipodomys spectabilis (largest kangaroo rat) 15,21Rodent removal, all Dipodomys species (kangaroo rats) 7,16Rodent removal, all seed-eating rodents 8,12Pogonomyrmex harvester ants 4,17All seed-eating ants 3,19All Dipodomys plus Pogonomyrmex ants 10,23All seed-eating rodents plus all seed-eating ants ___________________________________________________________ Munger, J. C. and J. H. Brown. 1981. Competition in desert rodents: an experiment with semipermeable enclosures. Science 211: 510-512.

40. open circles = rodents removed solid circles = controls

42. Defaunation Experiments in the Florida Keys Islands of mangrove trees were surveyed and numbers of arthropod species recorded Islands then covered in plastic tents and fumigated with methyl bromide Islands then resurveyed at intervals to document recolonization Simberloff and Wilson 1970

44. Evidence for Stability of Trophic Structure? First number is the number of species before defaunation, second in parentheses is the number after _______________________________________________________________________________________ Trophic Classes ______________________________________________________________________________ Island H S D W A C P ? Total _______________________________________________________________________________________ E1 9 (7)1 (0)3 (2)0 (0)3 (0)2 (1)2 (1)0 (0)20 (11) E211 (15)2 (2)2 (1)2 (2)7 (4)9 (4)3 (0)0 (1)36 (29) E3 7 (10)1 (2)3 (2)2 (0)5 (6)3 (4)2 (2)0 (0)23 (26) ST2 7 (6)1 (1)2 (1)1 (0)6 (5)5 (4)2 (1)1 (0)25 (18) E7 9 (10)1 (0)2 (1)1 (2)5 (3)4 (8)1 (2)0 (1)23 (27) E9 12 (7)1 (0)1 (1)2 (2)6 (5) 13 (10)2 (3)0 (1)37 (29) Totals 55 (55)7 (5) 13 (8)8 (6) 32 (23) 36 (31) 12 (9) 1 (3) 164 (140) _______________________________________________________________________________________ H = herbivore S = scavenger D = detritus feeder W = wood borer A = ant C = carnivorous predator P = parasite ? = undetermined

45. Wilson 1969