Gause ’ s and Park ’ s competition experiments Lotka-Volterra Competition equations

1. Gause’s and Park’s competition experiments Lotka-Volterra Competition equations dNi /dt = ri Ni ({Ki – Ni – S aij Nj }/Ki ) Summation is over j from 1 to n, excluding i Ni* = Ki – S aij Nj [Diffuse competition] 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, set dN/dt = 0 resultant vector analyses Four cases, depending on K/a’s compared to K’s Sp. 1 wins, sp. 2 wins, either/or, or coexistence

2. Four Possible Cases of Competition Under the Lotka–Volterra Competition Equations_____________________________________________________________________ Species 1 can contain Species 1 cannot contain Species 2 (K2/a21 < K1) Species 2 (K2/a21 > K1) ______________________________________________________________________ Species 2 can contain Case 3: Either species Case 2: Species 2 Species 1 (K1/a12 < K2) can win always wins______________________________________________________________________ Species 2 cannot contain Case 1: Species 1 Case 4: Neither species Species 1 (K1/a12 > K2) always wins can contain the other; stable coexistence______________________________________________________________________ Vito Volterra Alfred Lotka

3. Resultant Vectors

4. Saddle Point Point Attractor

5. Lotka-Volterra Competition Equations for 3 species: dN1 /dt = r1 N1 ({K1 – N1 – a12 N2 – a13 N3 }/K1) dN2 /dt = r2 N2 ({K2 – N2 – a21 N1 – a23 N3 }/K2) dN3 /dt = r3 N3 ({K3 – N3 – a31 N1 – a32 N2 }/K2) Isoclines: (K1 – N1 – a12 N2 – a13 N3 )/K1 = 0when N1 = K1 – a12 N2 – a13 N3 (K2 – N2 – a21 N1 – a23 N3 )/K2 = 0when N2 = K2 – a21 N1 – a23 N3 (K3 – N3 – a31 N1 – a32 N2 )/K3 = 0when N3 = K3 – a31 N1 – a32 N2 Lotka-Volterra Competition Equations for n species (i = 1, n): dNi /dt = riNi ({Ki – Ni – S aij Nj}/Ki) Ni* = Ki – S aij Nj where the summation is over j from 1 to n, excluding iDiffuse CompetitionS aij Nj

6. Mutualism Equations (pp. 234-235, Chapter 11) dN1 /dt = r1 N1 ({X1 – N1 + a12 N2 }/X1)dN2 /dt = r2 N2 ({X2 – N2 + a21 N1 }/X2) (X1 – N1 + a12 N2 )/X1 = 0when N1 = X1 + a12 N2 (X2 – N2 + a21 N1 )/X2 = 0when N2 = X2 + a21 N1 If X1 and X2 are positive and a12 and a21 are chosen so that isoclines cross, a stable joint equilibrium exists. Intraspecific self damping must be stronger than interspecific positive mutualistic effects.

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

10. 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

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

13. One Dimension: Distance between two points along a line: simply subtract smaller value from larger one x2-x1 = d Two Dimensions: Score position of each point on the first and second dimensions. Subtract smaller from larger on both dimensions. d1=x2-x1 d2=y2-y1 Square these differences, sum them and take the square root. This is the distance between the points in 2D: sqrt (d12+ d22) = d Three Dimensions —> n-dimensions: follow this same protocol summing over all dimensions i = 1, n: sqrt Sdi2 = d Euclid

14. Euclidean distance between two species in n-space n-dimensional hypervolume 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 = 1 to n . n is the number of resource dimensions n Euclid i = 1

16. 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

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

20. 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

21. 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

22. 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

23. 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

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 Anolis Thomas W. Schoener

29. 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

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)

33. Joseph Connell (1961)