Direct versus Indirect Interactions Exploitation vs Interference competition

Direct versus Indirect Interactions Exploitation vs. Interference competition Apparent Competition Competitive Mutualism Facilitation Food Chain Mutualism Trophic Cascades (top-down, bottom up) Complex Population Interactions (Colwell’s Plant-Pollinator System) Mutualisms Bioluminescence Euglossine bees and orchids Heliconius butterflies (larval nitrogen reserves) Cattle Egret Commensalism Lecture # 20 Gause’s competition lab experiments 7 th November 2019

Lotka-Volterra Competition Equations competition coefficient aij = per capita competitive effect of one individual of species j on the rate of increase of species i Alfred Lotka Vito Volterra d. N 1 /dt = r 1 N 1 ({K 1 – N 1 – a 12 N 2 }/K 1) d. N 2 /dt = r 2 N 2 ({K 2 – N 2 – a 21 N 1 }/K 2) Isoclines: (K 1 – N 1 – a 12 N 2 )/K 1 = 0 when N 1 = K 1 – a 12 N 2 (K 2 – N 2 – a 21 N 1 )/K 2 = 0 when N 2 = K 2 – a 21 N 1

Intercepts: N 1 = K 1 – a 12 N 2 if N 2 = K 1 / a 12, then N 1 = 0 N 2 = K 2 – a 21 N 1 if N 1 = K 2 / a 21, then N 2 = 0

r 1 No competitors / K 1 / _K competitors 1 N 2 _K 1 a 2 a competitors N 1

Zero isocline for species 1 N 1* = K 1 – a 12 N 2

Four Possible Cases of Competition Under the Lotka–Volterra Competition Equations Alfred Lotka Vito Volterra ___________________________________ Species 1 can contain Species 1 cannot contain Species 2 (K 2/a 21 < K 1) Species 2 (K 2/a 21 > K 1) ___________________________________ Species 2 can contain Case 3: Either species Case 2: Species 2 Species 1 (K 1/a 12 < K 2) can win always wins ___________________________________ Species 2 cannot contain Case 1: Species 1 Case 4: Neither species Species 1 (K 1/a 12 > K 2) always wins can contain the other; stable coexistence ___________________________________

Interspecific Competition leads to Niche Diversification Two types of Interspecific Competition: Exploitation competition is indirect, occurs when a resource is in short supply by resource depression Interference competition is direct and occurs via antagonistic encounters such as interspecific territoriality or production of toxins

Lotka-Volterra Competition Equations competition coefficient aij = per capita competitive effect of one individual of species j on the rate of increase of species i Alfred Lotka Vito Volterra d. N 1 /dt = r 1 N 1 ({K 1 – N 1 – a 12 N 2 }/K 1) d. N 2 /dt = r 2 N 2 ({K 2 – N 2 – a 21 N 1 }/K 2) Solve for Isoclines by setting d. N/dt‘s equal to zero: (K 1 – N 1 – a 12 N 2 )/K 1 = 0 when N 1 = K 1 – a 12 N 2 (K 2 – N 2 – a 21 N 1 )/K 2 = 0 when N 2 = K 2 – a 21 N 1

Resultant Vectors

Resultant Vectors

Resultant Vectors

Saddle Point Attractor

Lotka-Volterra Competition Equations for n species (i = 1, n): d. Ni /dt = ri. Ni ({Ki – Ni – S aij Nj}/Ki) Ni* = Ki – S aij Nj where the summation is over j from 1 to n, excluding i Diffuse Competition S aij Nj Robert H. Mac. Arthur

Lotka-Volterra Competition Equations for 3 species: d. N 1 /dt = r 1 N 1 ({K 1 – N 1 – a 12 N 2 – a 13 N 3 }/K 1) d. N 2 /dt = r 2 N 2 ({K 2 – N 2 – a 21 N 1 – a 23 N 3 }/K 2) d. N 3 /dt = r 3 N 3 ({K 3 – N 3 – a 31 N 1 – a 32 N 2 }/K 3) Isoclines: d. N/dt = 0 {curly brackets, above} (K 1 – N 1 – a 12 N 2 – a 13 N 3 ) = 0 when N 1 = K 1 – a 12 N 2 – a 13 N 3 (K 2 – N 2 – a 21 N 1 – a 23 N 3 ) = 0 when N 2 = K 2 – a 21 N 1 – a 23 N 3 (K 3 – N 3 – a 31 N 1 – a 32 N 2 ) = 0 when N 3 = K 3 – a 31 N 1 – a 32 N 2

Lotka-Volterra Competition Equations for n species (i = 1, n): d. Ni /dt = ri. Ni ({Ki – Ni – S aij Nj}/Ki) Ni* = Ki – S aij Nj where the summation is over j from 1 to n, excluding i Diffuse Competition S aij Nj

Alpha matrix of competition coefficients a 11 a 12 a 13 . . . a 1 n a 21 a 22 a 23 . . . a 2 n a 31 a 32 a 33 . . . a 3 n . . . an 3 . . . ann an 1 an 2 Self damping elements on the diagonal aii equal 1.

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

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, Hutchinsonian ratios 3. Incomplete biotas: niche shifts 4. Taxonomic composition of communities

Resource Matrix (m x n) Major Foods (Percentages) of Eight Species of Cone Shells, Conus, on Subtidal Reefs in Hawaii _______________________________ Gastro- Entero. Tere- Other Species pods pneusts Nereids Eunicea belids Polychaetes _______________________________ flavidus 4 64 32 lividus 61 12 14 13 pennaceus 100 abbreviatus 100 ebraeus 15 82 3 sponsalis 46 50 4 rattus 23 77 imperialis 27 73 _______________________________ Alan J. Kohn 4 Radula

Resource Matrix (m x n) Major Foods (Percentages) of Eight Species of Cone Shells, Conus, on Subtidal Reefs in Hawaii _______________________________ Gastro- Entero. Tere- Other Species pods pneusts Nereids Eunicea belids Polychaetes _______________________________ flavidus 4 64 32 lividus 61 12 14 13 pennaceus 100 abbreviatus 100 ebraeus 15 82 3 sponsalis 46 50 4 rattus 23 77 imperialis 27 73 _______________________________ Alan J. Kohn 4 Radula

Mac. Arthur’s Warblers (Dendroica) Robert H. Mac. Arthur

Time of Activity Ctenotus calurus Seasonal changes in activity times Ctenophorus isolepis

35. 3 ºC 26. 1 ºC 39. 5 ºC 27. 1 ºC 39. 1 ºC 29. 1 ºC 40. 0 ºC 31. 2 ºC

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

Prey size versus predator size

Galápagos Finches Woodpecker Finches Cocos Isand

Cocos Island

Galápagos Finches Peter R. Grant David Lack “Darwin’s Finches”

Character Displacement in Hydrobia mud snails in Denmark (Thomas Fenchel) Snail shell length, mm

Corixid Water Boatman G. E. Hutchinson

Hutchinsonian Ratios

Hutchinsonian Ratios Henry S. Horn Bob May

Hutchinsonian Ratios Henry S. Horn Bob May Recorders

Wind Instruments

Kitchen Knives

Kitchen Pots

Tricycles

Bikes

Hutchinsonian ratios among short wing Accipiter hawks Thomas W. Schoener

Hutchinsonian ratios among Australian Varanus lizards