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Mutualistic Interactions and Symbiotic Relationships Mutualism (obligate and facultative) Termite endosymbionts Commensalisms (Cattle Egrets) Examples: Bullhorn Acacia ant colonies (Beltian bodies) Caterpillars “sing” to ants (protection) Ants tend aphids for their honeydew, termites cultivate fungi Bacteria and fungi in roots provide nutrients (carbon reward) Bioluminescence (bacteria) Endozoic algae (Hydra), bleaching of coral reefs (coelenterates) Nudibranch sea slugs: Nematocysts, “kidnapped” chloroplasts Endosymbiosis (Lynn Margulis) mitochondria & chloroplasts Birds on water buffalo backs, picking crocodile teeth Figs and fig wasps (pollinate, lay eggs, larvae develop) Indirect Interactions Darwin — Lots of “Humblebees” around villages Indirect Interactions Darwin — Lots of “Humblebees” around villages bees —> clover Indirect Interactions Darwin — Lots of “Humblebees” around villages bees —> clover Indirect Interactions Darwin — Lots of “Humblebees” around villages bees —> clover Indirect Interactions Darwin — Lots of “Humblebees” around villages mice —o bees —> clover Indirect Interactions Darwin — Lots of “Humblebees” around villages cats —o mice —o bees —> clover Indirect Interactions Darwin — Lots of “Humblebees” around villages cats —o mice —o bees —> clover —> beef Indirect Interactions Darwin — Lots of “Humblebees” around villages cats —o mice —o bees —> clover —> beef —> sailors Indirect Interactions Darwin — Lots of “Humblebees” around villages cats —o mice —o bees —> clover —> beef —> sailors —> Britain’s naval prowess Indirect Interactions Darwin — Lots of “Humblebees” around villages spinsters —> cats —o mice —o bees —> clover —> beef —> sailors —> Britain’s naval prowess Indirect Interactions Darwin — Lots of “Humblebees” around villages —————————————————> spinsters —> cats —o mice —o bees —> clover —> beef —> sailors —> naval prowess Path length of seven! Longer paths take longer (delay) Longer paths are also weaker, but there are more of them Indirect Interactions Minus times minus = Plus Trophic “Cascades” Top-down, Bottom-up Competitive Mutualism 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 Verhulst-Pearl Logistic Equation dN/dt = rN [(K – N)/K] = rN {1– (N/K)} dN/dt = rN – rN (N/K) = rN – {(rN2)/K} dN/dt = 0 when [(K – N)/K] = 0 [(K – N)/K] = 0 when N = K dN/dt = rN – (r/K)N2 Inhibitory effect of each individual On its own population growth is 1/K Linear response to crowding No lag, instantaneous response rmax and K constant, immutable S - shaped sigmoidal population growth Lotka-Volterra Competition Equations Alfred J. Lotka Competition coefficient aij = per capita competitive effect of one individual of species j on the rate of increase of species i Vito Volterra dN1 /dt = r1 N1 ({K1 – N1 – a12 N2 }/K1) dN2 /dt = r2 N2 ({K2 – N2 – a21 N1 }/K2) (K1 – N1 – a12 N2 )/K1 = 0 when N1 = K1 – a12 N2 (K2 – N2 – a21 N1 )/K2 = 0 when N2 = K2 – a21 N1 N1 = K1 – a12 N2 if N2 = K1 / a12, then N1 = 0 N2 = K2 – a21 N1 if N1 = K2 / a21, then N2 = 0 N1 = K1 – a12 N2 N1 = K1 – a12 N2 Zero isocline for species 1 Four Possible Cases of Competition Under the Lotka–Volterra Competition Equations Vito Volterra Alfred J. Lotka ______________________________________________________________________ Species 1 can contain Species 1 cannot contain Species 2 (K2/a21 < K 1) Species 2 (K2/a21 > K 1) ______________________________________________________________________ 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 ______________________________________________________________________ Saddle Point Point Attractor 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 i Diffuse Competition Robert H. MacArthur Alpha matrix of competition coefficients a11 a12 a13 . . . a1n a21 a22 a23 . . . a2n a31 a32 a33 . . . a3n . . . . . . . . . . . . . . an1 an2 an3 . . . ann Elements on the diagonal aii equal 1. More realistic, curvilinear isoclines Competitive Exclusion in two species of Paramecium Georgi F. Gause Coexistence of two species of Paramecium Georgi F. Gause Coexistence of two species of Paramecium Two equations, two unknowns Georgi F. Gause 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 = 0 when N1 = X1 + a12 N2 (X2 – N2 + a21 N1 )/X2 = 0 when 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. Outcome of Competition Between Two Species of Flour Beetles ____________________________________________________________________ Relative Temp. Humidity Single Species (°C) (%) Climate Numbers Mixed Species (% wins) confusum castaneum ____________________________________________________________________ 34 70 Hot-Moist confusum = castaneum 0 100 34 30 Hot-Dry confusum > castaneum 90 10 29 70 Warm-Moist confusum < castaneum 14 86 29 30 Warm-Dry confusum > castaneum 87 13 24 70 Cold-Moist confusum < castaneum 71 29 24 30 Cold-Dry confusum > castaneum 100 0 ________________________________________________________ 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 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. Alpha matrix of competition coefficients N, K Vectors a11 a12 a13 . . . a1n N1 K1 a21 a22 a23 . . . a2n N2 K2 a31 a32 a33 . . . a3n N3 K3 . . . . . . . . . . . . . . . . . . an1 an2 an3 . . . ann Nn Kn Elements on the diagonal aii equal 1. Ni* = Ki – S aij Nj Matrix Algebra Notation: N = K – AN Lotka-Volterra Competition Equations for n species dNi /dt = riNi ({Ki – Ni – S aij Nj}/Ki) Ni* = Ki – S aij Nj at equilibrium Alpha matrix, vectors of N’s and K’s Diffuse competition – S aijNj summed over all j = 1, n (but not i) N1* = K1 – a12 N2 – a13 N3 – a14 N4 N2* = K2 – a21 N1 – a23 N3 – a24 N4 N3* = K3 – a31 N1 – a32 N2 – a34 N4 N4* = K4 – a41 N1 – a42 N2 – a43 N3 Vector Notation: N = K – AN where A is the alpha matrix Partial derivatives ∂Ni/ ∂Nj sensitivity of species i to changes in j Jacobian Matrix of partial derivatives (Lyapunov stability) 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 Major Foods (Percentages) of Eight Species of Cone Shells, Conus, on Subtidal Reefs in Hawaii _____________________________________________________________ Gastro- Entero- Species pods Tere- Other pneusts Nereids Eunicea belids Polychaetes ______________________________________________________________ flavidus 4 lividus 61 pennaceus abbreviatus 12 64 32 14 13 100 100 ebraeus 15 82 3 sponsalis 46 50 4 rattus 23 77 imperialis 27 73 ______________________________________________________________ Alan J. Kohn Major Foods (Percentages) of Eight Species of Cone Shells, Conus, on Subtidal Reefs in Hawaii _____________________________________________________________ Gastro- Entero- Species pods Tere- Other pneusts Nereids Eunicea belids Polychaetes ______________________________________________________________ flavidus 4 lividus 61 pennaceus abbreviatus 12 64 32 14 13 100 100 ebraeus 15 82 3 sponsalis 46 50 4 rattus 23 77 imperialis 27 73 ______________________________________________________________ Alan J. Kohn Resource Matrix Niche Breadth Niche Overlap Resource Matrix (n x m matrix) utilization coefficients and electivities Resource State 1 2 3 . . . m 1 u11 u21 u31 . . . um1 2 u12 u22 u32 . . . um2 Consumer Species 3 . . . u13 . . . u23 . . . u33 . . . . . . . . . . . . . . . um3 . . . n u1n u2n u3n . . . umn Cape May warbler Bay-breasted warbler MacArthur’s Warblers (Dendroica) Robert H. MacArthur John Terborgh John Terborgh Time of Activity Ctenotus calurus Seasonal changes in activity times Ctenophorus isolepis Active Body Temperature and Time of Activity