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Mutualistic Interactions and Symbiotic RelationshipsMutualism (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)
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Indirect Interactions
Darwin — Lots of “Humblebees” around villages
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Indirect Interactions
Darwin — Lots of “Humblebees” around villages
bees —> clover
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Indirect Interactions
Darwin — Lots of “Humblebees” around villages
bees —> clover
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Indirect Interactions
Darwin — Lots of “Humblebees” around villages
bees —> clover
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Indirect Interactions
Darwin — Lots of “Humblebees” around villages
mice —o bees —> clover
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Indirect Interactions
Darwin — Lots of “Humblebees” around villages
cats —o mice —o bees —> clover
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Indirect Interactions
Darwin — Lots of “Humblebees” around villages
cats —o mice —o bees —> clover —> beef
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Indirect Interactions
Darwin — Lots of “Humblebees” around villages
cats —o mice —o bees —> clover —> beef —> sailors
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Indirect Interactions
Darwin — Lots of “Humblebees” around villages
cats —o mice —o bees —> clover —> beef —> sailors —> Britain’s naval prowess
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Indirect Interactions
Darwin — Lots of “Humblebees” around villages
spinsters —> cats —o mice —o bees —> clover —> beef —> sailors —> Britain’s naval prowess
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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
—————————————————>
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Indirect Interactions
Trophic “Cascades” Top-down, Bottom-upMinus times minus = Plus
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Competitive Mutualism
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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
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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
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Inhibitory effect of each individualOn its own population growth is 1/K
Linear response to crowdingNo lag, instantaneous responsermax and K constant, immutable
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S - shaped sigmoidal population growth
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Lotka-Volterra
Competition Equations
Competition coefficient ij =
per capita competitive effect of
one individual of species j on
the rate of increase of species i
dN1 /dt = r1 N1 ({K1 – N1 – 12 N2 }/K1)
dN2 /dt = r2 N2 ({K2 – N2 – 21 N1 }/K2)
(K1 – N1 – 12 N2 )/K1 = 0 when N1 = K1 – 12 N2
(K2 – N2 – 21 N1 )/K2 = 0 when N2 = K2 – 21 N1
Vito VolterraAlfred J. Lotka
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N1 = K1 – 12 N2
if N2 = K1 / 12, then N1 = 0
N2 = K2 – 21 N1
if N1 = K2 / 21, then N2 = 0
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Zero isocline for species 1
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Four Possible Cases of Competition
Under the Lotka–Volterra
Competition Equations
______________________________________________________________________
Species 1 can contain Species 1 cannot contain
Species 2 (K2/21 < K 1) Species 2 (K2/21 > K 1)
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Species 2 can contain Case 3: Either species Case 2: Species 2
Species 1 (K1/12 < K2) can win always wins
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Species 2 cannot contain Case 1: Species 1 Case 4: Neither species
Species 1 (K1/12 > K2) always wins can contain the other;
stable coexistence
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Alfred J. Lotka
Vito Volterra
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Saddle Point
Point Attractor
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Lotka-Volterra Competition Equations
for n species (i = 1, n):
dNi /dt = riNi ({Ki – Ni – ij Nj}/Ki)
Ni* = Ki – ij Nj
where the summation is over j
from 1 to n, excluding i
Diffuse Competition
Robert H. MacArthur
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Alpha matrix of competition coefficients
11 12 13 . . . 1n
21 22 23 . . . 2n
31 32 33 . . . 3n
. . . . . . .
. . . . . . .
n1 n2 n3 . . . nn
Elements on the diagonalii equal 1.
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More realistic, curvilinear isoclines
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Competitive Exclusion in two species of Paramecium
Georgi F. Gause
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Coexistence of two species of Paramecium
Georgi F. Gause
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Coexistence of two species of Paramecium
Georgi F. Gause
Two equations, two unknowns
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Mutualism Equations (pp. 234-235, Chapter 11)
dN1 /dt = r1 N1 ({X1 – N1 + 12 N2 }/X1)
dN2 /dt = r2 N2 ({X2 – N2 + 21 N1 }/X2)
(X1 – N1 + 12 N2 )/X1 = 0 when N1 = X1 + 12 N2
(X2 – N2 + 21 N1 )/X2 = 0 when N2 = X2 + 21 N1
If X1 and X2 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.
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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 10034 30 Hot-Dry confusum > castaneum 90 1029 70 Warm-Moist confusum < castaneum 14 8629 30 Warm-Dry confusum > castaneum 87 1324 70 Cold-Moist confusum < castaneum 71 2924 30 Cold-Dry confusum > castaneum 100 0
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Evidence of Competition in Natureoften 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
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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, constant
Competition coefficients ij, 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/’s compared to K’sSp. 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.
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Alpha matrix of competition coefficients N, K Vectors
11 12 13 . . . 1n N1 K1
21 22 23 . . . 2n N2 K2
31 32 33 . . . 3n N3 K3
. . . . . . . . .
. . . . . . . . .
n1 n2 n3 . . . nn Nn Kn
Elements on the diagonalii equal 1.
Ni* = Ki – ij Nj
Matrix Algebra Notation: N = K – AN
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Lotka-Volterra Competition Equations for n species
dNi /dt = riNi ({Ki – Ni – ij Nj}/Ki)
Ni* = Ki – ij Nj at equilibrium
Alpha matrix, vectors of N’s and K’sDiffuse competition – ijNj summed over all j = 1, n (but not i)
N1* = K1 – 12 N2 – 13 N3 – 14 N4 N2* = K2 – 21 N1 – 23 N3 – 24 N4 N3* = K3 – 31 N1 – 32 N2 – 34 N4
N4* = K4 – 41 N1 – 42 N2 – 43 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)
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Evidence of Competition in Natureoften 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
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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
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flavidus 4 64 32
lividus 61 12 14 13
pennaceus 100
abbreviatus 100
ebraeus 15 82 3sponsalis 46 50 4rattus 23 77imperialis 27 73______________________________________________________________
Alan J. Kohn
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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 3sponsalis 46 50 4rattus 23 77imperialis 27 73______________________________________________________________
Alan J. Kohn
Resource MatrixNiche BreadthNiche Overlap
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Resource Matrix (n x m matrix)
utilization coefficients and electivities
Resource Consumer Species
State 1 2 3 . . . n
1 u11 u12 u13 . . . u1n
2 u21 u22 u23 . . . u2n
3 u31 u32 u33 . . . u3n
. . . . . . . .
. . . . . . . .
. . . . . . . .
m um1 um2 um3 . . . umn
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Cape May warbler Bay-breasted warbler
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MacArthur’s Warblers (Dendroica)
Robert H. MacArthur
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Time of Activity Seasonal changes in activity times
Ctenophorus isolepisCtenotus calurus
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Active Body Temperature and Time of Activity