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)

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

Trophic “Cascades” Top-down, Bottom-upMinus times minus = Plus

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 individualOn its own population growth is 1/K

Linear response to crowdingNo lag, instantaneous responsermax and K constant, immutable

S - shaped sigmoidal population growth

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

N1 = K1 – 12 N2

if N2 = K1 / 12, then N1 = 0

N2 = K2 – 21 N1

if N1 = K2 / 21, then N2 = 0

N1 = K1 – 12 N2

N1 = K1 – 12 N2

Zero isocline for species 1

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)

______________________________________________________________________

Species 2 can contain Case 3: Either species Case 2: Species 2

Species 1 (K1/12 < K2) can win always wins

______________________________________________________________________

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

______________________________________________________________________

Alfred J. Lotka

Vito Volterra

Saddle Point

Point Attractor

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

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.

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

Georgi F. Gause

Two equations, two unknowns

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.

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

________________________________________________________

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

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.

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

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)

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

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

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

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

Cape May warbler Bay-breasted warbler

MacArthur’s Warblers (Dendroica)

Robert H. MacArthur

John Terborgh

John Terborgh

Time of Activity Seasonal changes in activity times

Ctenophorus isolepisCtenotus calurus

Active Body Temperature and Time of Activity