Download 14-Competition

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