Download K 1

Competition
Photo of hyenas and lioness at a carcass from https://www.flickr.com/photos/davidbygott/4046054583
Pairwise Species Interactions
Influence of species A
Influence of Species B
A
-
- (negative)
0 (neutral/null)
-
0
-
A
-
B
Competition
Amensalism
-
0
A
B
0
A
B
A
B
Antagonism
(Predation/Parasitism)
+
A
B
0
0
Amensalism
Neutralism
(No interaction)
Commensalism
-
0
+
0
+
B
+ (positive)
+
A
B
+
Antagonism
(Predation/Parasitism)
A
B
+
Commensalism
Redrawn from Abrahamson (1989); Morin (1999, pg. 21)
A
+
Mutualism
B
Intra-specific vs. Inter-specific Competition
Interaction between individuals in which each is harmed by their
shared use of a limiting resource (which can be consumed or depleted)
for growth, survival, or reproduction
Photo of hyenas and lioness at a carcass from https://www.flickr.com/photos/davidbygott/4046054583
Intra-specific vs. Inter-specific Competition
“Complete competitors cannot coexist.”
(Hardin 1960)
Paramecium
aurelia
Paramecium
caudatum
Cain, Bowman & Hacker (2014), Fig. 12.11, after Gause (1934); photomicrographs from Wikimedia Commons
Intra-specific vs. Inter-specific Competition
Resource partitioning – differences in use of limiting resources –
can allow species to coexist
P. aurelia & P. caudatum ate mostly floating bacteria;
P. bursaria ate mostly yeast cells on the bottoms of the tubes
Cain, Bowman & Hacker (2014), Fig. 12.11, after Gause (1934)
Lotka – Volterra Phenomenological Competition Models
Alfred Lotka & Vito Volterra
(1880-1949)
(1860-1940)
Photo of Lotka from http://blog.globe-expert.info; photo of Volterra from Wikimedia Commons
Lotka – Volterra Phenomenological Competition Models
Lotka-Volterra Competition Equations:
Logistic population growth model – growth rate is
reduced by intraspecific competition:
Species 1: dN1/dt = r1N1[(K1-N1)/K1]
Species 2: dN2/dt = r2N2[(K2-N2)/K2]
Functions added to further reduce growth rate
owing to interspecific competition:
Species 1: dN1/dt = r1N1[(K1-N1-f(N2))/K1]
Species 2: dN2/dt = r2N2[(K2-N2-f(N1))/K2]
Lotka – Volterra Phenomenological Competition Models
Lotka-Volterra Competition Equations:
The function (f) could take on many forms, e.g.:
Species 1: dN1/dt = r1N1[(K1-N1-αN2)/K1]
Species 2: dN2/dt = r2N2[(K2-N2-βN1)/K2]
The competition coefficients α & β measure the per capita effect
of one species on the population growth of the other, measured
relative to the effect of intraspecific competition
If α = 1, then per capita intraspecific effects = interspecific effects
If α < 1, then intraspecific effects are more deleterious
to Species 1 than interspecific effects
If α > 1, then interspecific effects are more deleterious
Lotka – Volterra Phenomenological Competition Models
Find equilibrium solutions to the equations, i.e., set dN/dt = 0:
^
Species 1: N
1 = K1 - αN2
^
Species 2: N2 = K2 - βN1
This makes intuitive sense: The equilibrium for N1 is the carrying
capacity for Species 1 (K1) reduced by some amount owing to the
presence of Species 2 (αN2)
However, each species’ equilibrium depends on the equilibrium of
the other species! So, by substitution…
^
^
^
^
Species 1: N1 = K1 - α(K2 - βN1)
Species 2: N2 = K2 - β(K1 - αN2)
Lotka – Volterra Phenomenological Competition Models
The equations for equilibrium solutions become:
^
Species 1: N
1 = [K1 - αK2] / [1 - αβ]
^
Species 2: N
2 = [K2 - βK1] / [1 - αβ]
These provide some insights into the conditions required for
coexistence under the assumptions of the model
E.g., the product αβ must be < 1 for N to be > 0 for both species (a
necessary condition for coexistence)
But they do not provide much insight into the dynamics of
competitive interactions, e.g., are the equilibrium points stable?
Lotka – Volterra Phenomenological Competition Models
4 time steps
State-space graphs help to
track population trajectories
(and assess stability)
predicted by models
From Gotelli (2001)
Lotka – Volterra Phenomenological Competition Models
4 time steps
State-space graphs help to
track population trajectories
(and assess stability)
predicted by models
4 time steps
Mapping state-space
trajectories onto single
population trajectories
From Gotelli (2001)
Lotka-Volterra Model
Remember that equilibrium
solutions require dN/dt = 0
^
Species 1: N1 = K1 - αN2
Therefore:
When N2 = 0, N1 = K1
K1 / α
Isocline for Species 1
dN1/dt = 0
N2
When N1 = 0, N2 = K1/α
K1
N1
Lotka-Volterra Model
Remember that equilibrium
solutions require dN/dt = 0
^
Species 2: N2 = K2 - βN1
Therefore:
When N1 = 0, N2 = K2
K2
Isocline for Species 2
dN2/dt = 0
N2
When N2 = 0, N1 = K2/β
K2 / β
N1
Lotka-Volterra Model
Plot the isoclines for 2
species together to
examine population
trajectories
Competitive exclusion of
Species 2 by Species 1
K1/α > K2
K1 > K2/β
K1 / α
N2
For species 1:
K1 > K2α
(intrasp. > intersp.)
K2
For species 2:
K1β > K2
(intersp. > intrasp.)
= stable equilibrium
K2 / β
N1
K1
Lotka-Volterra Model
Plot the isoclines for 2
species together to
examine population
trajectories
Competitive exclusion of
Species 1 by Species 2
K2 > K1/α
K2/β > K1
K2
N2
For species 1:
K2α > K1
(intersp. > intrasp.)
For species 2:
K2 > K1β
(intrasp. > intersp.)
K1/ α
= stable equilibrium
K1
N1
K2 / β
Lotka-Volterra Model
Plot the isoclines for 2
species together to
examine population
trajectories
Competitive exclusion with an
unstable equilibrium
K2 > K1/α
K1 > K2/β
K1/ α
N2
For species 1:
K2α > K1
(intersp. > intrasp.)
K2
For species 2:
K1β > K2
(intersp. > intrasp.)
= stable equilibrium
= unstable equilibrium
K2 / β
K1
N1
Lotka-Volterra Model
Plot the isoclines for 2
species together to
examine population
trajectories
K1/α > K2
K2/β > K1
Coexistence at a stable equilibrium
K1 / α
N2
For species 1:
K1 > K2α
(intrasp. > intersp.)
For species 2:
K2 > K1β
(intrasp. > intersp.)
K2
= stable equilibrium
K2 / β
K1
N1
Mechanisms of Competition
Exploitation competition
Dissecting exploitation competition reveals its indirect nature
H
-
H
-
-
+
+
P
Interference competition (direct aggression, allelopathy, etc.)
H
-
H
P
-
P
Solid arrows = direct effects; dotted arrows = indirect effects
Redrawn from Menge (1995)
Mechanisms of Competition
David
Tilman
Synedra
Asterionella
Cain, Bowman & Hacker (2014), Fig. 12.4, after Tilman et al. (1981); photos of diatoms from Wikimedia Commons;
photo of Tilman from http://www.princeton.edu/morefoodlesscarbon/speakers/david-tilman/
Mechanisms of Competition
David
Tilman
Cain, Bowman & Hacker (2014), Fig. 12.4, after Tilman et al. (1981);
photo of Tilman from http://www.princeton.edu/morefoodlesscarbon/speakers/david-tilman/
Asymmetric vs. Symmetric Competition
Cain, Bowman & Hacker (2014), Fig. 12.7
Classic Pattern Interpreted as Evidence for
Competitively-Structured Assemblages
Robert MacArthur
(1930-1972)
Painting of “MacArthur’s warblers” by D. Kaspari for M. Kaspari (2008); anniversary reflection on MacArthur (1958)
Character Displacement
The “Ghost of Competition Past”
(sensu Connell 1980) is
hypothesized to be the cause of the
beak size difference on
Pinta Marchena
Cain, Bowman & Hacker (2014), Fig. 12.19, after Lack (1947)