Competition
George Williams described Mother Nature as a
“Wicked Old Witch”
This seems especially appropriate for negative interactions…
Competition
Competition (generally an intra-trophic level phenomenon)
occurs when each species negatively influences the population
growth rate (or size) of the other
This phenomenological definition is used in the modeling
framework proposed by Alfred Lotka (1880-1949)
& Vito Volterra (1860-1940)
Their goal was to determine the conditions under which
competitive exclusion vs. coexistence would occur
between two sympatric competitors
Population Dynamics
∆N
Exponential growth
Occurs when growth
rate is proportional to
population size;
Requires unlimited
resources
∆t
=r•N
N
Time
Population Dynamics
Density-dependent per capita birth (b) & death (d) rates
Notice that per capita
fitness increases with
decreases in
population size
from K
b
b
or
d
r
d
N
Equilibrium
(= carrying
capacity, K)
Population Dynamics
∆N
Logistic growth
∆t
= r • N • (1 –
N
K
K = carrying capacity
∆N
∆t
=0
N
∆N
∆t
∆N
∆t
is maximized
=0
Time
)
Competition
Lotka-Volterra Competition Equations:
In the logistic population growth model, the growth rate is
reduced by intraspecific competition:
Species 1: dN1/dt = r1N1[(K1-N1)/K1]
Species 2: dN2/dt = r2N2[(K2-N2)/K2]
Lotka & Volterra’s equations include functions to further reduce
growth rates as a consequence of interspecific competition:
Species 1: dN1/dt = r1N1[(K1-N1-f(N2))/K1]
Species 2: dN2/dt = r2N2[(K2-N2-f(N1))/K2]
Competition
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
2
2
1
2
1
1
2
1
1
1
1
1
1
Area within the frame represents carrying capacity (K) of either species
The size of each square is proportional to the resources an individual
consumes and makes unavailable to others (Sp. 1 = purple, Sp. 2 = green)
Individuals of Sp. 2 consume 4x resources consumed
by individuals of Sp. 1
For Species 1: dN1/dt = r1N1[(K1-N1-αN2)/K1]
Redrawn from Gotelli (2001)
… where α = 4.
2
2
1
2
1
1
2
1
1
1
1
1
1
Competition is occurring because both α & β > 0  α = 4 & β = ¼
In this case, adding an individual of Species 2 is more deleterious to
Species 1 than is adding an individual of Species 1…
but, adding an individual of Species 1 is less deleterious to
Species 2 than is adding an individual of Species 2
Redrawn from Gotelli (2001)
2
2
1
2
1
1
2
1
1
1
1
1
1
Asymmetric competition
In this case:
α>β
α>1
β<1
1
2
2
1
2
2
1
Asymmetric competition
In this case:
α>β
β=1
Asymmetric competition can occur throughout the spectrum of:
α  β, (α < = > 1, or β < = > 1)
What circumstances might the figure above represent?
Exclusively interspecific territoriality, intra-guild predation…
1
2
2
1
2
2
1
Symmetric competition
In this case:
α = β = 1, i.e., the special case of competitive equivalence
Symmetric competition can occur throughout the spectrum of:
(α = β) < = > 1
Lotka-Volterra Phenomenological Competition Model
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 (α N2)
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 Model
The equations for equilibrium solutions become:
^
Species 1: N
1 = [K1 - αK2] / [1 - α β]
^
Species 2: N
2 = [K2 - βK1] / [1 - α β]
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?
4 time steps
State-space graphs help to
track population trajectories
(and assess stability)
predicted by models
Mapping state-space
trajectories onto single
population trajectories
From Gotelli (2001)
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 in 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 in 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
Competition
Major prediction of the Lotka-Volterra competition model: Two species
can only coexist if intraspecific competition is stronger than
interspecific competition for both species
Earliest experiments within the Lotka-Volterra framework:
Gause (1932) – protozoans exploiting cultures of bacteria
The Lotka-Volterra models, coupled with the results of simple
experiments suggested a general principle in ecology:
The Lotka-Volterra-Gause Competitive Exclusion Principle
“Complete competitors cannot coexist” (Hardin 1960)
Competition
The Lotka-Volterra equations have been used extensively to model and
better understand competition, but they are phenomenological and
completely ignore the mechanisms of competition
In other words, they ignore the question: Why does a particular
interaction between species mutually reduce their population
growth rates and depress population sizes?
Competition
A commonly used, binary classification of mechanisms:
Exploitative / scramble (mutual depletion of shared resources)
Interference / contest (direct interactions between competitors)
More detailed classification of mechanisms (from Schoener 1983):
Consumptive (comp. for resources)
Preemptive (comp. for space; a.k.a. founder control)
Overgrowth (cf. size-asymmetric competition of Weiner 1990)
Chemical (e.g., allelopathy)
Territorial
Encounter
Exploitative / consumptive further divided by Byers (2000):
Resource suppression due to consumption rate
Resource-conversion efficiency
Competition
Case & Gilpin (1975) and Roughgarden (1983) claimed that interference
competition should not evolve unless exploitative competition
exists between two species
Why?
Interference competition is costly, and is unlikely to evolve under
conditions in which there is no payoff. If the two species do not
potentially compete for limiting resources (i.e., there is no opportunity for
exploitative competition), then there would be no reward for engaging
in interference competition.
Tilman’s Resource-Based Competition Models
Per capita reproductive
rate of Species 1
(dN/(N *dt)) is a function of
resource availability, R
R* = equilibrium resource
availability at which
reproduction and mortality
are balanced, and the level
to which species A can
reduce R in the
environment
dN/ N * dt (per capita)
Mortality rate, mA, is
assumed to remain
constant with changing R
Species A
mA
R*
Resource, R
Tilman’s Resource-Based Competition Models
Species B wins in this
case
Species A
dN/ N * dt (per capita)
When two species
compete for one limiting
resource, the species
with the lower R*
deterministically
outcompetes the other
mA
Species B
mB
RB*
RA*
Resource, R
Tilman’s Resource-Based Competition Models
Now consider the growth
response of one species to
two essential resources
R* divides the region into
portions favorable and
unfavorable to population
growth
dN/dt < 0
dN/dt > 0
R1*
R1
Tilman’s Resource-Based Competition Models
Now consider the growth
response of one species to
two essential resources
R* divides the region into
portions favorable and
unfavorable to population
growth
R2
dN/dt > 0
R2*
dN/dt < 0
R1*
R1
Tilman’s Resource-Based Competition Models
Now consider the growth
response of one species to
two essential resources
Zero Net Growth Isocline (ZNGI)
The two R*s divide the
region into portions
favorable and unfavorable
to population growth
R2
Consumption vectors can
be of any slope, but the
slope predicted under
optimal foraging would
equal R2*/R1*
If a population deviates
from the equilibrium along
the ZNGI, it will return to
the equilibrium
dN/dt > 0
R2*
dN/dt < 0
R1*
R1
Consumption
vector
Resource
supply point
Tilman’s Resource-Based Competition Models
Now consider two species
potentially competing for two
essential resources
In this case, species A
outcompetes species B in
habitats 2 & 3, and neither
species can persist in
habitat 1
A
R2
1
B
2
3
R1
Tilman’s Resource-Based Competition Models
In this case, species A wins
in habitat 2, species B wins in
habitat 6, and neither species
can persist in habitat 1
A
R2
1
B
2
?
6
R1
Consumption
vectors
Resource
supply points
Tilman’s Resource-Based Competition Models
There is also an equilibrium
point at which both species
can coexist
The extent to which that
equilibrium is stable depends
on the relative consumption
rates of the two species
R2
consuming the two
resources
A
1
B
2
?
6
R1
Tilman’s Resource-Based Competition Models
The extent to which that
equilibrium is stable depends
on the relative consumption
rates of the two species
consuming the two
resources
Slope of
consumption
vectors for A
A
B
Slope of
consumption
vectors for B
In this case, it is stable
R2
1
2
3
4
5
6
R1
Tilman’s Resource-Based Competition Models
The extent to which that
equilibrium is stable depends
on the relative consumption
rates of the two species
consuming the two
resources
Slope of
consumption
vectors for A
A
B
Slope of
consumption
vectors for B
In this case, it is stable
R2
Species A can only reduce
R2 to a level that limits
species A, but not species B,
whereas species B can only
reduce R1 to a level that
limits species B, but not
species A
Each species will return to its
equilibrium if displaced on its ZNGI
1
2
3
4
5
6
R1
Consumption
vectors
Resource
supply point
Tilman’s Resource-Based Competition Models
The extent to which that
equilibrium is stable depends
on the relative consumption
rates of the two species
consuming the two
resources
Slope of
consumption
vectors for B
A
B
Slope of
consumption
vectors for A
In this case, it is unstable
R2
1
2
3
4
5
6
R1
Tilman’s Resource-Based Competition Models
The extent to which that
equilibrium is stable depends
on the relative consumption
rates of the two species
consuming the two
resources
Slope of
consumption
vectors for B
A
B
Slope of
consumption
vectors for A
In this case, it is unstable
R2
Species A can reduce R1 to a
level that limits species A and
excludes species B, whereas
species B can reduce R2 to a
level that limits species B and
excludes species A
1
2
3
4
5
6
R1
Each species will return to its
equilibrium if displaced on its ZNGI
Consumption
vectors
Resource
supply point
Competition
The Lotka-Volterra competition model and Tilman’s R* model
are both examples of mean-field, analytical models (a.k.a.
“general strategic models”)
How relevant is the mean-field assumption to real
organisms?
“In sessile organisms such as plants, competition for resources
occurs primarily between closely neighboring individuals”
Antonovics & Levin (1980)
Neighborhood models describe how individual organisms respond to
variation in abundance or identity of neighbors
Competition
Spatially Explicit, Neighborhood Models of Plant Competition
There are many ways to formulate these models, and most require
computer-intensive simulations:
Cellular automata – Start with a grid of cells…
Spatially explicit individual-based models – Keep track of
the demographic fate and spatial location of every
individual in the population
Sometimes these are “empirical, field-calibrated
models”
Competition
Spatially Explicit, Neighborhood Models of Plant Competition
A key conclusion of these models:
At highest dispersal rates, i.e., “bath dispersal”, the predictions of the
mean-field approximations are often matched by the predictions of the
more complicated, spatially-explicit models
Low dispersal rates, however, lead to intraspecific clumping, which
tends to relax (broaden) the conditions under which two-species
coexistence occurs; this is similar to increasing the likelihood of
intraspecific competition relative to interspecific competition
Competition
Connell (1983)
Reviewed 54 studies
45/54 (83%) were consistent with competition
Of 54 studies, 33 (61%) suggested asymmetric competition
Schoener (1983)
Reviewed 164 studies
148/164 (90%) were consistent with competition
Of 61 studies, 51 (85%) suggested asymmetric competition
Kelly, Tripler & Pacala (ms. 1993) [But apparently never published!]
Only 1/4 of plot-based studies were consistent with competition,
whereas 2/3 of plant-centered studies were consistent
with competition
A classic competition study: MacArthur (1958)
Five sympatric warbler species with similar bill sizes and shapes broadly
overlap in arthropod diet, but they forage in different zones
within spruce crowns
Is this an example of the “ghost of competition past”
(sensu Connell [1980])?
Light and nutrient competition among rain forest tree seedlings
(Lewis and Tanner 2000)
Above-ground competition for light
is considered to be critical to seedling
growth and survival
Fewer studies exist of in situ belowground competition
Design:
Transplanted seedlings of two species (Aspidospermum - shade
tolerant; Dinizia - light demanding) into understory sites (1% light)
and small gaps (6% light) in nutrient-poor Amazonian forest
Reduced below-ground competition by “trenching” (digging a 50-cm
deep trench around each focal plant and lining it with plastic); this stops
neighboring trees from accessing nutrients and water
Relative height growth/day x 1000
2.0
gap
2.0
Dinizia
Aspidospermum
gap
1.5
1.5
understory
understory
1.0
1.0
0.5
0.5
0
No trench
Trench
No trench
Trench
0
No trench
Trench
No trench
Trench
Results and Conclusion:
Trenching had as big an impact as increased light did on seedling
growth
Seedlings are apparently simultaneously limited by (and
compete for) nutrients and light
Could allelopathy also be involved?
Effect of territorial honeyeaters on homopteran abundance
Loyn et al. (1983)
Flocks of Australian Bell Miners defend communal territories in eucalypt
forest, excluding other (sometimes much larger) species of birds
Up to 90% of miners’ diet is nymphs, secretions and lerps (shields) of
Homopterans (Psyllidae)
Experiment: Counted birds, counted lerps, removed miners
Results & conclusion: Invasion by a guild of 11 species of insectivorous
birds (competitive release), plus 3x increase in lerp removal rate,
reduction in lerp density, and 15% increase in foliage biomass
Competition between seed-eating rodents
and ants in the Chihuahuan Desert
Brown & Davidson (1977)
Strong resource limitation – seeds are the primary
food of many taxa (rodents, birds, ants)
Almost complete overlap in the sizes of seeds
consumed by ants and rodents – demonstrates the
potential for strong competition
Design:
Long-term exclosure experiments – fences to exclude rodents, and
insecticide to remove ants; re-censuses of ant and rodent populations
through time
Results and Conclusion:
Excluded rodents and the number of ant colonies increased 70%
Excluded ants and rodent biomass increased 24%
Competition can apparently occur between distantly related taxa
Competition between sexual and asexual species of gecko
Petren et al. (1993)
Humans have aided the dispersal of a sexual
species of gecko (Hemidactylus frenatus) to
several south Pacific islands and it is
apparently displacing asexual species
Experiment: Added H. frenatus and L.
lugubris alone and together to aircraft hanger
walls
Lepidodactylus lugubris,
asexual native on
south Pacific islands
Results and Conclusion: L. lugubris avoids
H. frenatus at high concentrations of insects
on lighted walls
Sometimes “obvious” hypothesized
reasons for competitive dominance are
incorrect
Competition among Anolis lizards
(Pacala & Roughgarden 1982)
What is the relationship between the strength of
interspecific competition and degree of
interspecific resource partitioning?
2 pairs of abundant insectivorous diurnal Anolis
lizards on 2 Caribbean islands:
St. Maarten: A. gingivinus & A. wattsi pogus
St. Eustatius: A. bimaculatus & A. wattsi schwartzi
Competition among Anolis lizards
(Pacala & Roughgarden 1982)
Body size (strongly correlated with prey size):
St. Maarten anoles: large overlap in body size
St. Eustatius anoles: small overlap in body size
Foraging location:
St. Maarten anoles: large overlap in perch ht.
St. Eustatius anoles: no overlap in perch ht.
Experiment:
Replicated enclosures on both islands, stocked
with one (not A. wattsi) or both species
Competition among Anolis lizards
(Pacala & Roughgarden 1982)
Results and Conclusions:
St. Maarten (similar resource use)
Growth rate of A. gingivinus was halved
in the presence of A. wattsi
St. Eustatius (dissimilar resource use)
No effect of A. wattsi on growth or
perch height of A. bimaculatis
Strength of present-day competition in these
species pairs is inversely related to resource
partitioning
Competition among Anolis lizards
(Pacala & Roughgarden 1982)
Why do these pairs of anoles on nearby islands
(30 km) differ in degree of resource partitioning?
Hypothesis: Character displacement occurred
on St. Eustatius during long co-evolutionary history
(i.e., the ghost of competition past), whereas
colonization of St. Maarten occurred much more
recently, and in both cases colonization was by
similarly sized Anolis species
Character displacement:
Evolutionary divergence of traits in response to
competition, resulting in a reduction in the intensity
of competition
Competition among Anolis lizards
(Pacala & Roughgarden 1985)
Pacala & Roughgarden (1985) presented evidence
to suggest that both species pairs have a long
history of co-occurrence on their respective islands
and that different colonization histories resulted in
the observed patterns of resource partitioning
Both islands may have been colonized by Anolis
species differing in size, yet on St. Maarten the
larger Anolis colonized later and has subsequently
converged in body size on the smaller resident
Character Displacement
Schluter & McPhail (1992) surveyed the literature on character
displacement and listed criteria necessary to exclude other potential
explanations for species that share similar traits in allopatry, but differ in
sympatry (similar to Connell’s [1980] requirements to demonstrate the
“ghost of competition past”):
1. Chance should be ruled out as an explanation for the pattern
(appropriate statistical tests, often involving null models)
2. Phenotypic differences should have a genetic basis
3. Enhanced differences between sympatric species should be the
outcome of evolutionary shifts, not simply the inability of similar-sized
species to coexist
4. Morphological differences should reflect differences in resource use
5. Sites of sympatry and allopatry should be similar in terms of physical
characteristics
6. Independent evidence should be obtained that similar phenotypes
actually compete for food