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The extent to which that equilibrium is stable depends on the relative consumption rates of the two species consuming the two

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 AlfredLotka (1880-1949)

& VitoVolterra (1860-1940)

Their goal was to determine the conditions under which competitive exclusion vs. coexistence would occur

between two sympatric competitors

∆N

Exponential growth

= r • N

∆t

Occurs when growth rate is proportional to population size;

Requires unlimited resources

N

Time

& death (d) rates

Density-dependent per capita birth (b)

Notice that per capita fitness increases with decreases in population size from K

b

b

r

or

d

d

Equilibrium

(= carrying capacity, K)

N

∆N

N

Logistic growth

= r • N • (1 – )

∆t

K

K = carrying capacity

= 0

N

is maximized

∆N

∆N

∆N

= 0

∆t

∆t

∆t

Time

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]

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 coefficientsα & 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

1

2

1

2

1

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] … where α = 4.

Redrawn from Gotelli (2001)

2

1

2

1

2

1

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

2

1

Asymmetric competition

In this case:

α > β

β = 1

Asymmetric competition can occur throughout the spectrum of:

αβ, (α < = > 1, or β < = > 1)

What circumstances might the figure above represent?

Exclusively interspecific territoriality, intra-guild predation…

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

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

Find equilibrium solutions to the equations, i.e., set dN/dt = 0:

Species 1: N1 = 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)

Lotka-Volterra Phenomenological Competition Model

The equations for equilibrium solutions become:

Species 1: N1 = [K1 - αK2] / [1 - α β]

Species 2: N2 = [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?

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)

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)

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

When N1 = 0, N2 = K1/α

N2

K1

N1

Remember that equilibrium solutions require dN/dt = 0

Species 2: N2 = K2 - βN1

^

Therefore:

When N1 = 0, N2 = K2

K2

When N2 = 0, N1 = K2/β

Isocline for Species 2

dN2/dt = 0

N2

K2 / β

N1

Plot the isoclines for 2 species together to examine population trajectories

K1/α > K2

K1 > K2/β

For species 1:

K1 > K2α

(intrasp. > intersp.)

For species 2:

K1β> K2

(intersp. > intrasp.)

Competitive exclusion of

Species 2 by Species 1

K1 / α

K2

N2

= stable equilibrium

K2 / β

K1

N1

Plot the isoclines for 2 species together to examine population trajectories

K2 > K1/α

K2/β > K1

For species 1:

K2α > K1

(intersp. > intrasp.)

For species 2:

K2 > K1β

(intrasp. > intersp.)

Competitive exclusion of

Species 1 by Species 2

K2

N2

K1/ α

= stable equilibrium

K2 / β

K1

N1

Plot the isoclines for 2 species together to examine population trajectories

K2 > K1/α

K1 > K2/β

For species 1:

K2α > K1

(intersp. > intrasp.)

For species 2:

K1β > K2

(intersp. > intrasp.)

Competitive exclusion in an

unstable equilibrium

K2

K1/ α

N2

= stable equilibrium

K1

K2 / β

= unstable equilibrium

N1

Plot the isoclines for 2 species together to examine population trajectories

K1/α > K2

K2/β > K1

For species 1:

K1 > K2α

(intrasp. > intersp.)

For species 2:

K2 > K1β

(intrasp. > intersp.)

Coexistence in a stable equilibrium

K1 / α

N2

K2

= stable equilibrium

K1

K2 / β

N1

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)

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?

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

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

Species A

Mortality rate, mA, is assumed to remain constant with changing R

mA

dN/ N * dt (per capita)

R* = equilibrium resource availability at which reproduction and mortality are balanced, and the level to which species A can reduce R in the environment

*

R

Resource, R

Tilman’s Resource-Based Competition Models

When two species compete for one limiting resource, the species with the lower R* deterministically outcompetes the other

Species A

mA

Species B wins in this case

Species B

dN/ N * dt (per capita)

mB

*

*

RB

RA

Resource, R

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

dN/dt < 0

dN/dt > 0

R1

*

R2

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

dN/dt < 0

R1

Consumption vectors can be of any slope, but the slope predicted under optimal foraging would equal R2/R1

*

*

*

*

R2

R1

Tilman’s Resource-Based Competition Models

Now consider the growth response of one species to two essential resources

The two R*s divide the region into portions favorable and unfavorable to population growth

Zero Net Growth Isocline (ZNGI)

R2

dN/dt > 0

dN/dt < 0

If a population deviates from the equilibrium along the ZNGI, it will return to the equilibrium

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

B

1

3

2

R2

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

B

1

2

R2

?

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 consuming the two

resources

A

B

1

2

R2

?

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

In this case, it is stable

Slope of

consumption

vectors for A

A

B

Slope of

consumption

vectors for B

1

3

2

R2

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

In this case, it is stable

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

Slope of

consumption

vectors for A

A

B

Slope of

consumption

vectors for B

1

3

2

R2

4

5

6

R1

Each species will return to its equilibrium if displaced on its ZNGI

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

In this case, it is unstable

Slope of

consumption

vectors for B

A

B

Slope of

consumption

vectors for A

1

3

2

R2

4

5

6

R1

Tilman’s Resource-Based Competition Models

resources

In this case, it is unstable

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

Slope of

consumption

vectors for B

A

B

Slope of

consumption

vectors for A

1

3

2

R2

4

5

6

R1

Each species will return to its equilibrium if displaced on its ZNGI

Consumption

vectors

Resource

supply point

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

Spatially Explicit, Neighborhood Models of Plant Competition

There are many ways to formulate these models, and most requirecomputer-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”

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 intraspecificcompetition relative to interspecificcompetition

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 below-ground 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

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 abundanceLoyn 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 geckoPetren 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

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

Lepidodactylus lugubris,

asexual native on

south Pacific islands

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

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

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