The quantitative theory of competition was developed by Vito Volterra and Alfred Lotka in 1925-6.

Once more there are separate differential equations - here for the growth of the two competing species. Each is a logistic equation. Begin with equations for each species growing alone.

The equation for species 1growing alone:

The equation for species 2growing alone:

dN

dtr N

K N

K1

1 11 1

1

dN

dtr N

K N

K2

2 22 2

2

The growth curves for the two species, each growing alone…

The equations basically indicate that a portion of the resource that could have contributed to the carrying capacity for each species growing alone is used up by the competing species. Competition is incorporated by specifying constants, called the competition coefficients, that represent the strength of the competitive interaction

There are two coefficients, each representing the effect of one species on the other.

We add them to the two logistic equations as follows:

dN

dtr N

K N N

K

and

dN

dtr N

K N N

K

11 1

1 1 1 2 2

1

22 2

2 2 2 1 1

2

,

,

Note that these are standard logistic growth equations with a new, additional term in each equation. The new terms are 1,2N2 and 2,1N1 .

The new terms in the equations are: in the growth equation for species 1 -

α1,2N2 which indicates how much of the needed resource is used by species 2 individuals. The competition coefficient indicates the relative amount used by species 2 individuals compared to species 1 (alternatively phrased: converting species 2 individuals into an equivalent number of species 1 individuals in resource use…

and in the growth equation for species 2 –α2,1N1 which indicates how much of the needed

resource is used by species 1 individuals, with the competition coefficient indicating the relative amount used by species 1 individuals compared to species 2.

A Graphical Analogy for Interspecific Competition

Frame represents thecarrying capacity forspecies 1 (K1)

Species 1

Species 2

• Each individual consumes a portion of the limitedresources and is represented by a tile.

• Individuals of sp. 2 reduce the carrying capacity 4 timesas much as sp. 1. Hence, the tiles of sp. 2 are 4 times

the size of sp. 1 AND = 4.0After Krebs 1985

α1,2 is a proportionality coefficient to indicate the effect of one individual of species 2 on the growth of species 1…

and α2,1 is a proportionality coefficient to indicate the effect of one individual of species 1 on the growth of species 2.

The intensity of competition increases as values for the coefficients increase. Note that if the α values are both 0, then both species grow logistically.

In parallel, values of α > 1 indicate very intense competition.

In the diagram, each individual of species 2 uses as much of the resource as 4 individuals of species 1, thus the competition coefficient for the effect of species 2 on species 1, 1,2 = 4.

Now we do the same thing we did to the predator and prey equations to determine what population sizes produce an equilibrium, i.e. both dN1/dt = 0 and dN2/dt = 0.

For each equation, the only way the right hand side can = 0 is if the terms in the numerator add to 0, i.e.

dN

dtr N

K N N

K

and

dN

dtr N

K N N

K

11 1

1 1 1 2 2

1

22 2

2 2 2 1 1

2

0

0

,

,

N1* = K1 – α1,2N2

andN2

* = K2 – α2,1N1

What these equations indicate is that competition has reduced the carrying capacity (the equilibrium value for the simple logistic equation) by a term that measures the resource use of the other species (population size x the appropriate proportionality coefficient).

We saw in the logistic that the per head growth rate declined from r at vanishingly small population density to 0 at the carrying capacity K. With competition, the whole line is lowered by an amount α1,2N2/K1 for species 1 and correspondingly for species 2. How far it is depressed clearly depends on the size of the α.

Effect of a high value for 1,2

We can also (as in predator-prey models) use a phase plane diagram to see what happens. The population size for species 2 is on the Y axis, and that for species 1 on the X axis. The line indicates how the equilibrium for species 1 (the N* from the equilibrium equations) changes as the size of species 2 changes. As arrows indicate, in the yellow area species 1 increases, outside the line it decreases…

N.B. Note the differencein the axes between thisand the previous graph-This graph is of population size, notgrowth rate.

The explanation:

Recall, N1* = K1 - N2

Set N1 = 0 Set N2 = 0

1,2 N2* = K1

N1* = K1

N2* = K1/ 1,2

dN1 /dt = 0

N1K1

N2

K1

Similarly, for species 2…

And the parallel explanation for species 2…

K2

N2

K2

N1

dN2 /dt = 0

Recall, N2* = K2 -

Set N1* = 0 Set N2

* = 0

2,1 N1 = K2 N2*

= K2

N1* = K2/ 2,1

2,1 N1

If we plot both zero growth isoclines together on the same plot, we can see that there are four possible outcomes…

In case it isn’t obvious, the four outcomes are:

1. Species 1 wins and species 2 is excluded (competitive exclusion)

2. Species 2 wins and species 1 is excluded

3. Both coexist with a stable equilibrium

4. Coexistence is possible, but unlikely because the equilibrium point is unstable (neutral stability). According to this prediction, either species can win. The ‘winner’ is determined by which species has a slight edge, for example a larger r, N, K, or α, or the specific starting populations.

Two types of equilibrium:

Stable

Unstable

Assumptions of the Lotka-Volterra model for competition:

(Remember that this model is based on the logistic equation, so its assumptions still apply.)

1. All individuals within each species are equivalent.2. Constant r and K3. No time lags in density responses4. α1,2 and α2,1 are constant, and independent of population

size.

The Lotka-Volterra model has been criticized for a number of its approaches and underlying assumptions:

1. It assumes competition itself is density-independent. The coefficients are constant across population sizes.

2. The equations really only function well for two species interaction.

3. The equations assume environmental constancy and logistic growth in the absence of competition.

4. The approach ignores other ecological processes/interactions and their possible effects on the competition.

The Lotka-Volterra model for competition uses the results, i.e. the population sizes of the species, but does not deal explicitly in the mechanism – resource use. David Tilman has developed an alternative modeling framework based directly on resource use. It is again graphical, but now the axes are abundances of resources.

To prevent obvious competitive exclusion due to total niche overlap, there must be two resources (1 and 2), and we can see how a species does. There are now isoclines (abundances) of each resource below which a population will decline.

In the purple area, population size of the species under study decreases. Along the left margin, resource 1 has been drawn down so low that decline occurs; along the bottom it is resource 2 that has been drawn down.

This is a diagram for one species. What if we look at two competing species on the same plot?

In these plots, representing different outcomes, the blue line is the zero growth isocline for species A, and the red dashed line the zero growth isocline for species B. In the plot on the left, species A can draw resources down to a level below that suitable for B; species A wins. In the middle panel the opposite occurs. On the right, the two species coexist, and, at the point where the isoclines cross, there is a stable equilibrium.

A Review of the Outcomes of Theory:

Both the Lotka-Volterra and Tilman models for competition have multiple outcomes.

In each model there are 2 outcomes in which one species wins and the other is excluded.

In both models there are outcomes in which both species persist.

In both the Lotka-Volterra and Tilman models, one outcome is a stable coexistence.

In the Lotka-Volterra model, there is also an outcome which predicts unstable coexistence.

The Populus model allows you to see the numerical changes in populations that predict these various outcomes. First, stable coexistence…

Species 1: N0 = 10, r = 0.9, K = 500, α1,2 = 0.6Species 2: N0 = 20, r = 0.5, K = 700, α2,1 = 0.7

Competitive exclusion (here the difference in K explains the result):

Species 1: N0 = 10,r = 0.7,K = 500,α1,2 = 0.6Species 2: N0 = 10,r = 0.5,K = 1000,α2,1 = 0.7

We can also get competitive exclusion when the α values are much different (assuming other values are the same or nearly so). The species with the larger effect (higher α) wins:

Species 1: N0 = 10, r = 0.7, K = 500, α1,2 = 0.6Species 2: N0 = 10, r = 0.7, K = 500, α2,1 = 1.3

Finally, unstable coexistence… either species could win. We determine the result by setting the initial values, here a difference of 1 in the starting populations.

Species 1: N0 = 11, r = 0.7, K = 500, α1,2 = 1.3Species 2: N0 = 10, r = 0.7, K = 500, α2,1 = 1.3

Now some examples of experiments documenting the kinds of results that theory predicts…

Example 1: Bruce Menge, from Oregon State University, examined the competition between two species of starfish.

Both species eat clams.They live together in tidepools at the edge of the Pacific.One of the two species is large, and the other species is small.

Menge hypothesized that the larger starfish was dominant in competition with the smaller species, and got more of the clams.

He did a removal experiment. From some tidepools he removed the larger species, and other pools were left as controls. Then he compared the weights of the smaller species in removal and control pools. The results…

Treatment Weight of the smaller species

Control No change in body weight

Removal 30% gain in body weight

His hypothesis was supported.

One of the classic experiments documenting competition, used in every first year text, is Connell’s study of competition between barnacles, Balanus balanoides and Chthamalus stellatus, in the intertidal zone on the rocky coast of Scotland.

Barnacles are sessile, filter-feeders.

For them, space is the limiting factor, not food.

When larvae settle on the rocks, the distributions of the two species almost totally overlap.

Typically, adult individuals of Chthamalus are found higher in the intertidal zone than are Balanus.

The two species differ in size. Balanus averages about 15mm, while Cthamalus is much smaller, around 6-7mm.

The reason for the different adult distributions between species, and compared to larval distributions, is interference competition.

Connell found that physiological tolerances differed little.

Balanus have heavier shells and grow more rapidly. As their shells grow, they edge underneath the shells of the Cthalamus and literally pry them off the rock.

Chthamalus persists in the upper region of the intertidal zone because it is more tolerant of the exposure and dessication that occurs there. Its access to the lower regions is subject to interference from Balanus.

Are there examples that seem to indicate unstable equilibria?

At least one classic study – Park’s studies of two species of flour beetles, Tribolium confusum and Tribolium castaneum.

When grown together in flour, the two species did not coexist for very long in any particular jar. However, which species won, eliminating the other, seemed to vary almost randomly among jars. His result was called competitive indeterminacy.

The result was probably determined by very small differences in r or K in individual cultures. We know N, since he started cultures with identical numbers of the two species.

The environment also played an important role.

Humidity and temperature in the flour were important in determining which species won.

Here’s a table of some of Park’s results:Temp. R.H. Climate Mixed species % win

T. confusum T. castaneum 34 70 hot-moist 0 100 34 30 hot-dry 90 10 29 70 temperate-moist 14 86 29 30 temperate-dry 87 13 24 70 cold-moist 71 29 24 30 cold-dry 100 0

One species wins when its warm & moist, the other when its cold & dry.