Transcript Slide 1 - University of Windsor

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 1
growing alone:
 K1  N 1 
dN1

 r1 N 1 
dt
 K1 
The equation for species 2
growing alone:
 K2  N 2 
dN 2

 r2 N 2 
dt
 K2 
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:
 K1  N 1  1,2 N 2 
dN 1

 r1 N 1 
dt
K1


and
 K 2  N 2   2 ,1 N 1 
dN 2

 r2 N 2 
dt
K2


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 the
carrying capacity for
species 1 (K1)
Species 1
Species 2
• Each individual consumes a portion of the limited
resources and is represented by a tile.
• Individuals of sp. 2 reduce the carrying capacity 4 times
as much as sp. 1. Hence, the tiles of sp. 2 are 4 times
the size of sp. 1 AND 
= 4.0
After 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.
 K1  N 1  1,2 N 2 
dN 1
 0
 r1 N 1 
dt
K1


and
 K2  N 2   2 ,1 N 1 
dN 2
 0
 r2 N 2 
dt
K2


N1* = K1 – α1,2N2
and
N2* = 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 difference
in the axes between this
and the previous graphThis graph is of
population size, not
growth rate.
The explanation:
K1

N2
N1
K1
Recall, N1* = K1 -  N2
Set N1 = 0
1,2 N2* = K1
N2* = K1/ 1,2
Set N2 = 0
N1* = K1
Similarly, for species 2…
And the parallel explanation for species 2…
K2
N2
N1
K2

Recall, N2* = K2 - 2,1 N1
Set N1* = 0
N2* = K2
Set N2* = 0
2,1 N1 = K2
N1* = K2/ 2,1
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 K
3. No time lags in density responses
4. 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 = 0.6
Species 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 = 0.6
Species 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 = 0.6
Species 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 = 1.3
Species 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.