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Transcript
Competition
• Different kinds of competition
• Modeling competition
• Examples of competition-case studies
• Understanding the role of competition
Competition
• The outcome of competition is that an
individual suffers a reduction in fecundity
(ability to reproduce), growth or
survivorship
• Competition is for a limiting resource
Intraspecific competition
(between members of own species)
Bald Eagles with disagreement over life mate
Olympic ski cross
Olympic short track skaters disagree on
potential podium position
Snails in deathmatch over habitat
Interspecific competition
(between members of different species)
Eagle vs fox - fight for control of deer carcass
Yoda vs Darth Vader
-fight for control of coffee shop aboard the
Death Star
Types of competition
Resource exploitation - indirect
White pine roots are
better able to take up
moisture and nutrients
compared to other plants.
This prevents other plants
from accessing the limited
resources. White pine
indirectly suppresses the
other plants.
fig 13.2 Molles and Cahill
Humans have become super efficient at taking up
fish from oceans and suppress other animals that
depend on fish resources.
Interference competition-direct competition
Damsel Fish aggressively maintain territories.
fig 13.1 Molles and Cahill
Humans aggressively maintain territories.
Interference competition
___________________________________________
Exploitative competition
• Competition is NOT always the most
important type of interaction between or
amongst members of species
•
•
•
Abiotic (environmental) stress
Predation
Parasites
May individually or in combination play a
greater role in limiting fitness, population
dynamics and community structure
•
Mathematical Modeling of Competition
Why??
What is a model?
• Simplification of a system (nature)
• It is NOT a working facsimile!
• Used to gain insight into how things work
• Should be testable and modifiable
“A model should be as simple as possible
but not simpler! “
Albert Einstein
What does this mean?
Make the irreducible basic elements of your model
as simple and as few as possible without having to
give up the adequate representation of what you are
modeling
From last class. . .
Molles 2008
(page 261)
cf. Molles & Cahill,
2008, p. 316
dN =r N
max
dt
( K-N
K )
(
)
(
)
dN =r N K1-N1
max 1
dt
K1
dN =r N K2-N2
max 2
dt
K2
Logistic model for population
growth
Change in population growth of
Species 1
Change in population growth of
Species 2
In these models population growth slows as N increases
since resource supplies decrease with as population
increases
N1
K1 = Relative level of intraspecific competition
But resource levels can also be reduced by
interspecific competition
Lotka Volterra Models of Interspecific Interations
Lotka Volterra Models
dN1
dt =rmax1 N1
dN2
dt =rmax2 N2
( K - N K- α
1
12N2
1
1
( K - N K- α
2
21N1
2
2
)
)
These equations are same as
before but now include the
effect of interspecific
competition (in red).
We now can simultaneously
model the effect of
individuals of the same
species and individuals of
competing species on the
rate of growth of a
population
Lotka Volterra Models
a12 and a21 are competition coefficients that modify the
effect of N2 and N1
(remember N1 and N2 are population sizes)
dN1
dt =rmax1 N1
( K - N K- α
dN2 =r
dt max2 N2
1
12N2
1
1
( K - N K- α
2
)
21N1
2
2
)
α12 = effect of an individual of species 2 on the
rate of population growth of species 1
α21= effect of an individual of species 1 on the
rate of population growth of species 2
a12 < 1 means that the individuals of species 2 have less effect on
individuals of species 1 than individuals of species 1 have on others of
their own species.
a12 >1 means that the individuals of species 2 have more effect on
individuals of species 1 than individuals of species 1 have on others of
their own species.
Lotka Volterra Models
Predicts:
i) coexistence of two species when
for both species interspecific
competition is weaker than
intraspecific competition
ii)If the above condition is NOT
met than one species will eventually
exclude the other
Lotka Volterra Models
Can we determine under what conditions these species are predicted to coexist and
under what conditions one species will exclude the other?
To do this we determine equilibria: population sizes for species 1 and 2 for which
population growth of both species will be zero.!
If population growth is zero, then the population sizes do not change over time, and
we have an equilibrium (a situation in which conditions remain the same over time.)
dN1
dt =rmax1 N1
( K - N K- α
12N2
)= 0
dN2 =r
max2 N2
dt
( K - N K- α
21N1
)= 0
1
1
1
2
2
2
Populations stop growing when:
dN1
dt =rmax1N1
( K - N K- α
dN2
dt =rmax2 N2
1
12N2
1
1
( K - N K- α
2
)= 0
21N1
2
2
)= 0
This occurs when
0 = (K1 - N1 - α12N2)
0 =(K2 - N2 - α21N1)
Populations stop growing when:
0 = (K1 - N1 - α12N2)
0 =(K2 - N2 - α21N1)
On further rearranging. . .
N1 = K1-α12N2
N2 = K2-α21N1
N1 = K1-α12N2 and N2 = K2-α21N1
are really simple equations to describe straight
lines! You may recall from high school . . .
y=slope(x)+ b
N1 is y
Effects of species 2 on species 1 (a12) is slope
N2 is the x value
K1 is y intercept (when x = 0)
At every point along these
lines growth is stopped
Arrow up means growth of species 1
Arrow down means decrease of
species 1
The line is called the zero growth isocline for species 1:
it represents all combinations of N1 and N2 for which
growth of N1 is zero.
Any combination of species 1 and 2 below the line means
that species 1 will increase
Any combination of species 1 and 2 above the line means
that species 1 will decrease
Similarly. . . .
Arrow right means growth of species 2
Arrow left means decrease of species 2
There are four ways to plot these two
species together
Species 1 increasing
while species 2 is
decreasing
Each red arrow is a vector of
the combined direction of
both species.
Example a:! The species 1 isocline is above the species 2 isocline.
Below both isoclines, species 1 and 2 both increase.!
In the range of the graph between the two isoclines, we are above the
species 2 isocline so it (sp2) decreases, but we are also below the species 1
isocline so it (sp1) continues to increase.!
The result is that species 2 declines to zero and species 1 increases to its
carrying capacity.! In this case species 1 has competitively excluded species
2.
K1>K2/α21
so! K1α21>K2
This means that when species 1 is at its carrying capacity, its impact
on species 2 (measured by K1 times a21) is greater than the impact of
K2 individuals of species 2.!
Thus, species 1 is affecting species 2 more negatively than species 2
affects itself.!
Interspecific competition regulates species 2 more than species 2 is
regulated by intraspecific competition.
Example b:! The species 2 isocline is above the species 1 isocline.
From the N2 axis:
K2>K1/α12
and so K2α12>K1
K1>K2/α21
so! K1α21>K2
Thus, species 2 is affecting species 1 more negatively than species 1 affects itself.!
Example c): Isoclines for the two species cross; the K values on each axis are higher than the K/a values
In this situation, from the N1 axis:
K1>K2/α21
thus K1α21>K2
Indicating higher impact of
interspecific competition than
intraspecific competition on species 2
BUT
From the N2 axis:
K2>K1/α12
so K2α12>K1
Indicating higher impact of
interspecific competition than
intraspecific competition on species 1.
Thus, either species could exclude the other species. Here interspecific
competition is stronger than is intraspecific competition.
Example d: Isoclines for the two species cross; the K values on each axis are lower than the K/a values
From N1 axis:
K2/a21>K1
thus
K2>K1a21
Indicating that species 2 is regulated more by intraspecific
competition than by interspecific competition
From the N2 axis:
K2<K1/α12
so K2α12<K1
Indicating that species 1 is regulated more by intraspecific
competition than by interspecific competition
So we can see that when each species is regulated more by
intraspecific competition rather than by competition with the other
species, the two species can coexist.