1. Competition
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Photo of hyenas and lioness at a carcass from

2. Pairwise Species Interactions
Influence of species A
- (negative)
0 (neutral/null)
+ (positive) A B
Competition - A B
Amensalism - A B
Antagonism
(Predation/Parasitism) + - - A B
Amensalism - A B
Neutralism
(No interaction) A B
Commensalism +
Influence of Species B
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Note that whereas +/- interactions are labeled “antagonism” in the table above, your textbook refers to these types of interactions as “exploitation.”
Abrahamson, Warren G., ed Plant-Animal Interactions. McGraw-Hill Publishing, New York, NY.
Morin, Peter J Community Ecology. Blackwell Science, Inc., Oxford, U.K. A B
Antagonism
(Predation/Parasitism) - + A B
Commensalism + A B
Mutualism + +
Redrawn from Abrahamson (1989); Morin (1999, pg. 21)

3. Intra-specific vs. Inter-specific Competition
Interaction between individuals in which each is harmed by their
shared use of a limiting resource (which can be consumed or depleted)
for growth, survival, or reproduction
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Note that resources contrast with other environmental factors that can limit organisms, but are not consumed nor are depleted, e.g., pH. Space can be a resource. Oxygen can be a resource in aquatic ecosystems.
Photo of hyenas and lioness at a carcass from

4. Intra-specific vs. Inter-specific Competition
“Complete competitors cannot coexist.”
(Hardin 1960)
Paramecium
aurelia
Paramecium
caudatum
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From your textbook: “Competing species are more likely to coexist when they use resources in different ways.”
Intraspecific competition causes logistic growth.
Interspecific competition causes competitive exclusion of P. caudatum when with P. aurelia.
Hardin, Garrett The competitive exclusion principle. Science 131:
Gause, G. F Experimental analysis of Vito Volterra’s mathematical theory of the struggle for existence. Science 79:16-17.
Cain, Bowman & Hacker (2014), Fig , after Gause (1934); photomicrographs from Wikimedia Commons

5. Intra-specific vs. Inter-specific Competition
Resource partitioning – differences in use of limiting resources –
can allow species to coexist
P. aurelia & P. caudatum ate mostly floating bacteria;
P. bursaria ate mostly yeast cells on the bottoms of the tubes
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From your textbook: “Competing species are more likely to coexist when they use resources in different ways.”
Intraspecific competition causes logistic growth.
Interspecific competition causes competitive exclusion of P. caudatum when with P. aurelia.
Gause, G. F Experimental analysis of Vito Volterra’s mathematical theory of the struggle for existence. Science 79:16-17.
Cain, Bowman & Hacker (2014), Fig , after Gause (1934)

6. Lotka – Volterra Phenomenological Competition Models
Alfred Lotka & Vito Volterra
( )
( )
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Wikipedia “Vito Volterra” page; accessed 02-X-2014
Photo of Lotka from photo of Volterra from Wikimedia Commons

7. Lotka – Volterra Phenomenological Competition Models
Lotka-Volterra Competition Equations:
Logistic population growth model – growth rate is
reduced by intraspecific competition:
Species 1: dN1/dt = r1N1[(K1-N1)/K1]
Species 2: dN2/dt = r2N2[(K2-N2)/K2]
Functions added to further reduce growth rate
owing to interspecific competition:
Species 1: dN1/dt = r1N1[(K1-N1-f(N2))/K1]
Species 2: dN2/dt = r2N2[(K2-N2-f(N1))/K2]
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Note that in the logistic population growth model, growth rate is reduced by something related to population density (predation, parasitism, competition) – that can be assumed to be “intraspecific competition” for the sake of the phenomenological model.
Note that the presentation of the logistic equation is a bit different from the textbook, which uses:
dN/dt = rN(1-(N/K))
K results from intraspecific competition (i.e., we model intraspecific competition with K).
Then, when terms for the other species are added to the population models, imagine what would happen at K for a given species (the addition of the competitor would make the pop. growth rate go negative).

8. Lotka – Volterra Phenomenological Competition Models
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
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Note: sometimes the first equation’s alpha is labeled α12 for the per capita effect on Species 1 of Species 2.
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

9. Lotka – Volterra Phenomenological Competition Models
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) ^
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)
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dN1/dt = r1N1[(K1-N1-αN2)/K1]
0 = r1N1[(K1-N1-αN2)/K1]
0 / r1N1 = [r1N1[(K1-N1-αN2)/K1]] / r1N1
0 = (K1-N1-αN2)/K1
0 * K1 = [(K1-N1-αN2)/K1] * K1
0 = K1-N1-αN2
N1 = K1-αN2

10. Lotka – Volterra Phenomenological Competition Models
The equations for equilibrium solutions become:
Species 1: N1 = [K1 - αK2] / [1 - αβ]
Species 2: N2 = [K2 - βK1] / [1 - αβ] ^ ^
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)
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Solve previous slide’s equations for N (use algebra).
But they do not provide much insight into the dynamics of competitive interactions, e.g., are the equilibrium points stable?

11. Lotka – Volterra Phenomenological Competition Models
4 time steps
State-space graphs help to track population trajectories (and assess stability) predicted by models
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Gotelli, N A Primer of Ecology. Sinauer, Sunderland, MA.
From Gotelli (2001)

12. Lotka – Volterra Phenomenological Competition Models
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
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From Gotelli (2001)

13. Lotka-Volterra Model
Remember that equilibrium solutions require dN/dt = 0
Species 1: N1 = K1 - αN2 ^
Therefore:
When N2 = 0, N1 = K1
K1 / α
When N1 = 0, N2 = K1/α
Isocline for Species 1
dN1/dt = 0 N2
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Zero net growth isocline = zero population growth isocline (textbook’s term) = isocline
When N2 = 0, N1 is at its carrying capacity.
If α>1 (i.e., intersp. > intrasp. comp.), then (K1/α) < K1
If 0<α<1 (i.e., intrasp. > intersp. comp.), then (K1/α) > K1
N1 = K1 - αN2
0 = K1 - αN2
αN2 = K1
N2 = K1 / α K1 N1

14. Lotka-Volterra Model
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
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K2 / β N1

15. Competitive exclusion of
Lotka-Volterra Model
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 N2 K2
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Solve the first 2 equations for K1 and K2, respectively, to give the inequality for Sp. 1 and Sp.2, respectively. Remember that K represents intraspecific competition.
= stable equilibrium
K2 / β K1 N1

16. Competitive exclusion of
Lotka-Volterra Model
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
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= stable equilibrium K1
K2 / β N1

17. Competitive exclusion with an
Lotka-Volterra Model
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
Competitive exclusion with an
unstable equilibrium K2
K1/ α N2
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Starting conditions determine the winner.
= stable equilibrium
K2 / β K1
= unstable equilibrium N1

18. Coexistence at a stable equilibrium
Lotka-Volterra Model
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
Coexistence at a stable equilibrium
K1 / α N2 K2
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Note that intraspecific competition must be greater than interspecific competition in both species for this to happen.
Note also that the equilibrial densities are less than the carrying capacities of the two species.
= stable equilibrium K1
K2 / β N1

19. Mechanisms of Competition Exploitation competition
Dissecting exploitation competition reveals its indirect nature H - H - - + + P
Interference competition (direct aggression, allelopathy, etc.)
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For this figure I am using P=plant & H=herbivore.
From the phenomenological definition of competition, if 1 herbivore increases and the other decreases (and vice versa) we would consider this competition.
So, under exploitation competition, indirect competition results in these reciprocal changes mediated by the resource (plant [or could be prey]) species.
Menge, Bruce A Indirect effects in marine rocky intertidal interaction webs: Patterns and importance. Ecological Monographs 65:21-74. H - H P - P
Solid arrows = direct effects; dotted arrows = indirect effects
Redrawn from Menge (1995)

20. Mechanisms of Competition
David
Tilman
Synedra
Asterionella
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Tilman, David et al Competition and nutrient kinetics along a temperature gradient: an experimental test of a mechanistic approach to niche theory. Limnology & Oceanography 26:
Cain, Bowman & Hacker (2014), Fig. 12.4, after Tilman et al. (1981); photos of diatoms from Wikimedia Commons;
photo of Tilman from

21. Mechanisms of Competition
David
Tilman
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Tilman, David et al Competition and nutrient kinetics along a temperature gradient: an experimental test of a mechanistic approach to niche theory. Limnology & Oceanography 26:
Cain, Bowman & Hacker (2014), Fig. 12.4, after Tilman et al. (1981);
photo of Tilman from

22. Asymmetric vs. Symmetric Competition
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Cain, Bowman & Hacker (2014), Fig. 12.7

23. Classic Pattern Interpreted as Evidence for Competitively-Structured Assemblages
Robert MacArthur
( )
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Wikipedia “Robert MacArthur” page; accessed 02-X-2014
Kaspari, Michael Knowing your warblers: thoughts on the 50th anniversary of MacArthur (1958). Bulletin of the Ecological Society of America. October:
MacArthur, Robert H Population ecology of some warblers of northeastern coniferous forests. Ecology 39:
Painting of “MacArthur’s warblers” by D. Kaspari for M. Kaspari (2008); anniversary reflection on MacArthur (1958)

24. Character Displacement
The “Ghost of Competition Past”
(sensu Connell 1980) is hypothesized to be the cause of the beak size difference on
Pinta Marchena
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G. fuliginosa was absent from Daphne Island & G. fortis was absent from Los Hermanos Island when these data were collected.
Character displacement occurs when directional selection operates in each species, but in opposite directions in the two species.
Connell, Joseph H Diversity and the coevolution of competitors, or the ghosts of competition past. Oikos 35:
Lack, David Darwin’s Finches. Cambridge University Press, Cambridge, UK.
Cain, Bowman & Hacker (2014), Fig , after Lack (1947)