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competition.txt
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UNIT 6: Competition
----------------------------------------------------------------------
SEC Introduction
----------------------------------------------------------------------
OFFSLIDE Inter-species interactions
Competition: interaction hurts the growth rate of both species
NOANS
Exploitation: interaction is good for one species but bad for the
other
NOANS
Mutualism: interaction is good for both species
NOANS
Commensalism: interaction is good for one species, and close to
neutral for the other
NOANS
CHANGE Ask for examples.
----------------------------------------------------------------------
Competition
Competition occurs when two species both depend on the same
resource, or resources
Each species' ability to reproduce successfully is reduced by the
presence of the other
Via effects on any component of successful reproduction:
ANS Survival, growth, producing offspring
Species may be very similar, or very different
ANS Oaks and maples competing for light
ANS Ants and mammals competing for leaves
ANS Mussels and algae competing for space in the intertidal zone
----------------------------------------------------------------------
PSLIDE Competition
FIG webpix/larches.jpg
----------------------------------------------------------------------
PSLIDE Competition
BC
FIG webpix/leaf_cutters.jpg
NC
WIDEFIG webpix/tapir.jpg
EC
----------------------------------------------------------------------
PSLIDE Competition
FIG webpix/mussels_algae.jpg
----------------------------------------------------------------------
Competition in ecology
What factors determine which species survive in which habitats?
What factors determine how many similar species can co-exist?
Why do similar species coexist at all?
----------------------------------------------------------------------
Flour beetles
There is a series of experiments where researchers allow two
species of flour beetles to compete in different laboratory
environments
The larger species survives better in drier conditions, and the
smaller species reproduces faster in moister conditions
POLL What outcomes do you expect under wet vs dry conditions?
ANS Each species wins when conditions are better for it
POLL What if I ``tune" the conditions to something in between? What if I tune the conditions to something in between?
ANS The species could both survive together
ANS Sometimes one survives, and sometimes the other
ANS Whichever species got a ``head start'' would survive
----------------------------------------------------------------------
PSLIDE Outcomes of competition
FIG webpix/confusum.jpg
COMMENT The confused flour beetle
----------------------------------------------------------------------
Outcomes of competition
In a given stable environment, we generally expect the competitive
interaction between two species to have one of the following results
{\bf Dominance}: one species wins every time
{\bf Co-existence}: if both species are present, they will both
persist
{\bf Founder control}: whichever species gets established first
will exclude the other
----------------------------------------------------------------------
TSEC Population model with competition
We modeled a single species using the equation:
$\ds \frac{dN}{dt} = (b(N)-d(N)) N$
We want to modify this for a species which is competing
with another species
$\ds \frac{dN_1}{dt} = ? $
The amount of competition seen by species 1 is $\tilde N_1 = N_1 +
\alpha_{21} N_2$
How should our single-species equation change?
ANS $\ds \frac{dN_1}{dt} = (b_1(\tilde N_1) - d_1(\tilde N_1))
N_1 $
ANS $\ds \frac{dN_2}{dt} = (b_2(\tilde N_2) - d_2(\tilde N_2))
N_2 $
----------------------------------------------------------------------
Carrying capacity
For this unit, we will mostly ignore Allee effects
Therefore, we expect each species to converge to its \emph{carrying
capacity} $K$ (or $K_1$ and $K_2$) when it is alone
How do we define carrying capacity?
ANS The birth rate equals the death rate: $b(K) = d(K)$
----------------------------------------------------------------------
Carrying capacity with competition
$\ds \frac{dN_1}{dt} = (b_1(\tilde N_1) - d_1(\tilde N_1))
N_1 $
How can this population be at equilibrium?
ANS $\tilde N_1 = K_1$ (carrying capacity): the species has the
right amount of competitive pressure to make $\R=1$
ANS $N_1 = 0$ (extinction): the species is not present
----------------------------------------------------------------------
Logistic model
One popular approach (as we discussed in non-linear models)
As before, our model is similar to the logistic model, except:
Birth and death are tracked separately
We don't assume functions are straight lines
----------------------------------------------------------------------
SS Balanced competition
----------------------------------------------------------------------
Equal competition
If the $\alpha$s are both equal to one, we have equal competition.
This means that the competitive effect of an individual from either
species is the same.
If $\bar N = N_1 + N_2$, then:
$\ds \frac{dN_1}{dt} = (b_1(\bar N) - d_1(\bar N)) N_1 $
$\ds \frac{dN_2}{dt} = (b_2(\bar N) - d_2(\bar N)) N_2 $
What happens in this case?
ANS Competition is mediated by only one quantity, $\bar N$.
ANS Whichever species has a higher value of $K$ can survive at a density
where the other one can't
ANS Dominance!
----------------------------------------------------------------------
CSLIDE Dominance time plot
FIG competition/first.comp.Rout-2.pdf
----------------------------------------------------------------------
PSLIDE Dominance phase plot
DOUBLEFIG competition/first.comp.Rout-2.pdf competition/first.comp.Rout-0.pdf
----------------------------------------------------------------------
Dominance
DOUBLEFIG competition/second.comp.Rout-4.pdf competition/second.comp.Rout-0.pdf
----------------------------------------------------------------------
CSLIDE Dominance
DOUBLEFIG competition/second.comp.Rout-2.pdf competition/second.comp.Rout-0.pdf
----------------------------------------------------------------------
Time plots and phase plots
\emph{Time plots} have time on one axis and show population
quantities on another
Fixed parameters (usually)
Single starting points
\emph{Phase plots} have population quantities on both axes
Fixed parameters (usually)
Multiple starting points (usually)
Better for seeing overall pattern of results
Worse for seeing rates (how quickly things change)
----------------------------------------------------------------------
Reading phase plots
Log or linear (per capita vs.~total perspective)
Open circles are starting points
Closed circles are ending points
Arrows show direction of time
----------------------------------------------------------------------
Dominance again
DOUBLEFIG competition/third.comp.Rout-4.pdf competition/third.comp.Rout-0.pdf
----------------------------------------------------------------------
REPSLIDE What's the difference?
DOUBLEFIG competition/second.comp.Rout-4.pdf competition/second.comp.Rout-0.pdf
----------------------------------------------------------------------
Log plots and linear plots
We will look at \emph{population} quantities on either a \emph{log}
or \emph{linear} scale
Log plots show \emph{proportional} differences
Linear plots show \emph{absolute} differences
----------------------------------------------------------------------
Different scales
DOUBLEPDF competition/second.comp.Rout
----------------------------------------------------------------------
Units of $\alpha$
$\tilde N_1 = N_1 + \alpha_{21} N_2$;
$\tilde N_2 = N_2 + \alpha_{12} N_1$
$\alpha_{21}$ measures the strength of the competitive effect
\emph{of} individuals of species 2 \emph{on} the growth rate
of species 1.
What are the units of $\alpha_{21}$?
ANS $\indiv_1/\indiv_2$ (usually; depending on units of the $N$s)
Since $\alpha$ has units, we don't expect there to be anything
special about $\alpha=1$
Equal competition (both species have the same effect on each other)
is a special case of balanced competition (both species have the
same \emph{relative} effect on each other)
----------------------------------------------------------------------
Balanced competition example
Two plants compete with each other for water. The value of
$\alpha_{21}$ is $4 \indiv_1/\indiv_2$
POLL Which species is bigger? 1; 2
ANS $4 \indiv_1$ have as much impact as $1 \indiv_2$
ANS Species 2 individuals are bigger
If they're only competing for water, what's the value of
$\alpha_{12}$?
ANS $\alpha_{12} = 1\indiv_2/4\indiv_1$
ANS $1 \indiv_2$ has as much impact as $4 \indiv_1$
ANS The two ratios convey the same information in this case
----------------------------------------------------------------------
PSLIDE Balanced competition example
FIG webpix/water_compete.jpg
----------------------------------------------------------------------
Balanced competition
POLL What results do we expect from balanced competition? Bigger species
wins; founder effects; coexistence; something else
ANS Balanced competition works just like equal competition
ANS Both species experience total density in the same way
ANS So the species with the higher carrying capacity (compared
using the same units) will dominate
ANS This is not necessarily the bigger species
If competition is balanced, there is no tendency for founder
control or for coexistence
----------------------------------------------------------------------
Measuring competitive effects
It makes sense that we have a range of parameters that give us
balanced competition, because we know qualitative changes in
dynamics are explained by unitless parameters
What's the unitless parameter here?
ANS $C = \alpha_{21}\alpha_{12}$
$C$ measures the relative effect of between-species and
within-species competition
$C=1$ means competition is balanced
$C<1$ means there is more competition within species (tendency
for coexistence)
$C>1$ means there is more competition between species (tendency
for founder control)
----------------------------------------------------------------------
TSS Unbalanced competition
BC
If two species are competing by using a simple resource, we expect
competition to be balanced
Both $\alpha$s measure the relative effect of the two species on
the resource
In more realistic situations, competition may not be balanced
NC
SIDEFIG webpix/toe_balance.jpg
EC
----------------------------------------------------------------------
Coexistence
Coexistence \emph{may} occur when $C<1$
POLL Why might individuals have relatively weaker competitive interactions with members of the other species?
ANS They may compete for mates or mating sites
ANS Example: birds with different nesting preferences
ANS Organisms may use resources in different ways
ANS Trees may produce leaves at different times
----------------------------------------------------------------------
PSLIDE Coexistence
FIG webpix/cliff_swallows.jpg
----------------------------------------------------------------------
PSLIDE Coexistence
FIG webpix/robins_nest.jpg
----------------------------------------------------------------------
CSLIDE Coexistence
FIG competition/coexist.comp.Rout-7.pdf
----------------------------------------------------------------------
Coexistence
DOUBLEFIG competition/coexist.comp.Rout-2.pdf competition/coexist.comp.Rout-7.pdf
----------------------------------------------------------------------
PSLIDE Coexistence phase plot
FIG competition/coexist.comp.Rout-0.pdf
----------------------------------------------------------------------
PSLIDE Coexistence phase plot (log scale)
FIG competition/coexist.comp.Rout-1.pdf
----------------------------------------------------------------------
Coexistence phase plots
DOUBLEFIG competition/coexist.comp.Rout-0.pdf competition/coexist.comp.Rout-1.pdf
----------------------------------------------------------------------
Founder control
Founder control \emph{may} occur when $C>1$
POLL Why might individuals have relatively stronger competitive interactions with members of the other species?
ANS Conspecifics might co-operate to defend resources
ANS Example: dogs and leopards
ANS Organisms might change the environment in a way that favors
their own species
ANS Example: trees and grasses
----------------------------------------------------------------------
PSLIDE Founder control
FIG webpix/wild_dogs.jpg
COMMENT Co-operation
----------------------------------------------------------------------
PSLIDE Founder control
FIG webpix/leopard.jpg
COMMENT Co-operation
----------------------------------------------------------------------
PSLIDE Founder control
FIG webpix/savanna_burn.jpg
COMMENT Changing the environment
----------------------------------------------------------------------
PSLIDE Founder control
FIG competition/mutual.comp.Rout-3.pdf
----------------------------------------------------------------------
PSLIDE Founder control
FIG competition/mutual.comp.Rout-5.pdf
----------------------------------------------------------------------
Founder control
DOUBLEFIG competition/mutual.comp.Rout-3.pdf competition/mutual.comp.Rout-5.pdf
----------------------------------------------------------------------
PSLIDE Founder control phase plot
FIG competition/mutual.comp.Rout-0.pdf
----------------------------------------------------------------------
PSLIDE Founder control phase plot (log scale)
FIG competition/mutual.comp.Rout-1.pdf
----------------------------------------------------------------------
Founder control phase plots
DOUBLEFIG competition/mutual.comp.Rout-0.pdf competition/mutual.comp.Rout-1.pdf
----------------------------------------------------------------------
Founder control can be complicated
WFIG 0.6 competition/mutual.comp.Rout-4.pdf
Founder control really means each species can win with a \emph{big
enough} head start
----------------------------------------------------------------------
REPSLIDE Founder control phase plots
DOUBLEFIG competition/mutual.comp.Rout-0.pdf competition/mutual.comp.Rout-1.pdf
----------------------------------------------------------------------
Results of competition
$C$ measures the relative effect of each species on each other, but
it doesn't reflect growth rates or how strongly each species is
affected by competition
$C$ may stay (about) the same, even as we switch conditions so that
one or the other species dominates
POLL $C$ tells us what will happen \emph{if} neither species is dominating. What are the possible results, other than dominance? What are the possible results, other than dominance?
ANS Founder effects, neutrality or coexistence
ANS Like when we tune the conditions so that neither species of flour
beetle wins
----------------------------------------------------------------------
SEC Population-level interactions
----------------------------------------------------------------------
TSS Invasion theory
The competitive relationship between two species can be
investigated by studying two \textbf{invasion} scenarios:
What happens if one species is established, and the other one tries
to invade (ie., some individuals are introduced)?
ANS Dominance occurs when one species can invade the other
ANS Coexistence occurs when each species can invade the other
ANS Founder control occurs when neither species can invade the
other
ANS In the absence of Allee effects, invasion is all you need to know
----------------------------------------------------------------------
ONSLIDE Allee effects
This analysis assumes that species that can be successful under a
certain competitive environment can also invade that environment
That is, it neglects Allee effects
Would this assumption work with Allee effects?
ANS No. With Allee effects a species may be able to do well if established, but not be able to ``invade" if it's rare
----------------------------------------------------------------------
Competitive results
The competitive effect felt by species 1 is measured by $\tilde
N_1$
The \emph{amount} of competition needed for species 1 to be at
equilibrium is:
ANS $\tilde N_1 = K_1$
The amount of competition species 1 feels when trying to invade a
population of species 2 is:
ANS $\tilde N_1 = \alpha_{21} N_2$
ANS = $\alpha_{21} K_2$, if species 2 is at equilibrium
If species 1 feels more competition from invading species two than
it feels at its own equilibrium, it cannot invade. And
\textbf{conversely}.
----------------------------------------------------------------------
Population-level competitive effects
The population-level competitive effect of species 2 on species one
is $E_{21} \equiv \alpha_{21} K_2/K_1$
This is the unitless ratio of the two measures of effect on
species 1 from the previous slide.
The two values of $E$ determine the competitive dynamics between
the two species.
If $E_{21} > 1$ species 2 can exclude species 1 (species 1 cannot
invade). And \textbf{conversely}.
----------------------------------------------------------------------
Results of competition
If both $E$s are $<1$, neither can exclude the other
ANS We expect coexistence
If both $E$s are $>1$, they both exclude each other
ANS which species wins will depend on starting conditions:
founder control
If one $E$ is $>1$, the large-$E$ species can exclude the other
ANS We expect that species to always win: dominance
----------------------------------------------------------------------
PSLIDE Results of competition
FIG competition/bifurcation.Rout-0.pdf
----------------------------------------------------------------------
PSLIDE Results of competition
FIG competition/bifurcation.Rout-1.pdf
----------------------------------------------------------------------
PSLIDE Results of competition
FIG competition/bifurcation.Rout-2.pdf
----------------------------------------------------------------------
PSLIDE Results of competition
FIG competition/bifurcation.Rout-3.pdf
----------------------------------------------------------------------
PSLIDE Results of competition
FIG competition/bifurcation.Rout-4.pdf
----------------------------------------------------------------------
PSLIDE Results of competition
FIG competition/bifurcation.Rout-5.pdf
----------------------------------------------------------------------
Results of competition
FIG competition/bifurcation.Rout-6.pdf
----------------------------------------------------------------------
Measuring competition
$\alpha$ measures competitive effects at the individual level
has units (ratios of types of individuals)
$E$ measures competitive effects at the population level, using
equilibrium populations
unitless
$C = \alpha_{21}\alpha_{12} = E_{21} E_{12}$
$C$ tells us: do the species have a \emph{tendency} for founder
control or coexistence?
For specific conditions, we also need to know values of $E$
Each species may dominate when conditions are good for it
We see the tendency for founder control or coexistence in
intermediate conditions
----------------------------------------------------------------------
OFFSLIDE Neutral competition
If competition is balanced, and neither species dominates, this is
called neutral competition
No tendency for either species to win
No tendency for founder control or for coexistence
If there's any small difference between the species, one may
dominate
Even if there's no difference, one should win eventually, by random
``drift"
----------------------------------------------------------------------
ONSLIDE Founder control
Up until now, we've thought of founder control as a single outcome
But from the point of view of the competing species, it's pretty
important which one of them gets control
POLL What factors determine who gets control?
ANS Who gets there first
ANS Initial maximum growth rate
ANS How strongly they affect each other
----------------------------------------------------------------------
ONSLIDE PSLIDE Growth rate and founder control
FIG competition/rmutual.comp.Rout-0.pdf
----------------------------------------------------------------------
ONSLIDE PSLIDE Growth rate and founder control
FIG competition/rmutual.comp.Rout-1.pdf
----------------------------------------------------------------------
ONSLIDE Growth rate and founder control
DOUBLEFIG competition/rmutual.comp.Rout-0.pdf competition/rmutual.comp.Rout-1.pdf
----------------------------------------------------------------------
TSS Colonization and co-existence
Up until now, we've thought about the question of which species
controls a particular area in the long term
But if available habitat is changing, it also matters what happens
in the short term
$rK$ tradeoff
$r$ strategists do better in the short term; $K$ strategists do
better do better in the long term
POLL When can you survive by doing better in the short term?
ANS When new opportunities (empty habitat) keep coming available
ANS When there is a lot of disturbance: fire, flood \ldots
----------------------------------------------------------------------
Growth rates
The maximum growth rate (for each species) is $r_0 = (b(0) - d(0))$:
ANS The net growth rate when crowding pressure is very low
The species with the better $r_0$ should do better in the short run
Faster exponential growth
If patches are very stable, then $K$ species wins
If they are very unstable, then $r$ species wins
In between, we get coexistence at the level of multiple populations
i.e., at the landscape level species may coexist
----------------------------------------------------------------------
rK tradeoff
FIG competition/rK.comp.Rout-4.pdf
----------------------------------------------------------------------
SEC Niches and coexistence
----------------------------------------------------------------------
Ecological niches
An ecological niche refers to the way an organism makes a living:
What resources does it need?
What sort of environmental conditions does it need?
----------------------------------------------------------------------
Fundamental niches
A {\bf fundamental} niche is defined as the conditions under which an
organism could make a living (in other words, survive with $\R>1$)
{\em in the absence of competition}.
Many plants have very large fundamental niches
The reason spruce trees don't grow in Cootes Paradise is not that
they can't grow there
ANS They can't compete with the other trees that
grow there
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Realized niche
BC
The realized niche is defined as the conditions under which an
organism can make a living, including the effects of competing
species
The realized niche of spruce trees does not include Cootes
Paradise
NC
SIDEFIG webpix/spruce.jpg
EC
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PSLIDE Example: chipmunks
FIG webpix/chipmunk.jpg
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Example: chipmunks
There are several species of chipmunks in the Sierra Nevada mountains
The most aggressive can only survive where the rainfall is good,
and it out-competes all the other species
The least aggressive can survive anywhere in the mountain range,
but it cannot co-exist with any of the other species
What are the fundamental and realized niches of these species?
ANS The aggressive species has the same fundamental and realized
niches: the places where rainfall is good
ANS The mild-mannered species has a large fundamental niche, but
its realized niche is the area that's too dry for the other
species
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TSS The competitive exclusion principle
If two species use resources in the same way, we expect that
$C=1$.
The effect of an individual of each species can be measured by
its impact on resources. If individuals of species one have
(e.g.) twice the impact, this should be seen by both species
equally.
If two species use resources in the same way, we do not expect them
to co-exist
One species will use the resources more efficiently (nothing in
biology is exactly equal)
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OFFSLIDE Exclusion and drift
Even if the two species were \emph{exactly} equal
in efficiency, we expect one species to go extinct at random
Due to the randomness of births and deaths, we expect the
proportions to fluctuate at random until one proportion reaches 0
and gets stuck there
We call this process ``drift", and it is strongly analogous to
genetic drift
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Competitive exclusion and biodiversity
Two species that use resources the same way cannot co-exist in a
stable environment in the long term due to their competitive
dynamics
This statement can be justified mathematically, and it has
important implications for real populations \ldots\
\ldots but it must also break down
POLL How? Why do we observe species co-existing?
ANS Species may not use resources in the same way
ANS The environment may not be stable
ANS Co-existence may not be ``long term"!
ANS There may be stabilizing factors outside competitive dynamics
(e.g., natural enemies, mutualists, competition for mates)