Transcript Powerpoint

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SELECTION
& DRIFT
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IN ECOLOGICAL COMMUNITIES
D. s = 0.05, J = 50
. s = 0.05, J = 500
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Time
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Frequency of Species A
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Frequency of Species A
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In the beginning…
We always had selection:
Per capita
competitive
effect of
sp. 2 on sp. 1
a11N1 + a12N2
dN1
= r1N1 1 K1
dt
Lotka-Volterra competition
If aij < 1, stable coexistence possible
K1/a12
N2
K2
K1
N1
K2/a21
Species
The Community Matrix (Levins 1968)
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a21
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a31
a32
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a41
a42
a43
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a51
a52
a53
a54
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Species
Some stuff people said based on analyses of
this type of model:
• There is a limit to the similarity in resource use
between coexisting species (MacArthur &
Levins 1967)
• Diversity (i.e., more species) destabilizes a
community (May 1972)
All based on analysis of equilibrium points in
deterministic models; focus on species differences
But there’s more than one way to be “different”…
“Fitness” differences: ri > rj, Ki > Kj
“Niche” differences: aij < 1
If large, coexistence less likely
If large, coexistence more likely
a11N1 + a12N2
dN1
= r1N1 1 K1
dt
a22N2 + a21N1
dN2
= r2N2 1 K2
dt
small fitness differences
Formalization in Chesson’s “Equalizing” vs.
“Stabilizing” mechanisms of coexistence
large niche differences
Coexistence happens when each
species tends to increase in relative
abundance when rare, so…
Population
growth rate
when rare
for species i
Chesson (2000, Ann. Rev. Ecol. Syst)
Fitness difference
between species i
and community
average
Niche difference
(1 – overlap)
Bottom line: A big fitness difference means you need a big
niche difference for coexistence (and vice versa)
Niche difference
(1 – overlap)
Population
growth rate
when rare
for species i
Chesson (2000, Ann. Rev. Ecol. Syst)
Fitness difference
between species i
and community
average
A simpler way to think about it (I think)…
FORMS OF SELECTION BETWEEN TWO SPECIES, A & B
FitnessA – FitnessB
A. Constant selection
+
expected dynamics (sp. A wins)
0
-
unstable
equilibrium
0
stable
equilibrium
Frequency of Species A
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FORMS OF SELECTION BETWEEN TWO SPECIES
A. Constant
unstable
equilibrium
stable
equilibrium
+
D. Complex frequency-dependence
FitnessA – FitnessB
-
+
B. Negative frequency-dependence
-
+
E. Neutral
-
+
C. Positive frequency-dependence
-
+
-
Frequency of Species A
Equalizing and Stabilizing Mechanisms of Coexistence
How much do you need to bend these lines (while maintaining average
height) to get stable coexistence?
FitnessA – FitnessB
Fitness difference large
Fitness difference small
+
+
-
only a little
a lot
+
+
-
-
Equalizing and Stabilizing Mechanisms of Coexistence
How much do you need to raise these lines (while maintaining bendiness)
to get rid of stable coexistence?
FitnessA – FitnessB
Niche difference large
Niche difference small
+
+
-
only a little
a lot
+
+
-
-
Prone to the influence of drift
Combining selection and drift
(Starting with pure drift)
# NEUTRAL MODEL (single community, no speciation)
# Set initial community (e.g., 25 individuals of sp. 1 + 25 of sp. 2; J = 50)
J <- 50 # must be an even number
COM <- vector(length=J)
COM[1:J/2] <- 1
COM[(J/2+1):J] <- 2
# set number of “years” to run simulations & empty matrix for data
num_years <- 50
prop_1 <- matrix(0,nrow=J*num_years,ncol=1)
# run model
for (i in 1:(J*num_years)) {
COM[ceiling(J*runif(1))] <- COM[ceiling(J*runif(1))]
prop_1[i] <- sum(COM[COM==1])/J
}
# plot results
plot(prop_1, type="l")
Combining selection and drift
# SELECTION-DRIFT MODEL (single community, no speciation)
# Set initial community (e.g., 25 individuals of sp. 1 + 25 of sp. 2; J = 50)
J <- 50 # must be an even number
COM <- vector(length=J)
COM[1:J/2] <- 1
COM[(J/2+1):J] <- 2
s <- 0.02 # the selection coefficient
# set number of “years” to run simulations & empty matrix for data
num_years <- 50
prop_1 <- matrix(0,nrow=J*num_years,ncol=1)
prop_1 <- 0.5
# run model
for (i in 2:(J*num_years)) {
death_cell <- ceiling(J*runif(1))
prob_1 <- (1+s)*prop_1[i-1]
if (runif(1) < prob_1) COM[death_cell] <- 1 else COM[death_cell] <- 2
prop_1[i] <- sum(COM[COM==1])/J
}
# plot results
plot(prop_1, type="l")
B. Pure drift, J = 50
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D. s = 0.05, J = 50
C. s = 0.05, J = 500
Time
Selection
makes some
outcomes
more likely
than others,
but it does not
guarantee any
particular
outcome
Time
A. Pure drift, J = 500
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Frequency of Species A
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Frequency of Species A
An example of bringing together drift & selection:
Effects of finite community size on invasion probabilities
The analogies:
new mutation = new species
selection coefficient, s = population growth rate, r
effective population size, Ne = effective community size, Je
initial frequency of allele/species = p
From Kimura (1962):
Pr(inv) = (1 – exp(-2Jerp)) / (1 – exp(-2Jer))
Vellend & Orrock (2010, In: Theory of Island Biogeography Revisited
From Kimura (1962):
Pr(inv) = (1 – exp(-2Jerp)) / (1 – exp(-2Jer))
Key (obvious) results:
Invasion more likely with higher r
Invasion more likely with higher p
But how does Je (i.e., drift) modulate these effects?
Vellend & Orrock (2010, In: Theory of Island Biogeography Revisited
For a given initial frequency,
invasion less likely in small communities
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Invasion
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Probability
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Pr(inv)
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Initial frequency, p = Ninit/J
Vellend & Orrock (2010, In: Theory of Island Biogeography Revisited
For a given initial population size,
invasion more likely in small communities
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Invasion
Probability 0.10
Pr(inv)
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Population growth rate, r
Vellend & Orrock (2010, In: Theory of Island Biogeography Revisited
Wild speculations
Frequency
A. Mutations
Lethal
From Hedrick (2000)
Selection coefficient, s
Frequency
B. Introduced species
Pop. growth rate when rare, r
Invader is
superior
Based on the
Kimura
equations…
Invader
is inferior
Other selection-drift models for communities…
Stochastic Niche Theory: “Classic” Tilman with some
stochasticity
Invasion (establishment) modeled probabilistically based
on small initial population that might to extinct
(demographic stochasticity) despite being
deterministically favored
Resource competition with environmental (temperature)
heterogeneity
Other selection-drift models for communities (this one with dispersal)…