1. IN ECOLOGICAL COMMUNITIES
SELECTION & DRIFT
IN ECOLOGICAL COMMUNITIES
Time

2. We always had selection:
In the beginning…
We always had selection:
Per capita competitive effect of sp. 2 on sp. 1
dN1 dt
= r1N1 1 -
a11N1 + a12N2 K1
Lotka-Volterra competition

3. If aij < 1, stable coexistence possible
K1/a12 K2 N2 K1
K2/a21 N1

4. The Community Matrix (Levins 1968)2 3 4 5 1
a21
a31
a32
a41
a42
a43
a51
a52
a53
a54
Species
Species

5. 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

6. But there’s more than one way to be “different”…
“Fitness” differences: ri > rj, Ki > Kj
If large, coexistence less likely
“Niche” differences: aij < 1
If large, coexistence more likely
dN1 dt
= r1N1 1 -
a11N1 + a12N2 K1
dN2 dt
= r2N2 1 -
a22N2 + a21N1 K2

7. 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…
Niche difference (1 – overlap)
Fitness difference between species i and community average
Population growth rate when rare for species i
Chesson (2000, Ann. Rev. Ecol. Syst)

8. 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
Fitness difference between species i and community average
Chesson (2000, Ann. Rev. Ecol. Syst)

9. + - A. Constant selection FitnessA – FitnessB
A simpler way to think about it (I think)…
FORMS OF SELECTION BETWEEN TWO SPECIES, A & B
A. Constant selection +
expected dynamics (sp. A wins)
FitnessA – FitnessB
unstable
equilibrium
stable
equilibrium - 1
Frequency of Species A

10. FORMS OF SELECTION BETWEEN TWO SPECIES
A. Constant
unstable
equilibrium
stable + + -
D. Complex frequency-dependence - + -
B. Negative frequency-dependence + -
E. Neutral
FitnessA – FitnessB + -
C. Positive frequency-dependence
Frequency of Species A

11. Equalizing and Stabilizing Mechanisms of Coexistence How much do you need to bend these lines (while maintaining average height) to get stable coexistence?
Fitness difference large
Fitness difference small + + - - - +
a lot - +
only a little
FitnessA – FitnessB

12. Prone to the influence of drift
Equalizing and Stabilizing Mechanisms of Coexistence How much do you need to raise these lines (while maintaining bendiness) to get rid of stable coexistence?
Niche difference large
Niche difference small + - + - - +
a lot - +
only a little
FitnessA – FitnessB
Prone to the influence of drift

13. Combining selection and drift (Starting with pure drift)
# NEUTRAL MODEL (single community, no speciation)
# Set initial community (e.g., 25 individuals of sp 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")

14. Combining selection and drift
# SELECTION-DRIFT MODEL (single community, no speciation)
# Set initial community (e.g., 25 individuals of sp 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 < # 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")

15. Selection makes some outcomes more likely than others, but it does not guarantee any particular outcome

16. Pr(inv) = (1 – exp(-2Jerp)) / (1 – exp(-2Jer))
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
If we use the analogies that a new species in a community is equivalent to a new allele in a population, the population growth rate is equivalent to the selection coefficient, and community size stands in for population size, then if we also know the initial frequency of our new species we can apply Kimura’s general equation predicting the probability of invasion or establishment. These pictures are here to remind me of the important assumption that life is a zero-sum game. As we might imagine in a closed canopy forest or in a community of cavity nesting birds, you can’t add new individuals until others are removed. The Kimura equation nicely ties together the attributes of an invading species (r and s) with one key attribute of the receiving community (J) in a simple way, and it’s this effect of J I want to focus on.
Fixation models nicely tie together, in a very simple way, the key elements of invasion success: r, p and J.
Need to explain why fixation = mutation.
Nice b/c p not neca 1/N (like invaders)
From Kimura (1962):
Pr(inv) = (1 – exp(-2Jerp)) / (1 – exp(-2Jer))
Vellend & Orrock (2010, In: Theory of Island Biogeography Revisited

17. Pr(inv) = (1 – exp(-2Jerp)) / (1 – exp(-2Jer))
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

18. For a given initial frequency,
invasion less likely in small communities
0.9
0.8
0.7
r = 0.01, J = 1000
0.6
Invasion
Probability
Pr(inv)
0.5
0.4
0.3
The Kimura equation tells us that for a given initial frequency of an invader, that is its initial population size divided by the community size, invasion is less likely in small communities than large ones. These results are for the case when the invader has a small but positive rate of population growth. The reason is that an initial frequency of 2% in a population of 100 means there are only 2 individuals initially, and with only a small advantage over residents they are likely to quickly die out. 2% of 1000 gives you 20 individuals, which are much more likely to establish given even a relatively small advantage. A situation like this I imagine applying to something like lakes where small and large lakes might receive the same amount of boat traffic per surface area, which translates into similar initial frequencies of exotic invaders and therefore very different initial population sizes.
0.2
r = 0.01, J = 100
0.1
0.0
0.02
0.04
0.06
0.08
0.1
0.12
Initial frequency, p = Ninit/J
Vellend & Orrock (2010, In: Theory of Island Biogeography Revisited

19. For a given initial population size,
invasion more likely in small communities
0.20
Ninit = 5
0.15
Invasion
Probability
Pr(inv)
J = 100
0.10
J = 1000
In contrast, for a given intial population size, the small community is more likely to be invaded that the large one when the degree of species advantage is small. This is because if you add 5 individuals to a small community the initial frequency is large relative to adding the same 5 individuals to a large community, and with a small degree of selective advantage the initial frequency approximates the ultimate probability of invasion. So, if someone wants to get rid of 5 Norway maple trees, they might be more likely to come to dominate a little patch of forest than a large one.
Admittedly the importance of finite recipient community size is restricted to a somewhat narrow range of possible parameter space, but I think it’s just one example of a result from pop-gen that we hadn’t really derived in ecology, and that can basically be imported as is. No doubt there are many others.
Je is what determines fate; J is denominator of p
Throw in caveats of small r, etc.
At least mention possibility of deleterious invaders to fix.
0.05
0.00
0.000
0.005
0.010
0.015
0.020
Population growth rate, r
Vellend & Orrock (2010, In: Theory of Island Biogeography Revisited

20. Wild speculations Frequency Frequency A. Mutations
Lethal
Frequency
From Hedrick (2000)
Selection coefficient, s
Pop. growth rate when rare, r
Frequency
B. Introduced species
The fun part of drawing an analogy between mutations and introduced species comes with some more speculative thoughts. Population geneticists have paid lots of attention to the distribution of selection coefficients among new mutations given its importance in determining the role of mutation in evolution, and there are reasons to suspect a similar distribution of initial population growth rates for introduced species. Just as there are many lethal mutants, most house plants probably wouldn’t survive a week of winter in Boston. A whole might be able to deal with the climate but not on top of competition from natives and thus are equivalent to slightly deleterious mutants, and then a very few will enjoy an advantage over natives, and these of course are the ones to worry about.

21. Based on the Kimura equations…
Invader is superior
Invader is inferior
Based on the Kimura equations…

22. 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

23. Other selection-drift models for communities (this one with dispersal)…