Chapter 8. Coevolution, major genes, and spatial patterns of genetic variation
Biological Motivation
IN THIS CHAPTER WE EXPLORE WHEN COEVOLUTION GENERATES SPATIAL PATTERNS. Focused on very
simple models that illustrate how we can include spatial structure
START WITH MUTUALISM AND POINT OUT HOW WEIRD IT IS THAT THERE IS VARIATION WITHIN POPS
Until now we have focused on interactions in single pops. We know however, that spatial structure
matters… Let’s look at the snails and trematodes again… One way the importance of space has been
discerened is by conducting a reciprocal cross infection experiment… These experiments have revealed
very large levels of local adaptation. What is the underlying cause of these patterns?
Key Questions:
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Can spatial structure maintain genetic polymorphism within coevolving mutualistic populations?
Can coevolution generate genetic differentiation among populations?
How do coevolutionary selection and gene flow interact to shape rates of mutualistic
interaction?
Does coevolution maintain genetic polymorphism?
Can coevolution explain variation in infectivity within populations?
Should coevolution cause infection rates to differ among populations?
Building a model of SINGLE LOCUS LA
Analyzing the Model
Answers to Key Questions:
Can coevolution shape population dynamics?
Do population dynamics influence the coevolutionary process?
Mathematica Resources: http://www.webpages.uidaho.edu/~snuismer/Nuismer_Lab/the_theory_of_coevolution.htm
Will coevolution ever drive a species to extinction?
New Questions Arising:
raising several important questions:
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Other forms of interaction (mutualism)
Selection mosaics
Strong coevolution
In the next three sections, we will generalize our simple model in ways that allow us to answer these
questions.
Generalizations
Generalization 1: Integrating costs of resistance and virulence
Generalization 2: Matching alleles interaction
Generalization 3: Quantitative traits
Conclusions and Synthesis
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References
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Figure Legends
Figure 1. Population dynamics predicted by the Lotka-Volterra model for three different rust death
rates, d. In all cases, the dashed line indicates rust population size and the solid line indicates flax
population size. Parameters held constant across panels were 𝛼 = .1, 𝛽 = .03, 𝜉 = .01, 𝑟 = .1,
and 𝑘 = 100. Rust death rates were: A) 𝑑 = .005 B) 𝑑 = .081, and C) 𝑑 = .1.
Figure 2. Population sizes and allele frequencies for the gene-for-gene model over time for three
different rust death rates, d. In all cases, dashed lines indicate rust population size or frequency of the
virulent allele and solid lines indicate flax population size or frequency of the resistant allele. Parameters
held constant across panels were 𝛽 = .003, 𝜉 = .001, 𝑟 = .1, and 𝑘 = 100. Rust death rates were: A)
𝑑 = .005 B) 𝑑 = .081, and C) 𝑑 = .1.
Figure 3. Population sizes and allele frequencies for the gene-for-gene model with costs of resistance
and virulence over time for three different costs of virulence. In all cases, dashed lines indicate rust
population size or frequency of the virulent allele and solid lines indicate flax population size or
frequency of the resistant allele. Parameters held constant across panels were 𝛽 = .003, 𝜉 = .001,
𝑟𝐴 = .05, 𝑟𝑎 = .09, 𝑑𝑏 = .01 and 𝑘 = 100. Death rates of the virulent rust allele were: A) 𝑑𝐵 = .005 B)
𝑑𝐵 = .081, and C) 𝑑𝐵 = .1.
Figure 4. Population dynamics and allele frequencies for the matching alleles model over time for three
different P. ramosa death rates, d. In all cases, dashed lines indicate P. ramosa population size or
frequency of the B allele and solid lines indicate D. magna population size or frequency of the A allele.
Parameters held constant across panels were 𝛽 = .003, 𝜉 = .001, 𝑟 = .1, and 𝑘 = 100. Rust death
rates were: A) 𝑑 = .005 B) 𝑑 = .081, and C) 𝑑 = .1.
Figure 5. Population sizes and phenotypes over time for the phenotype matching model and two
different cuckoo death rates, d. In all cases, dashed lines indicate cuckoo population size or egg color
phenotype and solid lines indicate warbler population size or egg color phenotype. Parameters held
constant across panels were 𝛽 = .006, 𝜉 = .005, 𝑟 = .05, 𝑘 = 100, and the mutation rates of the two
species 𝜇𝑋 = .02 and 𝜇𝑌 = .02. Rust death rates were: A) 𝑑 = .06 and B) 𝑑 = .03.
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Table 1. Summary of stability conditions and ecological dynamics
Condition
Implications for stability
Equilibrium (3c) unstable;
𝑑 > 𝑘𝛼𝜉
Equilibrium (3b) stable
Equilibrium (3c) stable;
𝑑 < 𝑘𝛼𝜉
Equilibrium (3b) unstable
Equilibrium
(3c) stable and
4𝑘𝛼𝜉
oscillatory;
𝑑 < 𝑘𝛼𝜉 (
)
𝑟 + 4𝑘𝛼𝜉
Equilibrium (3b) unstable
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Biological consequences
Extinction of Rust. Flax at
carrying capacity
Coexistence of Flax and Rust
Coexistence of Flax and Rust.
Transient cycles likely.
Table 2. Equilibria corresponding to parasite extinction. Because we assume carrying the virulent allele is
costly, 𝑑𝐵 > 𝑑𝑏 .
Equilibrium
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 0, 𝑝𝑉 = 0
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 0, 𝑝𝑉 = 1
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 1, 𝑝𝑉 = 0
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 1, 𝑝𝑉 = 1
𝑁𝑋 = 𝑘, 𝑁𝑌 = 0, 𝑝𝑅 = 1 − (𝑑𝑏 ⁄𝑘𝜉 ), 𝑝𝑉 = 0
Conditions for instability
𝑑𝑏 < 𝑘𝜉
Always unstable
𝑑𝐵 − 𝑑𝑏 < 𝑘𝜉
𝑘𝜉 + 𝑑𝑏 < 𝑑𝐵 𝑜𝑟 𝑑𝐵 < 𝑘𝜉
𝑑𝐵 < 𝑘𝜉
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