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Lotka-Volterra dynamics kills the Red Queen: population size fluctuations and associated stochasticity dramatically change host-parasite coevolution.

Gokhale CS, Papkou A, Traulsen A, Schulenburg H - BMC Evol. Biol. (2013)

Bottom Line: This belief is founded on previous theoretical work, which assumes infinite or constant population size.To what extent are such sustained oscillations realistic?Our results call for a paradigm shift in our understanding of host-parasite coevolution, strongly suggesting that these are driven by recurrent selective sweeps rather than continuous allele oscillations.

View Article: PubMed Central - HTML - PubMed

Affiliation: Evolutionary Theory Group, Max Planck Institute for Evolutionary Biology, August Thienemann Str-2, 24306, Plön, Germany. gokhale@evolbio.mpg.de.

ABSTRACT

Background: Host-parasite coevolution is generally believed to follow Red Queen dynamics consisting of ongoing oscillations in the frequencies of interacting host and parasite alleles. This belief is founded on previous theoretical work, which assumes infinite or constant population size. To what extent are such sustained oscillations realistic?

Results: Here, we use a related mathematical modeling approach to demonstrate that ongoing Red Queen dynamics is unlikely. In fact, they collapse rapidly when two critical pieces of realism are acknowledged: (i) population size fluctuations, caused by the antagonism of the interaction in concordance with the Lotka-Volterra relationship; and (ii) stochasticity, acting in any finite population. Together, these two factors cause fast allele fixation. Fixation is not restricted to common alleles, as expected from drift, but also seen for originally rare alleles under a wide parameter space, potentially facilitating spread of novel variants.

Conclusion: Our results call for a paradigm shift in our understanding of host-parasite coevolution, strongly suggesting that these are driven by recurrent selective sweeps rather than continuous allele oscillations.

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Related in: MedlinePlus

The influence of initial allele frequency on fixation time and probability for the stochastic models. For an Navg=200, we plot the time until any of the four alleles goes to fixation (left column) and the probability of fixation of one of the host and parasite alleles for all possible initial conditions (middle and right columns) (averages over 106 realizations). Lotka-Volterra fluctuations lead to substantially faster allele fixations (top left panel) and high fixation probability for the host allele across a wide range of initial conditions (top middle panel). The simulations were always stopped when either one of the host or one of the parasite alleles reached fixation. Thus, the sum of the fixation probabilities of all alleles sums up to 1. The specific initial conditions used in Figure 2 are indicated.
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Figure 3: The influence of initial allele frequency on fixation time and probability for the stochastic models. For an Navg=200, we plot the time until any of the four alleles goes to fixation (left column) and the probability of fixation of one of the host and parasite alleles for all possible initial conditions (middle and right columns) (averages over 106 realizations). Lotka-Volterra fluctuations lead to substantially faster allele fixations (top left panel) and high fixation probability for the host allele across a wide range of initial conditions (top middle panel). The simulations were always stopped when either one of the host or one of the parasite alleles reached fixation. Thus, the sum of the fixation probabilities of all alleles sums up to 1. The specific initial conditions used in Figure 2 are indicated.

Mentions: Example of allele frequency dynamics with and without Lotka-Volterra population size fluctuations. Top: Lines show the deterministic Lotka-Volterra dynamics, as often considered in theoretical studies, cf. Eqs. (4). Middle: When stochasticity is included (thin lines show the results of 50 individual stochastic Gillespie simulations), then simulations may initially produce allele oscillations as above and below. However, alleles usually spread to fixation (or go extinct) at a much faster rate. Bottom: Dynamics without Lotka-Volterra cycles, fixing the average population size of both species to Navg=1000 by resetting it after every Navg reactions, while maintaining the ratio between the alleles. The 50 individual stochastic simulations now only rarely reach fixation. The figure illustrates the scenario where the rare host allele (H1) is more likely to reach fixation than the frequent host allele (see Figure 3). This fixation probability decreases with increasing initial frequency (cf. Figure 3). The simulation parameters are a=5, c=2.5, b=10/Navg=0.01 with H1=5%, H2=95%, P1=20%, P2=80% as initial condition.


Lotka-Volterra dynamics kills the Red Queen: population size fluctuations and associated stochasticity dramatically change host-parasite coevolution.

Gokhale CS, Papkou A, Traulsen A, Schulenburg H - BMC Evol. Biol. (2013)

The influence of initial allele frequency on fixation time and probability for the stochastic models. For an Navg=200, we plot the time until any of the four alleles goes to fixation (left column) and the probability of fixation of one of the host and parasite alleles for all possible initial conditions (middle and right columns) (averages over 106 realizations). Lotka-Volterra fluctuations lead to substantially faster allele fixations (top left panel) and high fixation probability for the host allele across a wide range of initial conditions (top middle panel). The simulations were always stopped when either one of the host or one of the parasite alleles reached fixation. Thus, the sum of the fixation probabilities of all alleles sums up to 1. The specific initial conditions used in Figure 2 are indicated.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4225518&req=5

Figure 3: The influence of initial allele frequency on fixation time and probability for the stochastic models. For an Navg=200, we plot the time until any of the four alleles goes to fixation (left column) and the probability of fixation of one of the host and parasite alleles for all possible initial conditions (middle and right columns) (averages over 106 realizations). Lotka-Volterra fluctuations lead to substantially faster allele fixations (top left panel) and high fixation probability for the host allele across a wide range of initial conditions (top middle panel). The simulations were always stopped when either one of the host or one of the parasite alleles reached fixation. Thus, the sum of the fixation probabilities of all alleles sums up to 1. The specific initial conditions used in Figure 2 are indicated.
Mentions: Example of allele frequency dynamics with and without Lotka-Volterra population size fluctuations. Top: Lines show the deterministic Lotka-Volterra dynamics, as often considered in theoretical studies, cf. Eqs. (4). Middle: When stochasticity is included (thin lines show the results of 50 individual stochastic Gillespie simulations), then simulations may initially produce allele oscillations as above and below. However, alleles usually spread to fixation (or go extinct) at a much faster rate. Bottom: Dynamics without Lotka-Volterra cycles, fixing the average population size of both species to Navg=1000 by resetting it after every Navg reactions, while maintaining the ratio between the alleles. The 50 individual stochastic simulations now only rarely reach fixation. The figure illustrates the scenario where the rare host allele (H1) is more likely to reach fixation than the frequent host allele (see Figure 3). This fixation probability decreases with increasing initial frequency (cf. Figure 3). The simulation parameters are a=5, c=2.5, b=10/Navg=0.01 with H1=5%, H2=95%, P1=20%, P2=80% as initial condition.

Bottom Line: This belief is founded on previous theoretical work, which assumes infinite or constant population size.To what extent are such sustained oscillations realistic?Our results call for a paradigm shift in our understanding of host-parasite coevolution, strongly suggesting that these are driven by recurrent selective sweeps rather than continuous allele oscillations.

View Article: PubMed Central - HTML - PubMed

Affiliation: Evolutionary Theory Group, Max Planck Institute for Evolutionary Biology, August Thienemann Str-2, 24306, Plön, Germany. gokhale@evolbio.mpg.de.

ABSTRACT

Background: Host-parasite coevolution is generally believed to follow Red Queen dynamics consisting of ongoing oscillations in the frequencies of interacting host and parasite alleles. This belief is founded on previous theoretical work, which assumes infinite or constant population size. To what extent are such sustained oscillations realistic?

Results: Here, we use a related mathematical modeling approach to demonstrate that ongoing Red Queen dynamics is unlikely. In fact, they collapse rapidly when two critical pieces of realism are acknowledged: (i) population size fluctuations, caused by the antagonism of the interaction in concordance with the Lotka-Volterra relationship; and (ii) stochasticity, acting in any finite population. Together, these two factors cause fast allele fixation. Fixation is not restricted to common alleles, as expected from drift, but also seen for originally rare alleles under a wide parameter space, potentially facilitating spread of novel variants.

Conclusion: Our results call for a paradigm shift in our understanding of host-parasite coevolution, strongly suggesting that these are driven by recurrent selective sweeps rather than continuous allele oscillations.

Show MeSH
Related in: MedlinePlus