### Abstract

- Top of page
- Abstract
- Introduction
- Fluctuating selection vs. arms race dynamics
- A simple coevolution model for the fluctuating selection dynamics
- Discussion
- Conclusion
- Acknowledgments
- References

The description of coevolutionary dynamics requires a characterization of the evolutionary dynamics of both the parasite and its host. However, a thorough description of the underlying genetics of the coevolutionary process is often extremely difficult to carry out. We propose that measures of adaptation (mean population fitness) across time or space may represent a feasible alternative approach for characterizing important features of the coevolutionary process. We discuss recent experimental work in the light of simple mathematical models of coevolution to demonstrate the potential power of this phenotypic experimental approach.

### Introduction

- Top of page
- Abstract
- Introduction
- Fluctuating selection vs. arms race dynamics
- A simple coevolution model for the fluctuating selection dynamics
- Discussion
- Conclusion
- Acknowledgments
- References

The study of host–parasite coevolution raises many experimental challenges. In particular, demonstrating evolutionary change of a focal species requires measurements of the phenotypes against a reference genotype of the interacting species. For example, the evolutionary change of the virulence of myxoma virus has been demonstrated by evaluating the case mortality of virus strains sampled at different points in time, with the same reference line of rabbits (Fenner & Fantini, 1999). Reciprocally, the coevolution of the rabbit population has been tracked by measuring the resistance of wild caught individuals at different points in time against reference virus strains (Fenner & Fantini, 1999).

Most coevolution studies therefore work with microbial organisms in the laboratory, taking advantage of their short generation times and cryopreservation (Levin & Lenski, 1983; Buckling & Rainey, 2002). In particular, this allows one to measure the performance of the parasite against hosts sampled in contemporaneous populations, in populations from the past and populations from the future. For example, Buckling & Rainey (2002) and Brockhurst *et al.* (2003) conducted cross-infection experiments across time to detect and measure the speed of coevolution between the bacteria *Pseudomonas fluorescens* and its phage φ2. Specifically, they measured the ability of phage populations from different time points to infect focal bacteria populations. They found that (i) the infectivity of phages from two transfers in the past was lower than the infectivity of contemporaneous phages and (ii) that the infectivity of phages from two transfers in the future was higher than the infectivity of contemporaneous phages (Fig. 1a).

In another recent experimental study, Decaestecker *et al.* (2007) used samples of dormant stages of *Daphnia* and their bacterial parasites archived in pond sediments to analyse, in the field, the emerging pattern of parasite adaptation across time. In contrast with the results obtained on the coevolution between bacteria and phage (Buckling & Rainey, 2002; Brockhurst *et al.*, 2003; Lopez Pascua & Buckling, 2008) they showed that, over approximately 30 years of coevolution, parasites were better able to infect contemporary hosts than hosts from either past or future generations (Fig. 1b).

Here we point out that simple mathematical arguments can (i) explain the contrasting patterns of adaptation across time obtained in these two studies and (ii) illustrate how one can use the observed pattern of adaptation across time to obtain important insights into the dynamics of coevolution.

### Fluctuating selection vs. arms race dynamics

- Top of page
- Abstract
- Introduction
- Fluctuating selection vs. arms race dynamics
- A simple coevolution model for the fluctuating selection dynamics
- Discussion
- Conclusion
- Acknowledgments
- References

Coevolution is often described using the metaphor of the ‘Red Queen’ (van Valen 1973), where the constant degradation of the environment (e.g. the evolution of interacting species) explains why ‘it takes all the running you can do, to keep in the same place’. Yet, very different coevolutionary processes may drive ‘Red Queen’ dynamics. Two different coevolutionary models are often distinguished (Woolhouse *et al.*, 2002). First, the arms race dynamics (ARD), with no frequency-dependent selection, where both species continually accumulate adaptive mutations. Second, the fluctuating selection dynamics (FSD), where host and parasite genotype frequencies oscillate over time because of negative frequency-dependent selection. Interestingly, numerical simulations can be used to show that these two coevolutionary dynamics yield very different patterns of adaptation over time (Gandon & Day, in press). Of course, these two dynamics are likely to represent two extremes in a continuum of more complex models of interactions (Agrawal & Lively, 2002). Furthermore, when multiple loci are involved in the interaction, some loci may evolve according to the FSD whereas others follow the ARD. Yet, the analysis of these two extreme cases is a necessary first step towards a better understanding of general coevolutionary scenarios.

Under the ARD, one expects the level of parasite adaptation to increase monotonically with the *time shift* (the time interval between the sampling of the parasite and the host). Indeed a larger time shift between the parasite and its host allows the parasite to have accumulated a larger number of adaptive mutations (Fig. 1c). In contrast, the FSD yields fluctuations of genotype frequencies and, consequently, generates patterns of adaptation that fluctuate with time shift (Fig. 1d). Note, however, that both ARD and FSD yield similar increasing patterns of adaptation with time shift, when the time shift is small (between the vertical dashed lines in Fig. 1c, d). Thus, distinguishing between these two alternative coevolutionary models require measures of adaptation across relatively long periods of time.

Unlike the other studies on the bacteria–phage system discussed above, which measured the time shift over approximately 14 bacterial generation intervals (or two experimental ‘transfers’), Buckling & Rainey (2002) measured interactions between bacteria and phage over a range of longer time intervals. The data are shown in Fig. 1a and the observed pattern is consistent with the ARD scenario. The pattern of coevolution between Daphnia and bacteria obtained by Decaestecker *et al.* (2007), however, is inconsistent with ARD (compare Fig. 1b and Fig. 1d). Decaestecker *et al.* (2007) conducted numerical simulations of FSD and showed that many patterns can be obtained, including some fitting the observed one. Broadly speaking, their simulation results demonstrate that coevolution in this system might be due to FSD, but below we illustrate that much more can be inferred about this system through the analysis of a simple mathematical model of coevolution leading to FSD.

### A simple coevolution model for the fluctuating selection dynamics

- Top of page
- Abstract
- Introduction
- Fluctuating selection vs. arms race dynamics
- A simple coevolution model for the fluctuating selection dynamics
- Discussion
- Conclusion
- Acknowledgments
- References

Suppose coevolution is governed by a single diallelic locus in both species. The dynamics of allele frequency oscillations in the host, *h* and the parasite, *p* are approximately (Gandon, 2002; Gandon & Otto, 2007):

- (1)

where *A* and *A*′ are the amplitudes, *B* is the frequency (period is *T* = 2*π*/*B* and *C* measures the lag between host and parasite genotype frequency oscillations. In this model, the parasite lags *CT/*2*π* time steps behind the host (Fig. 2a).

Under these assumptions, the mean fitness of a parasite population sampled *D* time steps away (i.e. *D* is the time shift between host and parasite populations) from the host population, where host and parasite samples are pooled from time windows of size *G*, is (Fig. 2b):

- (2)

Equation 2 can be used to clarify the simulation results obtained by Decaestecker *et al.* (2007). They point out that averaging over several generations of parasites can affect the pattern of adaptation across time (Fig. 2 in Decaestecker *et al.*, 2007). In the experiment and the simulations of Decaestecker *et al.* (2007), however, it is assumed that *D* = *G*. Equation 2 allows one to analyse the influence of *D* and *G* independently, and to see that there are two effects hidden in the simulation results presented in Fig. 2 of Decaestecker *et al.* (2007). First, the effect of increasing *G* only dampens the oscillations of the pattern of adaptation across time, without affecting its qualitative shape. Second, variations in *D*, can affect qualitatively the pattern of adaptation. For example, in Fig. 2b we show that the pattern observed by Decaestecker *et al.* (2007) is obtained when *D* = ±*T/*2 but not when *D* = ± *T*. We suspect that it is this latter effect that explains the change in the qualitative pattern presented in their Fig. 2. We also note that eqn 2 allows one to explore the effects of other factors as well, including virulence, mutation, number of alleles and migration rates, through their effects on *A*, *A*′, *B* and *C* (Gandon, 2002; Gandon & Otto, 2007).

More importantly, this simple model can be used to infer other important aspects of the coevolutionary process. In particular, the pattern observed in the data of Decaestecker *et al.* (2007), where the parasite is better able to infect contemporary hosts than hosts from either past or future generations (Fig. 1d), is expected to emerge only if *W*_{A,A′B,C} (0, *G*) > *W*_{A,A′B,C} (±*D*, *G*). Using eqn 2, this inequality reduces to the conditions:

- (3)

Conditions (3) can be satisfied only if (i) *C* < *π*/2 and (ii) *D* falls within the appropriate time interval. The condition *C* < *π*/2 means that the parasite population tracks host evolution very closely (Fig. 2). Several factors can produce such an outcome, but all involve increasing the relative strength of selection on the parasite (larger selection coefficients, shorter generation times, larger number of alleles in the matching-allele model), and/or increasing the genetic variance in the parasite population (higher mutation, recombination or migration rates than the host). This suggests that the pattern observed by Decaestecker *et al.* (2007) is generated by an asymmetry in favour of the parasite in one of the above factors.

### Conclusion

- Top of page
- Abstract
- Introduction
- Fluctuating selection vs. arms race dynamics
- A simple coevolution model for the fluctuating selection dynamics
- Discussion
- Conclusion
- Acknowledgments
- References

The importance of spatial structure on coevolutionary dynamics has been pointed by several authors (Thompson, 1994, 2005; Gomulkiewicz *et al.*, 2000; Lively & Dybdhal, 2000; Thompson & Cunningham, 2002). In particular, the study of spatial patterns of adaptation (i.e. local adaptation) has led to a better understanding of the coevolutionary process in many host–parasite systems (Dybdhal & Lively, 1996; Kaltz *et al.*, 1999; Hanifin *et al.*, 2008). Here we point out that the examination of patterns of adaptation across time in the light of numerical simulation (Fig. 1) and simple analytical models of coevolution (Fig. 2 and eqn 2) can be used to infer important information about: (i) the underlying model of coevolution (FSD vs. ARD), (ii) the asymmetry in selection pressures acting on the host and the parasite, and (iii) the asymmetry in the evolutionary potential of the host and the parasite. Hence, looking for patterns of adaptation across space and across time could help meet the challenges that often arise in the study of coevolution between parasites and their hosts. It might also be useful in the broader context of the coevolution between males and females (Rice, 1996), nuclear–cytoplasmic conflicts (Gigord *et al.*, 1998) or mutualistic interactions (Day *et al.*, 2008). Such phenotypic approaches cannot, however, answer all the questions about coevolutionary dynamics, and a thorough description of the underlying genetics of the host–parasite interaction (e.g. number of loci and alleles involved in the interaction) is required to complete the picture.