Short overview

Host-parasite interactions are acknowledged as a driving evolutionary force promoting biological diversity and sexual reproduction [10, 11], with the MA and GfG model being the most popular models to describe the genetic interaction for coevolving hosts and parasites [26–32]. Despite a number of important insights provided within their framework, the generality of findings often suffers from the complexity of the models employed and, as a consequence, the difficulty to fully understand them analytically [33].

In this study, we present a very general yet parsimonious model of host-parasite coevolution spanning from MA to GfG with either constant or interaction-driven changing population size. Derived analytical solutions revealed that the coevolution dynamics differs qualitatively between the models with constant and changing population sizes. Apart from the pure MA situation, the well known Red Queen dynamics with closed trajectories is only observed in models with constant population size. This implies that the patterns of host-parasite dynamics to be expected in real biological systems can be much more intricate than suggested by the most popular theoretical models.

Main results and analytical solution

Our study is based on a simplification of the model suggested by Agrawal and Lively [14] that explores a continuum between the MA and GfG models. We study the model in the context of haplotypes with a single locus, but relax the restriction to constant population size. With a coevolutionary system of two host and two parasite types we achieved an analytical characterization across the entire parameter space. To study ecological effects caused by the victim-exploiter interaction [34] between hosts and parasites, we consider models with changing population size aside of models with constant population size. Under the assumption of constant population size, the dynamics in MA and GfG models appear to be very similar, both showing sustained oscillations with only one oscillation frequency. Yet, introducing changing population size according to the Lotka-Volterra equations, we obtain distinct patterns of the population dynamics. For changing population sizes, a single oscillation frequency is present only in the MA model. An additional oscillation frequency arises for all other points on the MA-GfG continuum in that case. In other words, changing population size leads to a much more complex dynamics in GfG-like models, but not in the pure MA model.

In the stochastic model analyzed in [29], the analysis of allele fixation time for the MA model revealed that Lotka-Volterra dynamics in combination with the associated stochastic effects quickly break down the Red Queen circle. As the dynamics in GfG-like models take a completely different nature with changing population size, the influence of Lotka-Volterra dynamics on the Red Queen circle is yet unclear and remains to be assessed in more detail in the future, especially as our current analysis did not take stochastic effects into account.

Generality of results

To test the generality of our findings we additionally analyzed the interaction matrix suggested by Parker [13] (Eq. (37)). There, a factor that denotes the fitness reduction of the avirulent parasite encountering the resistant host and an advantage of the virulent parasite meeting the resistant host are assumed in addition. These two parameters together with the costs of resistance and virulence determine whether the model is MA or GfG. Again we obtain two distinct oscillation frequencies for the population dynamics with changing population sizes in GfG-like models (the ratio is shown in Eq. (38) in the Appendix A.3 Stability of the interior fixed point for general interaction matrices).

Despite the convincing biological relevance of the interaction matrix elements in [14], they do not change monotonically on the MA-GfG continuum, e.g., with a cost of virulence κ>0.5, \(\mathcal {M}_{21}^{p}\) in Eq. (1) first increases then decreases as α increases from 0 to 1. As an alternative interpolation, we therefore also considered interaction matrices that describe a linear transition from MA to GfG model, such that

$$\begin{array}{*{20}l} \begin{array}{ll} &\begin{array}{cc} \,\quad\mathcal{P}_{1} & \qquad\qquad\mathcal{P}_{2} \end{array}\\ \mathcal{M}^{h} = \begin{array}{l} \mathcal{H}_{1}\\ \mathcal{H}_{2} \end{array}& \left(\begin{array}{cc} -\sigma & - \alpha (1 - \kappa) \sigma\\ - \alpha \gamma & - \alpha \gamma - (1 - \alpha \kappa) \sigma \end{array} \right) \end{array} \end{array} $$ ((27a))

$$\begin{array}{*{20}l} \begin{array}{ll} &\begin{array}{cc} \,\qquad\mathcal{H}_{1} & \qquad\quad\mathcal{H}_{2} \end{array}\\ \mathcal{M}^{p} = \begin{array}{l} \mathcal{P}_{1}\\ \mathcal{P}_{2} \end{array}& \left(\begin{array}{cc} {\sigma} & 0\\ \alpha (1 - \kappa) {\sigma} & (1- \alpha \kappa) {\sigma} \end{array} \right). \end{array} \end{array} $$ ((27b))

The analysis in Appendix A.4 Linear interpolation between MA and GfG models shows that our conclusion also holds for the linear interpolation. One should keep in mind that both MA and GfG models and even the intermediate models proposed by Parker, Agrawal & Lively, or us are only a small subset of the possible models for host-parasite interaction. An observation that will hold for any such model is that as long as the population sizes are kept constant, the population dynamics follows a closed circle with a single oscillation frequency. However, with changing population size a second oscillation frequency arises when the model become GfG-like, which can lead to much more intricate dynamics. For a pure MA model or an inverse MA model (where the diagonal instead of the off-diagonal matrix elements are zero), there still is only one oscillation frequency (see Eq. (36) in Appendix A.3 Stability of the interior fixed point for general interaction matrices).

Impact of eco-evo feedback in genetically explicit models

In the last two decades it has been realized that evolutionary changes can be faster than previously thought and, thus, occurring on the same time-scale as ecological interactions, especially in case of coevolving hosts and parasites [35–38]. Population dynamics can influence the pace of coevolution via so called eco-evolutionary feedbacks, or even give rise to a new type of coevolutionary dynamics as we showed in our study. Interestingly enough, a comprehensive part of the theoretical studies on eco-evolutionary feedbacks is conducted within the framework of evolutionary game theory and adaptive dynamics [21, 39]. In contrast to our model, these approaches usually do not include an explicit definition of genetic interaction between the species, which limits their application for interpreting patterns of genetic variability in natural populations [40]. Rapid changes in genetic composition may lead to perturbation in host demography and disease dynamics, as was observed for the myxoma virus epidemic in Australian populations of European rabbit [41]. Genetic adaptation can improve overall population fitness and "buffer" the unfavorable impact of pathogens (evolutionary rescue) [42]. However, population perturbations may constrain adaptability, for example, via enhancing inbreeding, affecting trait heritabilities and disturbing allele composition irrespective of natural selection [43–46]. Thus, models accounting simultaneously for the genetic basis of host-parasite interaction and associated population dynamics may be necessary to fully understand ongoing coevolution among species and the effect it would have on genetic diversity. We are aware of only a few such models [29, 47–51], and most of them confirm that ecological parameters can have a very strong effect on coevolution.

Implications for maintenance of genetic diversity

Numerous field studies identified the presence of comprehensive heritable variation in resistance-infectivity patterns for plant and animal populations and their respective pathogens, suggesting that coevolution acts to maintain genetic diversity [3, 11, 52–55]. However, already the first studies, which attempted to explain such variation by cycling dynamics, encountered the problem of stability. This is especially true for the GfG model, as a parasite with the virulent allele would be quickly fixed, unless having a cost of virulence [3, 12, 56]. In addition to the cost, other factors have been examined for their potential role in maintaining variation, including epidemiological feedback [51, 57], spatial structure [48, 49, 58, 59], genetic drift [60], diffuse multi-species coevolution [61], models with multiple alleles and multiple loci [16, 60, 62]. Several studies proposed that multiple factors need to act jointly for long-term coexistence of multiple resisto- and infectotypes [33]. The view of a multifactorial basis of the maintenance of diversity creates an additional challenge for theoretical and empirical studies to disentangle them. As opposed to that, Tellier [34] presented a simple GfG framework showing that the general condition for stability is the presence of direct frequency-dependent selection (where fitness of an allele declines with increasing frequency of that allele itself). In this context, the distinction is made between direct frequency dependence and indirect frequency-dependent selection where fitness is mediated by the frequency of the corresponding antagonist. Direct frequency-dependent selection can be introduced in the model by incorporation of epidemiological or ecological factors ([32], Table 1). If we introduce a direct frequency-dependent element by applying competitive Lotka-Volterra equations or the concept of empty spaces [63] (implying the existence of a carrying capacity) into our model, the neutrally stable interior fixed point becomes stable. Instead of forming tori or moving along closed circles, the deterministic trajectory spirals inwards. In this case, the oscillation of allele frequencies lasts longer in stochastic simulations, hence the polymorphic state is more stable.

The stability analysis derived the condition for coexistence α γ<σ, suggesting that departing from the GfG end of the continuum would increase a range of parameters at which the oscillation of allele frequencies is maintained. Therefore, patterns of "partial" infectivity by a virulent parasite are more likely to result in cycling dynamics compared to a pure GfG situation. Agrawal and Lively [14] came to the same conclusion by evaluating computational simulations. This reinforces the importance of exploring dynamics for intermediate points on the MA-GfG continuum, especially as experimental studies provide some examples of such types of interaction [64]. In contrast to [34] and many other studies [14, 16, 59], our model is implemented on a continuous time-scale and, therefore, covers host and parasite systems with overlapping generations. Interestingly, it has been proposed that models with discrete generations would favor coevolutionary cycling by synchronizing ecological and epidemiological processes [51], while in [34] the condition for stable cycling is more restrictive for discrete generations when compared to the continuous model.