Institute for Integrative Biology, ETH Zurich, 8092 Zurich, Switzerland

Department of Infectious Disease Epidemiology, Imperial College, St Mary’s Campus, Norfolk Place, London W2 1PG, UK

Institute for Biogeochemistry and Pollutant Dynamics, ETH Zurich, 8092 Zurich, Switzerland

School of Biological Sciences, The University of Queensland, Brisbane, 4072, QLD, Australia

Department of Biology, Center for Infectious Disease Dynamics, Pennsylvania State University, University Park, 16802, PA, USA

Abstract

Background

Antagonistic species interactions can lead to coevolutionary genotype or phenotype frequency oscillations, with important implications for ecological and evolutionary processes. However, direct empirical evidence of such oscillations is rare. The rarity of observations is generally attributed to inherent difficulties of ecological and evolutionary long-term studies, to weak or absent interaction between species, or to the absence of negative frequency-dependence.

Results

Here, we show that another factor – non-genetic inheritance, mediated for example by epigenetic mechanisms – can completely eliminate oscillations in the presence of such negative frequency dependence, even if only a small fraction of offspring are affected. We analytically derive the threshold value of this fraction at which the dynamics change from oscillatory to stable, and investigate how selection, mutation and generation times differences between the two species affect the threshold value. These results strongly suggest that the lack of phenotype frequency oscillations should not be attributed to the lack of strong interactions between antagonistic species.

Conclusions

Given increasing evidence of non-genetic effects on the outcomes of antagonistic species interactions, we suggest that these effects should be incorporated into ecological and evolutionary models of interacting species.

Background

The phenotypes of species are generally assumed to be adapted to their environment by natural selection. A change in an environment can therefore lead to an evolutionary change in phenotypes as species adapt to new circumstances. Environments comprise both biotic and abiotic elements, and evolutionary change in one species is often driven by evolutionary change in another species. Indeed, ecology is dominated by species interactions such as predation, parasitism, mutualism and competition. If species interactions are antagonistic (i.e., one species benefits at the expense of another), the resulting patterns of adaptation and counter-adaptation can lead to cyclical dynamics typical of predator-prey or host-parasite systems. Understanding the causes and consequences of such fluctuating population dynamics is crucial in a number of biological phenomena, and particularly also in applied fields such as conservation biology and pest management.

The population dynamics of antagonistic species interactions can be captured with well-established models such as the Lotka-Volterra model

Phenotypic adaptations to changing environments need not be driven by natural selection alone. This is because many phenotypes are plastic and can change due to adverse environmental conditions, a property generally referred to as phenotypic plasticity. Interestingly, phenotypic change can be stably transmitted across generations at various levels of specificity. Transgenerational induction of defences has been reported in animals and plants

Our goal here is to understand the effect of non-genetic inheritance on patterns of antagonistic coevolution. We develop a simple model where two species (e.g., host and parasite) are interacting, and each species has two alternative phenotypes. If their phenotypes match, the outcome of the interaction has negative fitness consequences for one species (host) and positive for the other species (parasite). As a result of this, the phenotypes harmed by the interaction may switch to the alternative phenotype in the offspring. We are purposefully ignorant about the nature of the phenotype (e.g., molecular, developmental, behavioral) and about the underlying form of non-genetic inheritance responsible for the phenotype switch in the offspring. In the absence of non-genetic inheritance, this model reduces to the most basic model of antagonistic coevolution exhibiting negative frequency dependence and resulting in the classical Red Queen dynamics (i.e., oscillations of phenotypes). We find that non-genetic inheritance can strongly affect cycling behavior typical of Red Queen dynamics by dampening the phenotype frequency oscillations. To examine this in detail, we derive analytical expressions of the threshold rate at which this elimination occurs.

Methods

In order to understand how non-genetic inheritance affects the patterns of antagonistic coevolution, we consider a simple, discrete-generation, coevolutionary model of two species

Antagonistic interactions induce fitness costs on both species: successful interactions come at a cost for species _{1} and _{2}, and phenotype frequencies from species _{1}and _{2}. Only individuals from species _{
X
}and 1−_{
Y
}, respectively, survive in the next generation (see Table
_{
X
}for species _{
Y
}for species

where we have assumed that _{1} + _{2} =_{1} + _{2} = 1, _{1}, and _{1}. Equation (1) can be derived in three steps. First, one calculates the proportion of individuals of species _{
X
} of them switches to an alternative phenotype. This yields the first equation in (1), and the calculation for species _{
X
} becomes equivalent to a cost of induced switching, as seen in equation (1) above (see also Discussion).

**rel. fitness of species****
X
**

**
y
**

**
y
**

**rel. fitness of species ****
Y
**

**
x
**

**
x
**

_{1}

1−_{
X
}

1

_{1}

1

1−_{
Y
}

_{2}

1

1−_{
X
}

_{2}

1−_{
Y
}

1

In contrast, stochastic switching occurs independently of antagonistic interactions, and in proportion

This step can be also interpreted as mutation, and we generally assume that ^{−8} unless mentioned otherwise.

Finally, we allow for asymmetry in the generation time between the two species by defining a parameter ^{
′′
} and ^{
′′
}yield the phenotype frequencies after an entire generation of species

Results

It is generally expected that antagonistic interactions can result in cyclic allele frequency dynamics, reflecting a continuing arms race between the two species
_{
X
}=_{
Y
}= 0) our model reveals such a pattern (see Figure
_{2} and a decrease of the frequency _{1}. Such change will in turn drive the frequency change in species _{2} to increase, and so on. These oscillations are, in the absence of random genetic drift and mutation, expected to continue indefinitely, otherwise fixation or extinction of one of the two phenotypes occurs

Impact of induced phenotypic switching on the cyclic phenotype frequency dynamics

**Impact of induced phenotypic switching on the cyclic phenotype frequency dynamics.** (**A**): Coevolutionary dynamics between antagonistic phenotypes are predicted to continue indefinitely when no induced switching occurs due to time-lagged, negative frequency dependent selection (_{X }= 0). (**B**): When induced switching occurs (here in only in species _{Y }= 0) at a low rate (
**C**): When the switching rate exceeds a threshold value
**D-E**): Increased levels of induced switching also decrease the amplitude and increase the speed of the cycles. Note that
_{X }=_{Y }= 0.3, (D-E) _{X }=_{Y }= 0.65; (A) _{X }= 0, (B) _{X }= 0.03, (C) _{X }= 0.1; In all panels we used _{Y }= 0. Period is defined as a number of generations during which the phenotype frequency cycles around to its original value.

Consider now a situation where induced phenotypic switching is possible in a single species. Figure
_{
X
}> 0, _{
Y
}= 0), the cycles become faster and of lower amplitude, eventually leading to a stable state (^{∗},^{∗}) = (1/2,1/2); (Figure
_{
X
} exceeds a certain threshold value,
_{
X
}increases the amplitude gradually decreases to zero and the speed increases until the cycles disappear. This already illustrates that induced switching can fundamentally affect the oscillatory dynamics in the system.

In order to examine the persistence of cyclic dynamics in more detail, we derive an analytical expression for the stability of the cyclic behaviour as a function of _{
X
}, _{
Y
}, _{
X
}, and _{
Y
}.

In the case of the model considered here, the stability requires that ^{
′′
}=^{
′′
}=^{
′′
}≡^{
′′
}≡^{∗},^{∗}) = (0,0), (^{∗},^{∗}) = (0,1), (^{∗},^{∗}) = (1,0), (^{∗},^{∗}) = (1,1), and that (^{∗},^{∗}) = (1/2,1/2) is a non-trivial equilibrium with the Jacobian of the form

The corresponding eigenvalues are

where

This has been derived under the assumptions of 0 ≤_{
X
},_{
Y
}≤ 1, 0 ≤_{
X
},_{
Y
}≤ 1, and 0 ≤^{∗},^{∗}) = (1/2,1/2) requires that the absolute value of both eigenvalues be smaller than one, or

The inequality (5) yields constraints on the values of _{
X
} and _{
Y
} for which, given _{
X
}and _{
Y
}, the equilibrium (^{∗},^{∗}) = (1/2,1/2) is unstable, resulting in persisting phenotype frequency oscillations, or for which the equilibrium is stable, resulting in the cessation of the cycles.

The induced switching values for which the stability of the system is lost or regained can be calculated analytically for special cases of the stability condition (5), and otherwise either numerically or estimated from the simulation results. For example, when _{
X
}=_{
Y
}=_{
X
}=_{
Y
}=

It can be shown that the left-hand side of the inequality (6) is an increasing function of

which allows a precise calculation of the threshold levels of induced switching at which cycles disappear and reappear in this particular example.

To examine the persistence of cycles for a general case of _{
X
}≠_{
Y
}, we solve the relation (5) numerically and compare it with the simulation results. Figure
_{
X
}and _{
Y
}for which oscillations dampen, with different combinations of selection coefficients, based on the stability condition (5). Figure
^{−2}, and otherwise we consider the cycles to be absent. A comparison between Figure

Impact of the strength of selection on the persistence of cyclic phenotype frequency dynamics (A1–A4): Persistence of phenotype oscillations (black regions) for increased selection coefficients based on analytical predictions

**Impact of the strength of selection on the persistence of cyclic phenotype frequency dynamics (A1–A4): Persistence of phenotype oscillations (black regions) for increased selection coefficients based on analytical predictions.** When selection is weak (A1) cycles will continue indefinitely only for minimal or maximal levels of induced switching in both species. As selection becomes stronger (A2–A3), cyclic dynamics become more difficult to destroy and persist at higher levels of induced switching. For exceptionally high values of selection coefficients (A4), allele frequency oscillations will become unaffected by induced switching, including their speed and amplitude. As explained in the main text, the nature of cycles for minimal and maximal values of induced switching is different because of being driven by different evolutionary forces. (**B1–B4**): Numerical calculation of the amplitude of cycles from computer simulations confirms the analytical calculations in panels A1-A4. It also reveals that the amplitude of cycles decreases as the cycles reemerge at high values of induced switching in both species _{X }=_{Y }= 0.30, (A2,B2) _{X }=_{Y }= 0.57, (A3,B3) _{X }=_{Y }= 0.60, (A4,B4) _{X }=_{Y }= 0.90; panels A1-A4 were made in resolution 101×101; in panels B1-B4 were made in resolution 51×51, and cycles of amplitude 0.5×10^{−2} where considered non-existent.

The results in Figure
_{
X
}=_{
Y
}, the cycles can reemerge as

Interestingly, the nature of cycles for low and high levels of induced switching is very different. In the case of _{
X
}=_{
Y
}= 0, the oscillatory behaviour will persist due to time-delayed negative frequency-dependent selection (being rare is advantageous, being common is disadvantageous), whereas for _{
X
}=_{
Y
}= 1 oscillations will occur even in the absence of a selective force. The reason for this is that the latter situation represents the case where the phenotype frequency of one species in the next generation will be fully determined by the frequency of the phenotype of the other species. This will result in one species being constantly adapted to the other species population in the previous generation. However, since the other species does exactly the same, the two species will constantly cross-react even in the absence of any evolutionary force, leading to oscillatory “mirror dynamics” (see Discussion). Interestingly, in the parameter regime where these dynamics are dominant, increasing induced switching increases the amplitude of the allele cycles, while in the parameter regime where selection-induced cycles are dominant, increasing induced switching decreases the amplitude of the allele cycle, as seen in Figure

Stochastic switching can also destroy cyclic frequency dynamics. This is illustrated in Figure

Impact of stochastic phenotypic switching and asymmetric generation times on the persistence of phenotype oscillations

**Impact of stochastic phenotypic switching and asymmetric generation times on the persistence of phenotype oscillations.** (**A1–A4**): Persistence and amplitude of cycles for different stochastic switching values. Subsequent panels show the results for increased values of **B1-B4**): Persistence and amplitude of cycles for different asymmetric generation times between species _{X}) is much less prominent. The following parameter values were used: (A1-A4) _{X }=_{Y }= 0.66, (B1-B4) _{X }=_{Y }= 0.75; (A1) ^{−8}, (A2) ^{−2}, (A3) ^{−2}, (A4) ^{−2}; (B1) ^{−2}where considered non-existent.

Finally, we examined the effect of asymmetric generation times between species _{
X
} and _{
Y
}. This is because when one of the species evolves faster (here

Discussion

Antagonistic coevolution is pervasive in nature, and oscillatory dynamics are generally thought to be one of its key signatures. The stability of this pattern is of fundamental importance in biology because the dynamics of phenotypes and genotypes are central to evolutionary and ecological processes. Furthermore, the absence of oscillations could be interpreted as the absence of an antagonistic interaction. We have shown here that in a simple model of antagonistic coevolution between two species, phenotypic switching – transmitted to the next generation through non-genetic inheritance – can have a dramatic effect on the patterns of antagonistic coevolution. Minimal levels of induced phenotypic switching can completely eliminate oscillatory dynamics and result in stable frequencies. This therefore suggests that even in the presence of strong links between the two species (i.e., strong selection, high specificity, etc.), antagonistic coevolution need not result in fluctuations of genotypes and phenotypes.

We have identified three parameters that affect the threshold level of induced switching at which cycles disappear. The first is the strength of selection in an antagonistic species interaction. For the threshold level to be high, both species need to suffer large fitness costs, to the extent that when selection is strong enough cycles will never be affected. Parasites may indeed pay such costs because their reproduction often depends on a successful antagonistic interaction with a host (see e.g.,

What makes the cycles disappear? Fundamentally, cycles depend on time-lagged, negative frequency-dependent selection (see e.g.,

A more formal way to describe this phenomenon is to realize that under selection, the change of phenotype frequency in a single generation in one species, say

Impact of the three evolutionary forces of our model on the change in phenotype frequency

**Impact of the three evolutionary forces of our model on the change in phenotype frequency ****and ****during a single time step (one generation), as given by equations (**1**), and (**2**).** The beginning and end of each arrow marks the phenotype frequencies before and after the respective step. As can be seen, selection tends to produce oscillations, whereas both induced and stochastic phenotypic switching tend to dampen these oscillation. Parameters used are (**A**) _{X }=_{Y }= 0.5, (**B**) _{X }=_{Y }= 0.2, and (**C**)

Overall, one of the most striking findings of this study is just how little phenotypic switching, especially interaction-induced, is necessary to completely eliminate cycles. One is tempted to speculate that such a process could be one of the reasons why evidence of dynamic polymorphisms is so rare, apart from the fact that long-term observations are difficult
_{
X
}, would by substituted by _{
X
} + _{
X
}

One of the important assumptions of this study is that the model underlying the antagonistic interaction is of a ‘matching-alleles’ type. Such a model is mostly applicable in the case of hosts with a specific immune system, and antigenic parasites, which have to specifically match the host in order to infect it. By contrast, interactions in many plant-pathogen systems are usually thought to be of a ‘gene-for-gene’ type, where a host needs to recognise specific ‘effectors’ of the parasite in order to launch its defence

Conclusion

Environmentally induced phenotypic change that is stable across generations has recently been demonstrated in a number of cases, many of them involving stable epigenetic modifications

Competing interests

The authors declare no competing interests.

Authors’ contributions

RM conceived and designed the study, carried out the simulations, analysed the model and the results, and drafted the manuscript. JE analysed the results, and helped to draft the manuscript. MS conceived and designed the study, analysed the model and the results, and drafted the manuscript. All authors read and approved the final manuscript.

Acknowledgements

This work was supported by the Swiss National Science Foundation (RM & JE), and the Branco Weiss Fellowship (MS).