Suicidal Red Queen: Population dynamics and genetic drift accelerate diversity loss

Long term oscillations of genotype abundances in host-parasite systems are difficult to confirm experimentally. Therefore, much of our current understanding of these dynamics is based on theoretical concepts explored in mathematical models. However, the same biological assumptions can lead to very different mathematical models with diverging properties. The precise model can depend on the level of abstraction from reality, on the educational background and taste of the modeler, and on the current trends and conventions in the field. Here, we first review the current literature in the light of mathematical approaches. We then propose and compare our own framework of biologically similar, yet mathematical very different models that can all lead to host-parasite Red Queen dynamics. We highlight the different mathematical properties and use analytical and numerical tools to understand the long term dynamics. We focus on (i) the difference between deterministic and stochastic models and (ii) how ecological aspects, in our case population size, can influence the evolutionary dynamics. Our results show not only that stochastic effects can lead to extinction of subtypes, but that a changing population size speeds up this extinction. The loss of strain diversity can be counteracted with random mutations which then allow the populations to recurrently undergo fluctuating selection dynamics and selective sweeps.

ics are interpreted as oscillations in genotype abundances induced by antagonistic co-evolution Table 1: Literature overview. Mathematical models and properties discussed in this paper sorted by publication year. Many models deal with relative (allele) abundances without considering ecological dynamics -these have been categorised as constant population size models. Those models that include a changing population size and stochastic effects focus on completely different aspects than the possible extinction that are the focus of this paper. The mean population dynamics is ultimately driven by events on the individual level. These 88 individual based models can be written in the form of chemical reactions with a certain reaction 89 rate. All our stochastic processes are based on these individual interactions, where parasites have 90 negative fitness effects on the hosts, but beneficial effects on the parasite. Although our models 91 can be explained with the same words and biological relevance, the mathematics behind them 92 can be completely different. A verbal summary of the model is given in Table 2, for mathematical 93 details we refer to the supplementary material (Section S1 and Table S1). Whether parasites can 94 successfully infect a host or not is controlled by specificities. Parasites are often highly specific   algorithm, see Supplement S1.2). On the other end of the spectrum (and unrelated to Evolution-113 ary Game Theory) is the completely free independent reactions (IR) model (Supplement S1.3).

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Here, an interaction between matching host-parasite pairs directly results in parasite birth, e.g. but with reaction rates taken from Game Theory. We implement all models using the matching 120 allele interaction matrix, as described above. We could work with any other interaction matrix, 121 but as we are interested in a comparison of different dynamical process, it is simpler to focus on 122 a particular interaction mode.

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In total, these considerations define a microscopic process describing individual deaths and 124 births. To analyse these models, we can in some cases directly analyze the stochastic process, 125 but often we have to resort to deterministic limits or numerical simulations.

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The distribution of many independent simulations can be approximated by the stochastic process,    when population size is not constant. As an example with two host strains and two parasite 158 types (Fig. 1), we pick the Moran process and the logistic independent reactions (logIR), which 159 both have a pulling force as described above, but in the independent reactions model population 160 size is not fixed, merely constrained.

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These single simulations are only a snapshot and one specific realisation of the process.

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Ideally, we would analytically derive extinction times depending on the parameters of the model.    drift (intrinsic stochasticity) to govern the dynamics. We have found that flexible population 247 sizes lead to a faster extinction of subtypes when genetic drift is allowed (Figures 1 and 2). We 248 see that global competition stabilises the coexistence of types and leads to a prolonged period of 249 oscillations when selection is strong. This effect is greater for large population sizes (Figure 3).