On inferring evolutionary stability in finite populations using infinite population models.

Models of evolution by natural selection often make the simplifying assumption that populations are infinitely large. In this infinite population limit, rare mutations that are selected against always go extinct, whereas in finite populations they can persist and even reach fixation. Nevertheless, for mutations of arbitrarily small phenotypic effect, it is widely believed that in sufficiently large populations, if selection opposes the invasion of rare mutants, then it also opposes their fixation. Here, we identify circumstances under which infinite-population models do or do not accurately predict evolutionary outcomes in large, finite populations. We show that there is no population size above which considering only invasion generally suffices: for any finite population size, there are situations in which selection opposes the invasion of mutations of arbitrarily small effect, but favours their fixation. This is not an unlikely limiting case; it can occur when fitness is a smooth function of the evolving trait, and when the selection process is biologically sensible. Nevertheless, there are circumstances under which opposition of invasion does imply opposition of fixation: in fact, for the [Formula: see text]-player snowdrift game (a common model of cooperation) we identify sufficient conditions under which selection against rare mutants of small effect precludes their fixation-in sufficiently large populations-for any selection process. We also find conditions under which-no matter how large the population-the trait that fixes depends on the selection process, which is important because any particular selection process is only an approximation of reality.

Combating climate change with matching-commitment agreements.

Countries generally agree that global greenhouse gas emissions are too high, but prefer other countries reduce emissions rather than reducing their own. The Paris Agreement is intended to solve this collective action problem, but is likely insufficient. One proposed solution is a matching-commitment agreement, through which countries can change each other's incentives by committing to conditional emissions reductions, before countries decide on their unconditional reductions. Here, we study matching-commitment agreements between two heterogeneous countries. We find that such agreements (1) incentivize both countries to make matching commitments that in turn incentivize efficient emissions reductions, (2) reduce emissions from those expected without an agreement, and (3) increase both countries' welfare. Matching-commitment agreements are attractive because they do not require a central enforcing authority and only require countries to fulfil their promises; countries are left to choose their conditional and unconditional emissions reductions according to their own interests.

Difficulties in benchmarking ecological null models: an assessment of current methods.

Identifying species interactions and detecting when ecological communities are structured by them is an important problem in ecology and biogeography. Ecologists have developed specialized statistical hypothesis tests to detect patterns indicative of community-wide processes in their field data. In this respect, null model approaches have proved particularly popular. The freedom allowed in choosing the null model and statistic to construct a hypothesis test leads to a proliferation of possible hypothesis tests from which ecologists can choose to detect these processes. Here, we point out some serious shortcomings of a popular approach to choosing the best hypothesis for the ecological problem at hand that involves benchmarking different hypothesis tests by assessing their performance on artificially constructed data sets. Terminological errors concerning the use of Type I and Type II errors that underlie these approaches are discussed. We argue that the key benchmarking methods proposed in the literature are not a sound guide for selecting null hypothesis tests, and further, that there is no simple way to benchmark null hypothesis tests. Surprisingly, the basic problems identified here do not appear to have been addressed previously, and these methods are still being used to develop and test new null models and summary statistics, from quantifying community structure (e.g., nestedness and modularity) to analyzing ecological networks.

On selection in finite populations.

Two major forces shaping evolution are drift and selection. The standard models of neutral drift-the Wright-Fisher (WF) and Moran processes-can be extended to include selection. However, these standard models are not always applicable in practice, and-even without selection-many other drift models make very different predictions. For example, "generalised Wright-Fisher" models (so-called because their first two conditional moments agree with those of the WF process) can yield wildly different absorption times from WF. Additionally, evolutionary stability in finite populations depends only on fixation probabilities, which can be evaluated under less restrictive assumptions than those required to estimate fixation times or more complex population-genetic quantities. We therefore distill the notion of a selection process into a broad class of finite-population, mutationless models of drift and selection (including the WF and Moran processes). We characterize when selection favours fixation of one strategy over another, for any selection process, which allows us to derive finite-population conditions for evolutionary stability independent of the selection process. In applications, the precise details of the selection process are seldom known, yet by exploiting these new theoretical results it is now possible to make rigorously justifiable inferences about fixation of traits.

Evolutionary stability in continuous nonlinear public goods games.

We investigate a type of public goods games played in groups of individuals who choose how much to contribute towards the production of a common good, at a cost to themselves. In these games, the common good is produced based on the sum of contributions from all group members, then equally distributed among them. In applications, the dependence of the common good on the total contribution is often nonlinear (e.g., exhibiting synergy or diminishing returns). To date, most theoretical and experimental studies have addressed scenarios in which the set of possible contributions is discrete. However, in many real-world situations, contributions are continuous (e.g., individuals volunteering their time). The "n-player snowdrift games" that we analyze involve continuously varying contributions. We establish under what conditions populations of contributing (or "cooperating") individuals can evolve and persist. Previous work on snowdrift games, using adaptive dynamics, has found that what we term an "equally cooperative" strategy is locally convergently and evolutionarily stable. Using static evolutionary game theory, we find conditions under which this strategy is actually globally evolutionarily stable. All these results refer to stability to invasion by a single mutant. We broaden the scope of existing stability results by showing that the equally cooperative strategy is locally stable to potentially large population perturbations, i.e., allowing for the possibility that mutants make up a non-negligible proportion of the population (due, for example, to genetic drift, environmental variability or dispersal).

Game theory of pre-emptive vaccination before bioterrorism or accidental release of smallpox.

Smallpox was eradicated in the 1970s, but new outbreaks could be seeded by bioterrorism or accidental release. Substantial vaccine-induced morbidity and mortality make pre-emptive mass vaccination controversial, and if vaccination is voluntary, then there is a conflict between self- and group interests. This conflict can be framed as a tragedy of the commons, in which herd immunity plays the role of the commons, and free-riding (i.e. not vaccinating pre-emptively) is analogous to exploiting the commons. This game has been analysed previously for a particular post-outbreak vaccination scenario. We consider several post-outbreak vaccination scenarios and compare the expected increase in mortality that results from voluntary versus imposed vaccination. Below a threshold level of post-outbreak vaccination effort, expected mortality is independent of the level of response effort. A lag between an outbreak starting and a response being initiated increases the post-outbreak vaccination effort necessary to reduce mortality. For some post-outbreak vaccination scenarios, even modest response lags make it impractical to reduce mortality by increasing post-outbreak vaccination effort. In such situations, if decreasing the response lag is impossible, the only practical way to reduce mortality is to make the vaccine safer (greater post-outbreak vaccination effort leads only to fewer people vaccinating pre-emptively).

Modelling the spread of diseases in clustered networks.

It is now well appreciated that population structure can have a major impact on disease dynamics, outbreak sizes and epidemic thresholds. Indeed, on some networks, epidemics occur only for sufficiently high transmissibility, whereas in others (e.g. scale-free networks), no such threshold effect exists. While the effects of variability in connectivity are relatively well known, the effects of clustering in the population on disease dynamics are still debated. We develop a simple and intuitive model for calculating the reproductive number R(0) on clustered networks with arbitrary degree distribution. The model clearly shows that in general, clustering impedes epidemic spread; however, its effects are usually small and/or coupled with other topological properties of the network. The model is generalized to take into account degree-dependent transmissibility (e.g., relevant for disease vectors). The model is also used to easily rederive a known result concerning the formation of the giant component.