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Most models of ecological and eco‐evolutionary processes involve creating trajectories of something, be it population densities, average trait values, or environmental states, over time. This requires decision‐making regarding how to represent the flow of time in models. Most ecologists are exposed to continuous‐time models (typically in the form of ordinary differential equations) as part of their training, especially since the famous Lotka‐Volterra predator–prey dynamics are formulated this way. However, few appear sufficiently well trained to produce their own work with continuous‐time models and may lack exposure to the true versatility of available methods. Specifically, knowledge that discrete individuals can be modelled in continuous time using the Gillespie algorithm is not as widespread as it should be. I will illustrate the flexibility of continous‐time modelling methods such that researchers can make informed choices, and not resort to discretizing time as a ‘default’ without a clear biological motivation to do so. I provide three example‐based tutorials: (1) a comparison of deterministic and stochastic dynamics of the Lotka‐Volterra predator–prey model, (2) an evaluation of matelessness in a hypothetical insect population (and of selection to mate more often by either searching more efficiently or by shortening the ‘time out’ after each mating) and (3) within‐season density dependence followed by a birth pulse leading to Beverton‐Holt or Ricker dynamics depending on whether the deaths of conspecifics help reduce the mortality of others or not (compensatory mortality). I highlight properties of the exponential distribution that, while counter‐intuitive, are good to know when deriving expected lifetime reproductive success or other similar quantities. I also give guidance on how to proceed if the so‐called memorylessness assumption does not hold in a given situation, and show how continuous and discrete times can be freely mixed if the biological situation dictates this to be the preferred option. Continuous‐time models can also be empirically fitted to data, and I review briefly the insight this gives into the so‐called ‘do hares eat lynx?’ paradox that has been plaguing the interpretation of the Hudson Bay hare and lynx dataset.

Theory predicts that sexual reproduction is difficult to maintain if asexuality is an option, yet sex is very common. To understand why, it is important to pay attention to repeatably occurring conditions that favour transitions to, or persistence of, asexuality. Geographic parthenogenesis is a term that has been applied to describe a large variety of patterns where sexual and related asexual forms differ in their geographic distribution. Often asexuality is stated to occur in a habitat that is, in some sense, marginal, but the interpretation differs across studies: parthenogens might not only predominate near the margin of the sexuals' distribution, but might also extend far beyond the sexual range; they may be disproportionately found in newly colonizable areas (e.g. areas previously glaciated), or in habitats where abiotic selection pressures are relatively stronger than biotic ones (e.g. cold, dry). Here, we review the various patterns proposed in the literature, the hypotheses put forward to explain them, and the assumptions they rely on. Surprisingly, few mathematical models consider geographic parthenogenesis as their focal question, but all models for the evolution of sex could be evaluated in this framework if the (often ecological) causal factors vary predictably with geography. We also recommend broadening the taxa studied beyond the traditional favourites. This article is part of the themed issue ‘Weird sex: the underappreciated diversity of sexual reproduction’.

Polyandry, by elevating sexual conflict and selecting for reduced male care relative to monandry, may exacerbate the cost of sex and thereby seriously impact population fitness. On the other hand, polyandry has a number of possible population-level benefits over monandry, such as increased sexual selection leading to faster adaptation and a reduced mutation load. Here, we review existing information on how female fitness evolves under polyandry and how this influences population dynamics. In balance, it is far from clear whether polyandry has a net positive or negative effect on female fitness, but we also stress that its effects on individuals may not have visible demographic consequences. In populations that produce many more offspring than can possibly survive and breed, offspring gained or lost as a result of polyandry may not affect population size. Such ecological ‘masking’ of changes in population fitness could hide a response that only manifests under adverse environmental conditions (e.g. anthropogenic change). Surprisingly few studies have attempted to link mating system variation to population dynamics, and in general we urge researchers to consider the ecological consequences of evolutionary processes.

Reversible phenotypic plasticity, the ability to change one’s phenotype repeatedly throughout life, can be selected for in environments that do not stay constant throughout an individual’s lifetime. It might also mitigate senescence, as the mismatch between the environment and a non-plastic individual’s traits is likely to increase as time passes. To understand why reversible plasticity may covary with lifespan, studies tend to assume unidirectional causality: plasticity evolves under suitable rates of environmental variation with respect to life history. Here we show that if lifespan also evolves in response to plasticity, then long life is not merely a context that sets the stage for lifelong plasticity. Instead, the causality is bidirectional because plasticity itself can select for longevity. Highly autocorrelated environmental fluctuations predict low investment in reversible plasticity and a phenotype that is poorly matched to the environment at older ages. Such environments select for high reproductive effort and short lifespans. Reversible phenotypic plasticity is expected to be favoured by long lifespan, as this increases the environmental variation individuals experience. Here, the authors develop a model showing how phenotypic plasticity can drive selection on lifespan, leading to coevolution of these traits.

Fitness usually increases when a male mates with more females, but is the same true for females? A new meta-analysis in PLOS Biology shows that females, like males, tend to have a positive relationship between the number of mates and their reproductive output. But why?

Ubiquity of cancer across the tree of life yields opportunities to understand variation in cancer defences across species. Peto’s paradox, the finding that large-bodied species do not suffer from more cancer despite having more cells at risk of oncogenic mutations compared to small species, can be explained if large size selects for better cancer defences. Since birds live longer than non-flying mammals of equivalent size, and are descendants of moderate-sized dinosaurs, we ask whether ancestral cancer defences are retained if body size shrinks in a lineage. Our model derives selection coefficients and fixation events for changes in cancer defences over macroevolutionary time, based on known relationships between body size, cancer risk, extrinsic mortality, metabolic rate, and effective population size. We show that, if mutation rate is sufficiently high and cancer defences are costly, we expect birds to have lower cancer defences than their dinosaurian ancestors. However, if the evolution of cancer suppression is mutation limited, due to e.g. pleiotropy, birds may have kept excessive dinosaurian cancer defences, possibly explaining their low cancer risk. Counterintuitively, birds can then be ‘too robust’ for their own good, if excessive cancer suppression requires compromising reproductive rates. Yet, evolutionary innovations such as flight can increase longevity and keep selection for cancer suppression intact in birds, even if flight requires small body size. Retaining dinosaur-level cancer defences can then be adaptive, particularly if the evolution of flight is accompanied by an increase in cancer risk due to metabolic scaling. Overall, our study suggests that studying cancer suppression in birds can reveal alternative mechanisms to those found in mammals, possibly inherited from birds’ dinosaurian ancestors.

Theories for the evolution of sex rarely include facultatively sexual reproduction. Sexual harassment by males is an underappreciated factor: it should at first sight increase the relative advantage of asexual reproduction by increasing the cost of sex. However, if the same females can perform either sexual or asexual life cycles, then females trying to reproduce asexually may not escape harassment. If resisting male harassment is costly, it might be beneficial for a female to accept a mating and undertake a sexual life cycle rather than ‘insist’ on an asexual one. We investigate the effects of sexual harassment on the maintenance of sex under different population densities. Our model shows that resisting matings pays off at low population densities, which leads to the complete extinction of males, and thus to the evolution of completely asexual populations. Facultative sex persists in a narrow range of slightly higher densities. At high densities, selection favours giving up resisting male mating attempts and thus sexual reproduction takes over. These interactions between the outcomes of sexual conflict and population density suggest an explanation for the rarity of facultative sex and also patterns of geographical parthenogenesis, where marginal environments with potentially low densities are associated with asexuality.

Isogamy is a reproductive system where all gametes are morphologically similar, especially in terms of size. Its importance goes beyond specific cases: to this day non-anisogamous systems are common outside of multicellular animals and plants, they can be found in all eukaryotic super-groups, and anisogamous organisms appear to have isogamous ancestors. Furthermore, because maleness is synonymous with the production of small gametes, an explanation for the initial origin of males and females is synonymous with understanding the transition from isogamy to anisogamy. As we show here, this transition may also be crucial for understanding why sex itself remains common even in taxa with high costs of male production (the twofold cost of sex). The transition to anisogamy implies the origin of male and female sexes, kickstarts the subsequent evolution of sex roles, and has a major impact on the costliness of sexual reproduction. Finally, we combine some of the consequences of isogamy and anisogamy in a thought experiment on the maintenance of sexual reproduction. We ask what happens if there is a less than twofold benefit to sex (not an unlikely scenario as large short-term benefits have proved difficult to find), and argue that this could lead to a situation where lineages that evolve anisogamy—and thus the highest costs of sex—end up being associated with constraints that make invasion by asexual reproduction unlikely (the ‘anisogamy gateway’ hypothesis). This article is part of the themed issue ‘Weird sex: the underappreciated diversity of sexual reproduction’.

Searching for mates is a critical stage in the life cycle of most internally, and many externally, fertilizing species. Males usually invest more in this costly activity than females, but the reasons for this are poorly understood. Previous models have shown that female-biased parental investment, including anisogamy, does not by itself select for male-biased mate searching, so it requires additional explanations. Here, we correct and expand upon earlier models, and present two novel hypotheses that might explain the evolution of male-biased mate searching. The \"carry-over hypothesis\" states that females benefit less from searching if the associated costs affect other stages of the life cycle, rather than arising only while searching. It is relevant to the evolution of morphological traits that improve searching efficiency but are also expressed in other contexts. The \"mating window hypothesis\" states that females benefit less from searching if their life cycle includes intervals during which the exact timing of mating does not matter for the appropriate timing of reproduction (e.g., due to sperm storage or delayed embryo implantation). Such intervals are more likely to exist for females given the general pattern of greater female parental investment. Our models shed new light on classic arguments about sex role evolution.

1. It is widely accepted that the arrival order of migratory birds is correlated with the condition of the birds, which leads to high quality individuals occupying prime sites. However, the theoretical backgrounds for this argument have been lacking. A simple game-theoretic model of arrival timing is provided which investigates the evolutionary stability of condition-dependent arrival order in territorial migrant birds. 2. Competition for territories or other priority-dependent benefits can lead to arrival dates far preceding the cost-minimizing date (the optimum date in the absence of competition) for all but the weakest individuals. Increasing the number of competitors can generate a `cascading' competition for early arrival, which advances arrival dates further apart from the individual optimum dates for the onset of breeding. 3. At equilibrium, arrival order corresponds strictly to condition order only if marginal costs of advancing arrival are always larger for individuals in lower condition. If spring mortality vacates territories for later-arriving birds, the criterion for `honest' arrival order becomes still stricter: differential survival costs should exist, but survival differences among individuals (or, alternatively, territory quality differences) should not be very large. 4. If the habitat is saturated so that there is a risk of not obtaining a territory at all, or if worst territories are of much lower value than the rest, competition may lead to the majority of the population arriving within a fairly short interval, followed by a much later floating fraction. This synchrony in the arrival of breeders imposes an increasing cost for the lesser fit breeding birds. Thus, arrival costs paid are not necessarily highest for earliest arriving individuals, but for those who have the most to lose if they drop a few steps in the arrival order. 5. Competition for high quality territories can also lead to partial migration, in which case birds in good condition are expected to be most likely to remain resident.