Maintenance of variation in mutualism by screening

Models of partner choice leading to mutualism raise a conceptual problem: directional selection for high‐quality partners should ultimately erode variation in partner quality. How do we explain the persistence of variation in partner quality observed in nature? The problem arises in all models of partner choice, including screening models, in which a host induces potential symbionts of different quality to screen themselves by assigning them different costs and rewards. Using a screening model in which costs and rewards are sometimes assigned incorrectly, I show that a stable polymorphism can arise because rewards are higher when partners vary in quality than when there is only one type of partner. Partner quality, therefore, undergoes negative frequency‐dependent selection even though there is a preference for high‐quality partners. This also shows that partner choice by screening does not need to be totally accurate to be effective—inaccuracies enable both effective screening and the maintenance of variation.     

Like-with-like preference and sexual mixing models

Two new general methods for incorporating like-with-like preference into one-sex mixing models in epidemiology are presented. The first is a generalization of the preferred mixing equation, while the second comprises a transformation of a general preference function for partners of similar sexual activity levels. Both methods satisfy the constraints implicit in a mixing model. The behavior of the transformation preference method is illustrated, and it is compared with the standard proportionate mixing model.     

Sexual isolation in Drosophila. I. Theoretical models for multiple-choice experiments.

Several simple models are developed to calculate expected mating frequencies in ethological isolation experiments. They take into account the effect that the peculiar sexual behavior of Drosophila species can have in multiple-choice experiments. These models depend on only three basic parameters: male competitive ability (C), female receptivity (R) and the coefficient of females acceptance (A). Two types of model can be distinguished: (1) models with discrete preferences, in which A is a measure of the percentage of females accepting a particular kind of male and (2) models with continuous preferences, in which A represents the probability of acceptance for each courtship. It is demonstrated that the information rendered by just one experiment, although effective for determining whether sexual isolation exists, it is insufficient to estimate its degree or to demonstrate that it is asymmetrical. Further developments of the models under more complex conditions as well as their implications for reinforcement and founder effect theories are discussed.     

Likelihood-based inference for stochastic models of sexual network formation

Sexually-transmitted diseases (STDs) constitute a major public health concern. Mathematical models for the transmission dynamics of STDs indicate that heterogeneity in sexual activity level allow them to persist even when the typical behavior of the population would not support endemicity. This insight focuses attention on the distribution of sexual activity level in a population. In this paper, we develop several stochastic process models for the formation of sexual partnership networks. Using likelihood-based model selection procedures, we assess the fit of the different models to three large distributions of sexual partner counts: (1) Rakai, Uganda, (2) Sweden, and (3) the USA. Five of the six single-sex networks were fit best by the negative binomial model. The American women's network was best fit by a power-law model, the Yule. For most networks, several competing models fit approximately equally well. These results suggest three conclusions: (1) no single unitary process clearly underlies the formation of these sexual networks, (2) behavioral heterogeneity plays an essential role in network structure, (3) substantial model uncertainty exists for sexual network degree distributions. Behavioral research focused on the mechanisms of partnership formation will play an essential role in specifying the best model for empirical degree distributions. We discuss the limitations of inferences from such data, and the utility of degree-based epidemiological models more generally.     

Combining sources in stable isotope mixing models: alternative methods

Stable isotope mixing models are often used to quantify source contributions to a mixture. Examples include pollution source identification; trophic web studies; analysis of water sources for soils, plants; or water bodies, and many others. A common problem is having too many sources to allow a unique solution. We discuss two alternative procedures for addressing this problem. One option is a priori to combine sources with similar signatures so the number of sources is small enough to provide a unique solution. Aggregation should be considered only when isotopic signatures of clustered sources are not significantly different, and sources are related so the combined source group has some functional significance. For example, in a food web analysis, lumping several species within a trophic guild allows more interpretable results than lumping disparate food sources, even if they have similar isotopic signatures. One result of combining mixing model sources is increased uncertainty of the combined end-member isotopic signatures and consequently the source contribution estimates; this effect can be quantified using the IsoError model (). As an alternative to lumping sources before a mixing analysis, the IsoSource mixing model () can be used to find all feasible solutions of source contributions consistent with isotopic mass balance. While ranges of feasible contributions for each individual source can often be quite broad, contributions from functionally related groups of sources can be summed a posteriori, producing a range of solutions for the aggregate source that may be considerably narrower. A paleohuman dietary analysis example illustrates this method, which involves a terrestrial meat food source, a combination of three terrestrial plant foods, and a combination of three marine foods. In this case, a posteriori aggregation of sources allowed strong conclusions about temporal shifts in marine versus terrestrial diets that would not have otherwise been discerned.     

Fishes as models in studies of sexual selection and parental care

Fishes are by far the most diverse group of vertebrates. This fact is in no way, however, reflected in their use as model organisms for understanding sexual selection or parental care. Why is this so? Is it because fishes are actually poor models? The usefulness of fishes as models for sexual selection and parental care is discussed by emphasizing some problems inherent in fish studies, along with a number of reasons why fishes are indeed excellently suited. The pros and cons of fishes as models are discussed mainly by comparison with birds, the most popular model organisms in animal behaviour. Difficulties include a lack of background knowledge for many species, and the problems of marking and observing fishes in their natural environment. Positive attributes include the diversity of lifestyles among fishes, and the ease with which they can be studied experimentally in the laboratory. How useful fish models can be is briefly illustrated by the impressive and broadly relevant advances derived from studies of guppies Poecilia reticulata and three‐spined sticklebacks Gasterosteus aculeatus . A selection of topics is highlighted where fish studies have either advanced or could greatly enhance, the understanding of processes fundamental to animal reproductive dynamics. Such topics include sex role dynamics, the evolution of female ornamentation and mate choice copying. Finally, a number of potential pitfalls in the future use of fish as models for sexual selection and parental care are discussed. Researchers interested in these issues are recommended to make much more extensive use of fish models, but also to adopt a wider range of models among fishes.     

Using mathematical models to understand the AIDS epidemic

The most urgent public-health problem today is to devise effective strategies to minimize the destruction caused by the AIDS epidemic. This complex problem will involve medical advances and new public-health and education initiatives. Mathematical models based on the underlying transmission mechanisms of the AIDS virus can help the medical/scientific community understand and anticipate its spread in different populations and evaluate the potential effectiveness of different approaches for bringing the epidemic under control. Before we can use models to predict the future, we must carefully test them against the past spread of the infection and for sensitivity to parameter changes. The long and extremely variable incubation period and the low probability of transmitting the AIDS virus in a single contact imply that population structure and variations in infectivity both play an important role in its spread. The population structure is caused by differences between people in numbers of sexual partners and the use of intravenous drugs and because of the way in which people mix among age, ethnic, and social groups. We use a simplified approach to investigate the effects of variation in incubation periods and infectivity specific to the AIDS virus, and we compare a model of random partner choices with a model in which partners both come from similar behavior groups.     

Spatially implicit plankton population models: Transient spatial variability

Ocean plankton models are useful tools for understanding and predicting the behaviour of planktonic ecosystems. However, when the regions represented by the model grid cells are not well mixed, the population dynamics of grid cell averages may differ from those of smaller scales (such as the laboratory scale). Here, the ‘mean field approximation’ fails due to ‘biological Reynolds fluxes’ arising from nonlinearity in the fine-scale biological interactions and unresolved spatial variability. We investigate the domain-scale behaviour of two-component, 2D reaction-diffusion plankton models producing transient dynamics, with spatial variability resulting only from the initial conditions. Failure of the mean field approximation can be quite significant for sub grid-scale mixing rates applicable to practical ocean models. To improve the approximation of domain-scale dynamics, we investigate implicit spatial resolution methods such as spatial moment closure. For weak and moderate strengths of biological nonlinearity, spatial moment closure models generally yield significant improvements on the mean field approximation, especially at low mixing rates. However, they are less accurate given weaker transience and stronger nonlinearity. In the latter case, an alternative ‘two-spike’ approximation is accurate at low mixing rates. We argue that, after suitable extension, these methods may be useful for understanding and skillfully predicting the large-scale behaviour of marine ecosystems.     

The initialization and sensitivity of multigroup models for the transmission of HIV

We study two multigroup mathematical models of the spread of HIV. In the differential infectivity model, the infected population is divided into groups according to their infectiousness, and HIV is primarily spread by a small, highly infectious, group of superspreaders. In the staged-progression model, every infected individual goes through a series of infection stages and the virus is primarily spread by individuals in an initial highly infectious stage or in the late stages of the disease. We demonstrate the importance of choosing appropriate initial conditions, and define a new approach to distributing the initial population among the subgroups so as to minimize the artificial transients in the solutions due to unbalanced initial conditions. We demonstrate that the rate of removal in and out of a population is an important, yet often neglected, effect. We also illustrate the importance of distinguishing between the number of partners a person has and the number of contacts per partner. By assuming that people with many partners have fewer contacts per partner than people with few partners, we found that the epidemic is less sensitive to the partner acquisition rate than one might expect. However, because the probability of transmission of HIV per contact is low, the epidemic is very sensitive to the number of contacts per partner. Modeling this distinction is particularly important when estimating the impact of programs which encourage people to have fewer sexual partners.     

Disassortative Age-Mixing Does Not Explain Differences in HIV Prevalence between Young White and Black MSM: Findings from Four Studies

Objective

Age disassortativity is one hypothesis for HIV disparities between Black and White MSM. We examined differences in age mixing by race and the effect of partner age difference on the association between race and HIV status.

Design

We used data from four studies of MSM. Participants reported information about recent sexual partners, including age, race, and sexual behavior. Two studies were online with a US sample and two focused on MSM in Atlanta.

Methods

We computed concordance correlation coefficients (CCCs) by race across strata of partner type, participant HIV status, condom use, and number of partners. We used Wilcoxon rank-sum tests to compare Black and White MSM on partner age differences across five age groups. Finally, we used logistic regression models using race, age, and partner age difference to determine the odds ratio of HIV-positive serostatus.

Results

Of 48 CCC comparisons, Black MSM were more age-disassortative than White MSM in only two. Furthermore, of 20 comparisons of median partner age, Black and White MSM differed in two age groups. One indicated larger age gaps among the Black MSM (18-19). Prevalent HIV infection was associated with race and age. Including partner age difference in the model resulted in a 2% change in the relative odds of infection among Black MSM.

Conclusions

Partner age disassortativity and partner age differences do not differ by race. Partner age difference offers little predictive value in understanding prevalent HIV infection among Black and White MSM, including diagnosis of HIV-positive status among self-reported HIV-negative individuals.     

Testing homogeneity in Weibull-regression models

In survival studies with families or geographical units it may be of interest testing whether such groups are homogeneous for given explanatory variables. In this paper we consider score type tests for group homogeneity based on a mixing model in which the group effect is modelled as a random variable. As opposed to hazard-based frailty models, this model presents survival times that conditioned on the random effect, has an accelerated failure time representation. The test statistics requires only estimation of the conventional regression model without the random effect and does not require specifying the distribution of the random effect. The tests are derived for a Weibull regression model and in the uncensored situation, a closed form is obtained for the test statistic. A simulation study is used for comparing the power of the tests. The proposed tests are applied to real data sets with censored data.     

Replicator-dynamics models of sexual conflict

Evolutionary conflict between the sexes has been studied in various taxa and in various contexts. When the sexes are in conflict over mating rates, natural selection favors both males that induce higher mating rates and females that are more successful at resisting mating attempts. Such sexual conflict may result in an escalating coevolutionary arms race between males and females. In this article, we develop simple replicator-dynamics models of sexual conflict in order to investigate its evolutionary dynamics. Two specific models of the dependence of a female's fitness on her number of matings are considered: in model 1, female fitness decreases linearly with increasing number of matings and in model 2, there is an optimal number of matings that maximizes female fitness. For each of these models, we obtain the conditions for a coevolutionary process to establish costly male and female traits and examine under what circumstances polymorphism is maintained at equilibrium. Then we discuss how assumptions in previous models of sexual conflict are translated to fit to our model framework and compare our results with those of the previous studies. The simplicity of our models allows us to consider sexual conflict in various contexts within a single framework. In addition, we find that our model 2 shows more complicated evolutionary dynamics than model 1. In particular, the population exhibits bistability, where the evolutionary outcome depends on the initial state, only in model 2.     

Avoiding the bullies: The resilience of cooperation among unequals

Can egalitarian norms or conventions survive the presence of dominant individuals who are ensured of victory in conflicts? We investigate the interaction of power asymmetry and partner choice in games of conflict over a contested resource. Previous models of cooperation do not include both power inequality and partner choice. Furthermore, models that do include power inequalities assume a static game where a bully’s advantage does not change. They have therefore not attempted to model complex and realistic properties of social interaction. Here, we introduce three models to study the emergence and resilience of cooperation among unequals when interaction is random, when individuals can choose their partners, and where power asymmetries dynamically depend on accumulated payoffs. We find that the ability to avoid bullies with higher competitive ability afforded by partner choice mostly restores cooperative conventions and that the competitive hierarchy never forms. Partner choice counteracts the hyper dominance of bullies who are isolated in the network and eliminates the need for others to coordinate in a coalition. When competitive ability dynamically depends on cumulative payoffs, complex cycles of coupled network-strategy-rank changes emerge. Effective collaborators gain popularity (and thus power), adopt aggressive behavior, get isolated, and ultimately lose power. Neither the network nor behavior converge to a stable equilibrium. Despite the instability of power dynamics, the cooperative convention in the population remains stable overall and long-term inequality is completely eliminated. The interaction between partner choice and dynamic power asymmetry is crucial for these results: without partner choice, bullies cannot be isolated, and without dynamic power asymmetry, bullies do not lose their power even when isolated. We analytically identify a single critical point that marks a phase transition in all three iterations of our models. This critical point is where the first individual breaks from the convention and cycles start to emerge.     

Interval estimates for epidemic thresholds in two-sex network models

Epidemic thresholds in network models of heterogeneous populations characterized by highly right-skewed contact distributions can be very small. When the population is above the threshold, an epidemic is inevitable and conventional control measures to reduce the transmissibility of a pathogen will fail to eradicate it. We consider a two-sex network model for a sexually transmitted disease which assumes random mixing conditional on the degree distribution. We derive expressions for the basic reproductive number (R(0)) for one and heterogeneous two-population in terms of characteristics of the degree distributions and transmissibility. We calculate interval estimates for the epidemic thresholds for stochastic process models in three human populations based on representative surveys of sexual behavior (Uganda, Sweden, USA). For Uganda and Sweden, the epidemic threshold is greater than zero with high confidence. For the USA, the interval includes zero. We discuss the implications of these findings along with the limitations of epidemic models which assume random mixing.     

Evolutionary limits to the frequency of aggression between related or unrelated conspecifics in diploid species with simple mendelian inheritance

Game theory has been used by some authors to analyse evolutionary limits to the expression of aggression in theoretical haploid parthenogenetic species. Others have examined frequency dependent selection, of which aggression may be a case, by applying population genetic models to diploid species. A model is presented which attempts to combine these two approaches. Game theory is used to determine evolutionarily stable strategies and corresponding stable polymorphisms for a two-strategy game played by members of a diploid sexual species, when choice of strategy is determined by two alleles at a single locus. Results are given for dominant, co-dominant and recessive determination of choice of the more aggressive of two strategies, for two levels of relationship: unrelated players and sibs. It is found that for a range of models of single locus inheritance the evolutionarily stable strategy (ESS) determined for haploid species remains the stable population strategy for diploid sexual species, when players are unrelated. In sibling contestants aggression is reduced. The mixed strategy haploid ESS underestimates, but the pure strategy haploid ESS provides a good indication of the degree to which relatedness lessens aggression in diploid species. For both haploid and diploid species there may be a considerable advantage to confining conflicts to kin.     

Population models of sperm-dependent parthenogenesis

Organisms that reproduce by sperm-dependent parthenogenesis are asexual clones that require sperm of a sexual host to initiate egg production, without the genome of the sperm contributing genetic information to the zygote. Although sperm-dependent parthenogenesis has some of the disadvantages of sex (requiring a mate) without the counterbalancing advantages (mixing of parental genotypes), it appears amongst a wide variety of species. We develop initial models for the density-dependent dynamics of animal populations with sperm-dependent parthenogenesis (pseudogamy or gynogenesis), based on the known biology of the common Enchytraeid worm Lumbricillus lineatus. Its sperm-dependent parthenogenetic populations are reproductive parasites of the hermaphrodite sexual form. Our logistic models reveal two alternative requirements for coexistence at density-dependent equilibria: (i) If the two forms differ in competitive ability, the form with the lower intrinsic birth rate must be compensated by a more than proportionately lower competitive impact from the other, relative to intraspecific competition, (ii) If the two forms differ in their intrinsic capacity to exploit resources, the sperm-dependent parthenogen must be superior in this respect and must have a lower intrinsic birth rate. In general for crowded environments we expect a sperm-dependent parthenogen to compete strongly for limiting resources with the sexual sibling species. Its competitive impact is likely to be weakened by its genetic uniformity, however, and this may suffice to cancel any advantage of higher intrinsic growth rate obtained from reproductive investment only in egg production. We discuss likely thresholds of coexistence for other sperm-dependent parthenogens. The fish Poeciliopsis monacha-lucida likewise obtains an intrinsic growth advantage from reduced investment in male gametes, and so its persistence is likely to depend on it being a poor competitor. The planarian Schmidtea polychroa obtains no such intrinsic benefit because it produces fertile sperm, and its persistence may depend on superior resource exploitation.     

Incorporating uncertainty and prior information into stable isotope mixing models

Stable isotopes are a powerful tool for ecologists, often used to assess contributions of different sources to a mixture (e.g. prey to a consumer). Mixing models use stable isotope data to estimate the contribution of sources to a mixture. Uncertainty associated with mixing models is often substantial, but has not yet been fully incorporated in models. We developed a Bayesian-mixing model that estimates probability distributions of source contributions to a mixture while explicitly accounting for uncertainty associated with multiple sources, fractionation and isotope signatures. This model also allows for optional incorporation of informative prior information in analyses. We demonstrate our model using a predator–prey case study. Accounting for uncertainty in mixing model inputs can change the variability, magnitude and rank order of estimates of prey (source) contributions to the predator (mixture). Isotope mixing models need to fully account for uncertainty in order to accurately estimate source contributions.     

Stochastic models for aggregation processes

Three models are presented, which describe the aggregation of objects into groups and the distributions of groups sizes and group numbers within habitats. The processes regarded are pure accumulation processes which involve only formation and invasion of groups. Invasion represents the special case of fusion when only single objects - and not groups - join a group of certain size. The basic model is derived by a single parameter, the formation probability q, which represents the probability of an object to form a new group. A novel, discrete and finite distribution that results for the group sizes is deduced from this aggregation process and it is shown that it converges to a geometric distribution if the number of objects tends to infinity. Two extensions of this model, which both converge to the Waring distribution, are added: the model can be extended either with a beta distributed formation probability or with the assumption that the invasion probability depends on the group size. Relationships between the limiting distributions involved are discussed.     

Some generalized stochastic compartment models for digesta flow

Digesta flow models have been based on linear compartment theory that assumes exponential retention times, and on a generalized theory that incorporates nonexponential (Erlang) retention times (Matis, 1987, Journal of Theoretical Biology 124, 371-376). This paper develops a new family of passage models for heterogeneous digesta by mixing the previous models with assumed parametric, usually gamma, mixing distributions. The utility of the resulting models is demonstrated with experimental data on two treatments, namely a chopped and a ground straw, given to each of four cows. Treatment differences are apparent in the preferred model form and in the means of the estimated mean residence times. The models are relatively easy to fit to data using standard estimation procedures, and they should have broad application to other compartment modeling problems with "heterogeneous particles."