Pair formation models with maturation period

The standard model for pair formation is generalized to include a maturation period. This model in the form of three coupled delay equations is a special case of the general age-structured model for a two-sex population. The exact conditions for the existence of an exponential (persistent) two-sex solution are derived. It is shown that this solution is unique and locally stable. In order to achieve these results the theory of homogeneous differential equations is extended to a class of homogeneous delay equations.     

A standardized two-sex stable population

The “problem of the sexes” has been one of trying to reconcile inconsistent male and female demographic rates. The present paper deals with that question in the context of a two-sex stable population. A “rectangular” population, with equal numbers of persons in each age-sex group, is introduced as a standard, and a standardization relationship expressed in Eq. (3) relates changes in rectangular population rates to changes in age-sex composition. The standardization relationship is shown to satisfy a number of desirable properties and produce a realistic two-sex model. The standardization approach is then applied to data from Sweden 1973 and the United States 1963, and the results and their implications are discussed. In particular, it is seen that the intrinsic growth rate in a two-sex stable population is not necessarily bounded by the growth rate of the associated male and female one-sex stable populations.     

Are non-sexual models appropriate for predicting the impact of virus-vectored immunocontraception?

In response to the need to efficiently control mammal pest populations while avoiding unnecessary suffering, applied and theoretical ecologists have recently focused on virus-vectored immunocontraception (VVIC). So far, modellers have only considered a non-sexual approach (models of sexually reproducing populations without explicitly discerning between the sexes), which appears dubious in view of the sex-specificity of VVIC agents. In this paper, we derive and compare predictions of non-sexual and two-sex models of the spread of a VVIC agent in a host population in order to assess the adequacy of non-sexual models in this context. Our results show that predictions of non-sexual and two-sex models generally diverge and that non-sexual models often fail to predict the control impact of VVIC. We thus recommend using two-sex models, especially if the mating system and life history of the target species are known. Our analysis also shows that female-specific viruses generally give better results than male-specific ones, and suggests that virus choice should focus more on its sterilizing power rather than transmission efficiency.     

Pair formation

A multitype pair formation model for a one-sex population, without separation, with given type distribution of singles, produces a distribution of pairs with the given type distribution as a marginal distribution. The pair distribution can be seen as a nonnegative symmetric matrix. For this matrix representation formulas have been given years ago and have been widely used. The goal of the paper is to understand these formulas in probabilistic terms and give a meaning to their coefficients. Our approach connects the formulas to the problem of completing a substochastic matrix to a stochastic matrix. In this way the coefficients in the representation formula can be interpreted as preferences and insight can be gained into the set of distributions respecting given preferences. In order to put these questions into a wider perspective, the classical two-sex pair formation models are reviewed and embedded into the class of one-sex models, and dynamic models are designed that yield pair distributions as limit elements.     

A note on ‘A generalized two-sex logistic model’

We re-visit the recently published paper on a generalization of the two-sex logistic model by Maxin and Sega [A generalized two-sex logistic model, J. Biol. Dyn. 7(1) (2013), pp. 302–318]. We show that the logistic assumption of a non-increasing birth rate can be replaced by a more general assumption of a non-increasing ratio between the female/male birth and mortality rate. In this note we indicate the changes necessary in the proofs of the theorems in [D. Maxin and L. Sega, A generalized two-sex logistic model, J. Biol. Dyn. 7(1) (2013), pp. 302–318] and discuss several situations where this new assumption is useful.     

An interactive nonrandom-mating two-sex model whose intrinsic growth rate lies between one-sex rates.

In a recent paper, the author has presented an interactive two-sex model and applied it to the U.S. population, 1940–1971. While the model has removed many inconsistencies in the one-sex models for males and females, it has two major limitations. First, the model assumes random mating of men and women within their reproductive ages. Second, it gives an intrinsic growth rate that is not necessarily intermediate between the male and the female intrinsic growth rates, as obtained from the one-sex models. A new two-sex model is provided in this paper that removes both these limitations. The new model is applied to the U.S. population for the years 1940–1974.     

A two-sex demographic model with single-dependent divorce rate

Divorce appears to be one of the least studied demographic processes, both empirically and in two-sex demographic models. In this paper, we study mathematical as well as biological implications of the assumption that the divorce rate is positively affected by the amount of single (i.e., unmarried/unpaired) individuals in the population. We do that by modifying the classical exponential two-sex model accounting for pair formation and separation. We model the divorce rate as an increasing function of the single population size and show that the single population pressure on the established couples alters the exponential behavior of the classical model in which the divorce rate is assumed constant. In particular, the total population size becomes bounded and a unique positive equilibrium exists. In addition, a Hopf bifurcation analysis around the positive equilibrium shows that the modified model may exhibit sustained oscillations.     

Exponential growth in age-structured two-sex populations

We consider a continuous age-structured two-sex population model which is given by a semilinear system of partial differential equations with nonlocal boundary conditions and is a simpler case of Fredrickson-Hoppensteadt model. The non-linearity is introduced by a source term, called from its physical meaning, the marriage function. The explicit form of the marriage function is not known; however, there is an understanding among the demographers about the properties it should satisfy. We have shown that the homogeneity property of the non-linearity leads to the fact that the system supports exponentially growing persistent solutions using a general form of the marriage function and its properties. This suggests that the model can be viewed as a possible extension of the one-sex stable population theory to monogamously mating two-sex populations.     

A Reconsideration of Galton's Problem (Using a Two-Sex Population)

The main purposes of this paper are to promote and expound the bisexual Galton–Watson branching process as a relevant model for the consideration of Francis Galton's problem regarding the extinction of surnames of “men of note.” A scheme for adapting the bisexual process to consider Galton's problem is introduced. A necessary and sufficient condition for the certain extinction of a male-induced property in a two-sex species is presented. An approach for calculating the extinction of a male-generated characteristic in the two-sex species is proposed. That approach is then used to find the probability of the extinction of surnames in a bisexual population for Alfred Lotka's data based on an United States Census. Finally, these results are then compared with the classic extinction probabilities (from Lotka) associated with the traditional Galton–Watson branching process using asexual reproduction.     

A note on Das Gupta's two-sex population model

Some critical comments are made on the two-sex deterministic population model recently put forward by P. Das Gupta in this Journal, both in its interpretation and relation to any possible stochastic version of the model, and in regard to the validity of solutions other than the asymptotic limiting case.     

A generalized two-sex logistic model

We provide a generalization of the logistic two-sex model with ephemeral pair-bonds and with stable couples without assuming any specific mathematical form for fertility, mortality and the mating function. In particular, we establish a necessary and sufficient condition on the fertility/mortality density-dependent ratio that ensures the existence of the logistic behaviour. Several differences and similarities between the two models are also provided.     

A model of HIV-1 transmission for urban areas of Africa

A model of the transmission dynamics of HIV-1, appropriate to urban areas of Africa, is presented and its behaviour explored through numerical studies. The model is a two-sex model with age-dependent demographic and behavioural parameters. Adults are classified by age, sex, risk group, and epidemiologic status. HIV-1 is transmitted to adults heterosexually, and to infants and children vertically and parenterally. Numerical studies show that, while AIDS will slow population growth, growth rates do not become negative for reasonable parameter values. The sex and age patterns of infection are explored, as is the potential economic impact of changes in the sex and age composition of the population.     

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.     

The two-sex problem with persistent unions: a generalization of the birth matrix-mating rule model

The birth matrix-mating rule (BMMR) model solves the two-sex problem of classical stable population theory by allowing births to adjust to changes in a population's age-sex composition. To avoid complications the BMMR model assumes that unions last for only a single period. This paper drops that assumption and presents the BMMRPU (BMMR persistent unions) model. To establish the existence of equilibrium in the BMMRPU model, the existence proof used in the BMMR model is supplemented by a fixed-point argument.     

Persistent age-distributions for a pair-formation model

Conditions on the vital rates of a two-sex population are presented which imply the existence or nonexistence of exponentially growing persistent age-distributions.     

A stochastic model for the spread of a sexually transmitted disease which results in a scale-free network

A stochastic model for the spread of a sexually transmitted disease (STD) is presented. To reflect varying degrees of promiscuity among individuals it is assumed that the infectivity of any infected individual is proportional to the number of previous contacts the individual has had with other infected individuals. In both the simple single-sex model and in the more complex two-sex model, the tree graphs of the infection exhibit scale-free network behaviour (i.e. power-law behaviour in the upper tail of the degree distribution). The distributions of the size of the infection and of the ring number (distance from the original source of the infection) are determined.     

随机配对种群的两性模型的动力学性态

基于Kendall-Goodman模型，提出了一个两性具有不同生理性态的随机配对的两性模型．如果不考虑密度制约因素，那么模型存在一个全局渐近稳定的指数解；如果考虑密度制约因素，对于给定的一个出生函数，得到了唯一正平衡态存在及全局稳定的充要条件．结论表明，无论是否考虑密度制约因素，种群的性比总是稳定的．     

Mate Choice and the Origin of Menopause

Human menopause is an unsolved evolutionary puzzle, and relationships among the factors that produced it remain understood poorly. Classic theory, involving a one-sex (female) model of human demography, suggests that genes imparting deleterious effects on post-reproductive survival will accumulate. Thus, a ‘death barrier’ should emerge beyond the maximum age for female reproduction. Under this scenario, few women would experience menopause (decreased fertility with continued survival) because few would survive much longer than they reproduced. However, no death barrier is observed in human populations. Subsequent theoretical research has shown that two-sex models, including male fertility at older ages, avoid the death barrier. Here we use a stochastic, two-sex computational model implemented by computer simulation to show how male mating preference for younger females could lead to the accumulation of mutations deleterious to female fertility and thus produce a menopausal period. Our model requires neither the initial assumption of a decline in older female fertility nor the effects of inclusive fitness through which older, non-reproducing women assist in the reproductive efforts of younger women. Our model helps to explain why such effects, observed in many societies, may be insufficient factors in elucidating the origin of menopause.     

Mistaken Gender: 5-Alpha Reductase Hermaphroditism and Biological Reductionism in Sexual Identity Reconsidered

Biomedical research on a rare hermaphroditic condition among males of the rural Dominican Republic has supported biologically reductionist explanations of male gender identity development. I reinterpret this research by comparison to a parallel case among the Sambia of Papua New Guinea. Meanings of mistaken gender are reviewed to contrast sex assignment and socialization in two-sex and three-sex cultural systems. I refute the unicausal biological model and suggest that psychocultural factors in these cultures are more salient in the explanation of hermaphroditic sex-role change.     

Time-dependent solution to age-structured equations for sexual populations

The continuous age-time theory of a two-sex population, due to Fredrickson and based on M'Kendrick's equation, has been solved by means of the generation expansion. Difficulties due to the nonlinear birth condition which are encountered when the theory is cast in the form of ordinary differential equations for the total male and female populations are thereby avoided. The expansion can be expected to be useful when the time span over which the solution is required is of the order of only a few generations. Thus, it is a “short time” solution, supplementary to the “long time” solution, the steady state of exponential growth.