On two-sex models leading to stable populations

A mathematical, 2-sex, stable-population model that treats sex and age simultaneously was developed. The birth function is expressed in the form of an integral formula which when solved yields the intrinsic growth rate. Some of the concepts involved include the intrinsic sex ratio, the intrinsic age-specific birthrates, and the gross and net reproduction rates for both sexes. The new model demonstrates that both sexes can coexist in a stable population, whereas in the 1-sex model the intrinsic rates are internally inconsistent in regard to sex. 7 forms of the model are discussed and applied to data for the United States from 1963.     

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.     

From homogeneous eigenvalue problems to two-sex population dynamics

Journal of Mathematical Biology - Enclosure theorems are derived for homogeneous bounded order-preserving operators and illustrated for operators involving pair-formation functions introduced by...     

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.     

A two-sex life history model of handicap signaling

The males of many species (such as peacocks) develop excessively large traits, which appear to interfere with their agility and hence survival probability. Moreover, it is also observed that females seem to prefer mating males with such clumsy traits. Zahavi (1975) proposed a handicap theory to explain this phenomenon, suggesting that this trait/preference interaction is a way in which strong males can signal their viability by yielding a handicap in terms of a clumsy trait size. This paper presents a two-sex model of selfish genes that generates this particular male-female interaction, and characterizes the conditions behind a handicap equilibrium. We first show the female dominance result of Bateman (1948) in this two-sex model, and then specify the relevant equilibrium conditions, including the incentive compatibility condition for females, the individual rationality condition for males, and the stability condition of population composition. Identifying these conditions helps us understand the various features of the searching/signaling of sex selection in evolution.     

Relationships in a simple harmonic mean two-sex fertility model

A continuous time two-sex stable population model that does not recognize age is examined under the “harmonic mean” consistency condition of equation (2). Solutions for the stable population intrinsic growth rate r and the sex composition 5 are presented in equations (5) and (6). The process of stabilization is examined, and it is shown that, given two basic constraints, any initial population sex composition will eventually converge to the stable population value. An algebraic solution for the discrete case where the sex ratio at birth is unity is presented and used to describe the trajectory to stability of several hypothetical populations. A closed form algebraic expression for the trajectory to stability is presented for the continuous model in the special case of no mortality.     

Blood group and height in a multiethnic population

In a sample of 4,472 boys, aged 17-18 years, resident in Jerusalem, those with blood groups B or AB tended to be slightly shorter than groups O and A (p = 0.011). Participants were classified into 8 groups according to father's country of origin: Israel, Southern Europe/Balkans, rest of Europe, North Africa, Iraq, Iran, Yemen and the rest of Asia. The association of ABO blood group, classified according to the presence of the B antigen (groups B and AB) or its absence (O and A), with height differed in the 8 origin groups (p = 0.026 for interaction). In 7 of the 8 groups, subjects with the B allele were either shorter or of equal height to groups O and A and in only one instance were they taller. These findings do not support the generalizability of a positive association of the presence of the B antigen with height suggested by Borecki et al. [1985].     

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.     

Generalized stable population theory

In generalizing stable population theory we give sufficient, then necessary conditions under which a population subject to time dependent vital rates reaches an asymptotic stable exponential equilibrium (as if mortality and fertility were constant). If x ₀(t) is the positive solution of the characteristic equation associated with the linear birth process at time t, then rapid convergence of x ₀(t) to x ₀ and convergence of mortality rates produce a stable exponential equilibrium with asymptotic growth rate x ₀–1. Convergence of x ₀(t) to x ₀ and convergence of mortality rates are necessary. Therefore the two sets of conditions are very close. Various implications of these results are discussed and a conjecture is made in the continuous case.     

Y-chromosomal diversity in the population of Guinea-Bissau: a multiethnic perspective

Background  

The geographic and ethnolinguistic differentiation of many African Y-chromosomal lineages provides an opportunity to evaluate human migration episodes and admixture processes, in a pan-continental context. The analysis of the paternal genetic structure of Equatorial West Africans carried out to date leaves their origins and relationships unclear, and raises questions about the existence of major demographic phenomena analogous to the large-scale Bantu expansions. To address this, we have analysed the variation of 31 binary and 11 microsatellite markers on the non-recombining portion of the Y chromosome in Guinea-Bissau samples of diverse ethnic affiliations, some not studied before.     

The rate of convergence of a generalized stable population

In an age-structured population that grows exponentially, each age groupP _i(t) at periodt is asymptotically equivalent tox ₀ ^t for some positive number x₀. In this paper we show that the speed at which the ith age group reaches its exponential state of equilibrium can be measured by the rate at which the ratio v_i(t)=P_i(t)/p_i(t–1) converges tox ₀. The age specific rate of convergence is determined by considering a quantityr satisfyingv _i(t)-x ₀ ¦ r ^t whent is large;R _i=Infr (over all initial populations,r satisfying the above inequality) is the R-factor used in numerical analysis to measure the rate at which the sequencev _i (t) converges tox ₀;S _i =- In R_i is then defined as the rate of convergence to stability of the ith age group. The case of constant net maternity rates is studied in detail; in this contextS ₀ is compared to the population entropyH, which was proposed by Tuljapurkar (1982) as a measure of the rate of convergence to stability.     

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.     

水稻早世代稳定718群体的获得及分析

水稻早世代稳定现象是近年来发现的一种特殊现象,其中多个早世代稳定群体是在利用来自水稻双胚苗中的多倍体与不同二倍体材料杂交得到的。本实验利用一自然突变的多倍体,与二倍体水稻太平369杂交,F_1代大部分为非整倍体,其中5株为二倍体,收取这5株二倍体上的种子,种下得到 F_2群体且定名为718群体。该群体田间观察整齐一致,经过对田间调查数据和微卫星结果的分析,表明该群体为一早世代稳定群体。     

Sex-biased dispersal and the speed of two-sex invasions

Population models that combine demography and dispersal are important tools for forecasting the spatial spread of biological invasions. Current models describe the dynamics of only one sex (typically females). Such models cannot account for the sex-related biases in dispersal and mating behavior that are typical of many animal species. In this article, we construct a two-sex integrodifference equation model that overcomes these limitations. We derive an explicit formula for the invasion speed from the model and use it to show that sex-biased dispersal may significantly increase or decrease the invasion speed by skewing the operational sex ratio at the invasion's low-density leading edge. Which of these possible outcomes occurs depends sensitively on complex interactions among the direction of dispersal bias, the magnitude of bias, and the relative contributions of females and males to local population growth.     

Population differentiation and migration: coalescence times in a two-sex island model for autosomal and X-linked loci

Evolutionists have debated whether population-genetic parameters, such as effective population size and migration rate, differ between males and females. In humans, most analyses of this problem have focused on the Y chromosome and the mitochondrial genome, while the X chromosome has largely been omitted from the discussion. Past studies have compared F_ST values for the Y chromosome and mitochondrion under a model with migration rates that differ between the sexes but with equal male and female population sizes. In this study we investigate rates of coalescence for X-linked and autosomal lineages in an island model with different population sizes and migration rates for males and females, obtaining the mean time to coalescence for pairs of lineages from the same deme and for pairs of lineages from different demes. We apply our results to microsatellite data from the Human Genome Diversity Panel, and we examine the male and female migration rates implied by observed F_ST values.     

Self-reported ethnicity, genetic structure and the impact of population stratification in a multiethnic study

It is well-known that population substructure may lead to confounding in case–control association studies. Here, we examined genetic structure in a large racially and ethnically diverse sample consisting of five ethnic groups of the Multiethnic Cohort study (African Americans, Japanese Americans, Latinos, European Americans and Native Hawaiians) using 2,509 SNPs distributed across the genome. Principal component analysis on 6,213 study participants, 18 Native Americans and 11 HapMap III populations revealed four important principal components (PCs): the first two separated Asians, Europeans and Africans, and the third and fourth corresponded to Native American and Native Hawaiian (Polynesian) ancestry, respectively. Individual ethnic composition derived from self-reported parental information matched well to genetic ancestry for Japanese and European Americans. STRUCTURE-estimated individual ancestral proportions for African Americans and Latinos are consistent with previous reports. We quantified the East Asian (mean 27%), European (mean 27%) and Polynesian (mean 46%) ancestral proportions for the first time, to our knowledge, for Native Hawaiians. Simulations based on realistic settings of case–control studies nested in the Multiethnic Cohort found that the effect of population stratification was modest and readily corrected by adjusting for race/ethnicity or by adjusting for top PCs derived from all SNPs or from ancestry informative markers; the power of these approaches was similar when averaged across causal variants simulated based on allele frequencies of the 2,509 genotyped markers. The bias may be large in case-only analysis of gene by gene interactions but it can be corrected by top PCs derived from all SNPs.