MHC heterozygosity and survival in red junglefowl

Genes of the major histocompatibility complex (MHC) form a vital part of the vertebrate immune system and play a major role in pathogen resistance. The extremely high levels of polymorphism observed at the MHC are hypothesised to be driven by pathogen‐mediated selection. Although the exact nature of selection remains unclear, three main hypotheses have been put forward; heterozygote advantage, negative frequency‐dependence and fluctuating selection. Here, we report the effects of MHC genotype on survival in a cohort of semi‐natural red junglefowl (Gallus gallus) that suffered severe mortality as a result of an outbreak of the disease coccidiosis. The cohort was followed from hatching until 250 days of age, approximately the age of sexual maturity in this species, during which time over 80% of the birds died. We show that on average birds with MHC heterozygote genotypes survived infection longer than homozygotes and that this effect was independent of genome‐wide heterozygosity, estimated across microsatellite loci. This MHC effect appeared to be caused by a single susceptible haplotype (CD_c) the effect of which was masked in all heterozygote genotypes by other dominant haplotypes. The CD_c homozygous genotype had lower survival than all other genotypes, but CD_c heterozygous genotypes had survival probabilities equal to the most resistant homozygote genotype. Importantly, no heterozygotes conferred greater resistance than the most resistant homozygote genotype, indicating that the observed survival advantage of MHC heterozygotes was the product of dominant, rather than overdominant processes. This pattern and effect of MHC diversity in our population could reflect the processes ongoing in similarly small, fragmented natural populations.     

Pathogen burden,co‐infection and major histocompatibility complex variability in the European badger (Meles meles)

Pathogen‐mediated selection is thought to maintain the extreme diversity in the major histocompatibility complex (MHC) genes, operating through the heterozygote advantage, rare‐allele advantage and fluctuating selection mechanisms. Heterozygote advantage (i.e. recognizing and binding a wider range of antigens than homozygotes) is expected to be more detectable when multiple pathogens are considered simultaneously. Here, we test whether MHC diversity in a wild population of European badgers (Meles meles) is driven by pathogen‐mediated selection. We examined individual prevalence (infected or not), infection intensity and co‐infection of 13 pathogens from a range of taxa and examined their relationships with MHC class I and class II variability. This population has a variable, but relatively low, number of MHC alleles and is infected by a variety of naturally occurring pathogens, making it very suitable for the investigation of MHC–pathogen relationships. We found associations between pathogen infections and specific MHC haplotypes and alleles. Co‐infection status was not correlated with MHC heterozygosity, but there was evidence of heterozygote advantage against individual pathogen infections. This suggests that rare‐allele advantages and/or fluctuating selection, and heterozygote advantage are probably the selective forces shaping MHC diversity in this species. We show stronger evidence for MHC associations with infection intensity than for prevalence and conclude that examining both pathogen prevalence and infection intensity is important. Moreover, examination of a large number and diversity of pathogens, and both MHC class I and II genes (which have different functions), provide an improved understanding of the mechanisms driving MHC diversity.     

Are there Ubiquitous Parasite-driven Major Histocompatibility Complex Selection Mechanisms in Gray Mouse Lemurs?

A major goal of evolutionary biology is to understand how selection drives local adaptation. For example, the major histocompatibility complex (MHC) plays an important role in the immune system, and high levels of MHC variation are thought to be a form of adaptation in natural populations. Individual MHC composition may influence parasite resistance via advantages associated with 1) heterozygosity, because heterozygotes recognize a broader range of different antigens than homozygotes (heterozygote advantage); 2) highly variable amino acid sequences in MHC alleles, allowing individuals to bind a broader spectrum of parasite-derived peptides (divergent-alleles advantage, a mechanistic variant of the heterozygote advantage model); or 3) specific MHC alleles (rare allele advantage or frequency dependent selection). We investigated relationships between gastrointestinal nematode burden and both adaptive immune gene variability (MHC class II DRB) and neutral microsatellites in free-living gray mouse lemurs (Microcebus murinus) native to a dry deciduous forest population in western Madagascar to test these hypotheses. The individual MHC composition was related to parasite infestation. Specific MHC alleles were involved in parasite resistance and the presence of common alleles negatively influenced infestation intensity. We found no support for the heterozygote advantage hypothesis, but we did find support for the divergent-MHC allele advantage hypothesis: Individuals with very divergent MHC alleles carried fewer and less intense nematode infestations than individuals with more similar alleles in the more variable dry deciduous forest population. These results indicate that intestinal parasites are important selection pressures under natural conditions and suggest that different selection mechanisms are not mutually exclusive. In contrast, we detected no association between neutral overall individual genetic diversity (measured via 17 microsatellites) and parasite load. Finally, we investigated the ubiquity of parasite-driven selection mechanisms by comparing our results with a previous study of a mouse lemur population from the climatically different littoral forest in southeastern Madagascar, ca. 500 km away. This revealed that different specific MHC alleles were involved in parasite resistance in the 2 habitats, showing that gene-parasite associations are not consistent between populations.     

Major histocompatibility complex heterozygosity reduces fitness in experimentally infected mice

It is often suggested that heterozygosity at major histocompatibility complex (MHC) loci confers enhanced resistance to infectious diseases (heterozygote advantage, HA, hypothesis), and overdominant selection should contribute to the evolution of these highly polymorphic genes. The evidence for the HA hypothesis is mixed and mainly from laboratory studies on inbred congenic mice, leaving the importance of MHC heterozygosity for natural populations unclear. We tested the HA hypothesis by infecting mice, produced by crossbreeding congenic C57BL/10 with wild ones, with different strains of Salmonella, both in laboratory and in large population enclosures. In the laboratory, we found that MHC influenced resistance, despite interacting wild-derived background loci. Surprisingly, resistance was mostly recessive rather than dominant, unlike in most inbred mouse strains, and it was never overdominant. In the enclosures, heterozygotes did not show better resistance, survival, or reproductive success compared to homozygotes. On the contrary, infected heterozygous females produced significantly fewer pups than homozygotes. Our results show that MHC effects are not masked on an outbred genetic background, and that MHC heterozygosity provides no immunological benefits when resistance is recessive, and can actually reduce fitness. These findings challenge the HA hypothesis and emphasize the need for studies on wild, genetically diverse species.     

Major histocompatibility complex (MHC) heterozygote superiority to natural multi-parasite infections in the water vole (Arvicola terrestris)

The fundamental role of the major histocompatibility complex (MHC) in immune recognition has led to a general consensus that the characteristically high levels of functional polymorphism at MHC genes is maintained by balancing selection operating through host–parasite coevolution. However, the actual mechanism by which selection operates is unclear. Two hypotheses have been proposed: overdominance (or heterozygote superiority) and negative frequency-dependent selection. Evidence for these hypotheses was evaluated by examining MHC–parasite relationships in an island population of water voles (Arvicola terrestris). Generalized linear mixed models were used to examine whether individual variation at an MHC class II DRB locus explained variation in the individual burdens of five different parasites. MHC genotype explained a significant amount of variation in the burden of gamasid mites, fleas (Megabothris walkeri) and nymphs of sheep ticks (Ixodes ricinus). Additionally, MHC heterozygotes were simultaneously co-infected by fewer parasite types than homozygotes. In each case where an MHC-dependent effect on parasite burden was resolved, the heterozygote genotype was associated with fewer parasites, and the heterozygote outperformed each homozygote in two of three cases, suggesting an overall superiority against parasitism for MHC heterozygote genotypes. This is the first demonstration of MHC heterozygote superiority against multiple parasites in a natural population, a mechanism that could help maintain high levels of functional MHC genetic diversity in natural populations.     

“Balancing” balancing selection? Assortative mating at the major histocompatibility complex despite molecular signatures of balancing selection

In vertebrate animals, genes of the major histocompatibility complex (MHC) determine the set of pathogens to which an individual's adaptive immune system can respond. MHC genes are extraordinarily polymorphic, often showing elevated nonsynonymous relative to synonymous sequence variation and sharing presumably ancient polymorphisms between lineages. These patterns likely reflect pathogen‐mediated balancing selection, for example, rare‐allele or heterozygote advantage. Such selection is often reinforced by disassortative mating at MHC. We characterized exon 2 of MHC class II, corresponding to the hypervariable peptide‐binding region, in song sparrows (Melospiza melodia). We compared nonsynonymous to synonymous sequence variation in order to identify positively selected sites; assessed evidence for trans‐species polymorphisms indicating ancient balancing selection; and compared MHC similarity of socially mated pairs to expectations under random mating. Six codons showed elevated ratios of nonsynonymous to synonymous variation, consistent with balancing selection, and we characterized several alleles similar to those occurring in at least four other avian families. Despite this evidence for historical balancing selection, mated pairs were significantly more similar at MHC than were randomly generated pairings. Nonrandom mating at MHC thus appears to partially counteract, not reinforce, pathogen‐mediated balancing selection in this system. We suggest that in systems where individual fitness does not increase monotonically with MHC diversity, assortative mating may help to avoid excessive offspring heterozygosity that could otherwise arise from long‐standing balancing selection.     

Natural selection on a leaf-shape polymorphism in the ivyleaf morning glory (Ipomoea hederacea)

Leaf shape is one of the most variable plant traits. Previous work has provided much indirect evidence that leaf-shape variation is adaptive and that leaf shape influences thermoregulation, water balance, and resistance to natural enemies. Nevertheless, there is little direct evidence that leaf shape actually affects plant fitness. In this study, we first demonstrate that populations of the ivyleaf morning glory, Ipomoea hederacea, in North and South Carolina are frequently polymorphic at a locus that influences leaf shape. We then employ several field experiments to show that this polymorphism is subject to selection. In two of the experiments, at different sites, heterozygotes enjoyed a fitness advantage over both homozygotes. At a third site, in one year directional selection favored lobed leaves, whereas in a second year the pattern of fitnesses was consistent with similar directional selection or heterozygote superiority. Computer simulations of heterozygote advantage under the high selfing rates of I. hederacea indicate that balancing selection of the magnitude observed can by itself stabilize the polymorphism, although spatially and temporally variable selection may also contribute to its long-term maintenance.     

Stable Equilibria at Two Loci in Populations with Large Selfing Rates

The equilibrium structure of two-locus, two-allele models with very large selfing rates is found using perturbation techniques. For free recombination, r = 1/2, the following results hold. If the heterozygotes do not have at least an approximate 30% advantage in fitness relative to homozygotes, a stable equilibrium with all alleles present is possible only if all of the homozygote fitnesses differ at most by approximately the outcrossing rate, t, and all stable polymorphic equilibria have disequilibrium values, D, that are at most on the order of the outcrossing rate. Once the heterozygote fitnesses are above the threshold, there are stable equilibria possible with D near its maximum possible value. The results show that the observed disequilibria in highly selfed plant populations are not likely to result from selection leading to an equilibrium.     

MHC allele frequency distributions under parasite-driven selection: A simulation model

Background  

The extreme polymorphism that is observed in major histocompatibility complex (MHC) genes, which code for proteins involved in recognition of non-self oligopeptides, is thought to result from a pressure exerted by parasites because parasite antigens are more likely to be recognized by MHC heterozygotes (heterozygote advantage) and/or by rare MHC alleles (negative frequency-dependent selection). The Ewens-Watterson test (EW) is often used to detect selection acting on MHC genes over the recent history of a population. EW is based on the expectation that allele frequencies under balancing selection should be more even than under neutrality. We used computer simulations to investigate whether this expectation holds for selection exerted by parasites on host MHC genes under conditions of heterozygote advantage and negative frequency-dependent selection acting either simultaneously or separately.     

Marginal Overdominance in Drosophila

A reanalysis of Drosophila viability data was undertaken to determine the role of genotype-environment interactions in the maintenance of polymorphism. Between-replicate variances of viabilities in chromosomal homozygotes and heterozygotes with the same mean fitnesses were compared, with the expectation that if the heterozygote variance were on the average greater, conditional overdominance would be prevalent; if it were less, partial dominance would be prevalent; and if it were the same, marginal overdominance of the type considered by Wallace (1968) would be the prevalent type of variation. In fact, heterozygote variance was slightly less. The work of Dempster (1955) and of Gillespie and Langley (1974) is cited to show that this situation can still lead to balanced polymorphisms. Their general model for genetic variation in populations, consistent with the viability data, is reinforced.     

Heterozygote advantage fails to explain the high degree of polymorphism of the MHC

Major histocompatibility (MHC) molecules are encoded by extremely polymorphic genes and play a crucial role in vertebrate immunity. Natural selection favors MHC heterozygous hosts because individuals heterozygous at the MHC can present a larger diversity of peptides from infectious pathogens than homozygous individuals. Whether or not heterozygote advantage is sufficient to account for a high degree of polymorphism is controversial, however. Using mathematical models we studied the degree of MHC polymorphism arising when heterozygote advantage is the only selection pressure. We argue that existing models are misleading in that the fitness of heterozygotes is not related to the MHC alleles they harbor. To correct for this, we have developed novel models in which the genotypic fitness of a host directly reflects the fitness contributions of its MHC alleles. The mathematical analysis suggests that a high degree of polymorphism can only be accounted for if the different MHC alleles confer unrealistically similar fitnesses. This conclusion was confirmed by stochastic simulations, including mutation, genetic drift, and a finite population size. Heterozygote advantage on its own is insufficient to explain the high population diversity of the MHC.Electronic Supplementary Material Supplementary material is available in the online version of this article at     

CYCLICAL ENVIRONMENTAL CHANGES AS A FACTOR MAINTAINING GENETIC POLYMORPHISM. 2. DIPLOID SELECTION FOR AN ADDITIVE TRAIT

The subject of this paper is polymorphism maintenance due to stabilizing selection with a moving optimum. It was shown that in case of two-locus additive control of the selected trait, global polymorphism is possible only when the geometric mean fitnesses of double homozygotes averaged over the period are lower than that of the single heterozygotes and of the double heterozygote (with a multiplier [1 – r]^p, which depends on recombination rate r and period length p). But local stability of polymorphism cannot be excluded even if geometric mean fitnesses of all double homozygotes are higher than that of all heterozygotes. We proved, that for logarithmically convex fitness functions, cyclical changes of the optimum cannot help in polymorphism maintenance in case of additive control of the selected trait by two equal loci. However, within the same class of fitness functions, nonequal gene action and/or dominance effect for one or both loci may lead to local polymorphism stability with large enough polymorphism attracting domain. The higher the intensity of selection and closer the linkage between selected loci the larger is this domain. Note that even simple cyclical selection could result in two forms of polymorphic limiting behavior: (a) usually expected forced cycle with a period equal to that of environmental changes; and (b) “supercycles,” nondumping auto-oscillations with a period comprising of hundreds of forced oscillation periods.     

Heterozygote advantage at MHC DRB may influence response to infectious disease epizootics

The effect of MHC polymorphism on individual fitness variation in the wild remains equivocal; however, much evidence suggests that heterozygote advantage is a major determinant. To understand the contribution of MHC polymorphism to individual disease resistance or susceptibility in natural populations, we investigated two MHC class II B loci, DQB and DRB, in the New Zealand sea lion (NZSL, Phocarctos hookeri). The NZSL is a threatened species which is unusually susceptible to death by bacterial infection at an early age; it has suffered three bacterial induced epizootics resulting in high mortality levels of young pups since 1997. The MHC DQB and DRB haplotypes of dead NZSL pups with known cause of death (bacteria, enteritis or trauma) were sequenced and reconstructed, compared to pups that survived beyond 2 months of age, and distinct MHC DRB allele frequency and genotype differences were identified. Two findings were striking: (i) one DRB allele was present only in dead pups, and (ii) one heterozygous DRB genotype, common in live pups, was absent from dead pups. These results are consistent with some functional relationship with these variants and suggest heterozygote advantage is operating at DRB. We found no association between heterozygosity and fitness at 17 microsatellite loci, indicating that general heterozygosity is not responsible for the effect on fitness detected here. This result may be a consequence of recurrent selection by multiple pathogen assault over recent years and highlights the importance of heterozygote advantage at MHC as a potential mechanism for fitness differences in wild populations.     

Stochastic selection in large and small populations

We describe two models of stochastic variation in selection intensity. In both models the arithmetic mean fitness of all genotypes is equal; in both models the geometric mean fitness of the heterozygous genotype is greater than that of both homozygous genotypes. In one model the correlation between the fitnesses of the homozygous genotypes is +1; in the other it is −1. We show that the expected time to absorption of an allele in a finite population is significantly retarded for all initial gene frequencies in the former model. The expected time to absorption of an allele in the latter model is retarded only at extreme initial gene frequencies; at intermediate initial gene frequencies the expected time to absorption is accelerated. We conclude that the criterion for polymorphism based on the geometric mean of the heterozygote being greater than that of both homozygotes provides only limited information about the fate of gene frequency.     

Strikingly high levels of heterozygosity despite 20 years of inbreeding in a clonal honey bee

Inbreeding (the mating between closely related individuals) often has detrimental effects that are associated with loss of heterozygosity at overdominant loci, and the expression of deleterious recessive alleles. However, determining which loci are detrimental when homozygous, and the extent of their phenotypic effects, remains poorly understood. Here, we utilize a unique inbred population of clonal (thelytokous) honey bees, Apis mellifera capensis, to determine which loci reduce individual fitness when homozygous. This asexual population arose from a single worker ancestor approximately 20 years ago and has persisted for at least 100 generations. Thelytokous parthenogenesis results in a 1/3 of loss of heterozygosity with each generation. Yet, this population retains heterozygosity throughout its genome due to selection against homozygotes. Deep sequencing of one bee from each of the three known sub‐lineages of the population revealed that 3,766 of 10,884 genes (34%) have retained heterozygosity across all sub‐lineages, suggesting that these genes have heterozygote advantage. The maintenance of heterozygosity in the same genes and genomic regions in all three sub‐lineages suggests that nearly every chromosome carries genes that show sufficient heterozygote advantage to be selectively detrimental when homozygous.     

Red Queen Processes Drive Positive Selection on Major Histocompatibility Complex (MHC) Genes

Major Histocompatibility Complex (MHC) genes code for proteins involved in the incitation of the adaptive immune response in vertebrates, which is achieved through binding oligopeptides (antigens) of pathogenic origin. Across vertebrate species, substitutions of amino acids at sites responsible for the specificity of antigen binding (ABS) are positively selected. This is attributed to pathogen-driven balancing selection, which is also thought to maintain the high polymorphism of MHC genes, and to cause the sharing of allelic lineages between species. However, the nature of this selection remains controversial. We used individual-based computer simulations to investigate the roles of two phenomena capable of maintaining MHC polymorphism: heterozygote advantage and host-pathogen arms race (Red Queen process). Our simulations revealed that levels of MHC polymorphism were high and driven mostly by the Red Queen process at a high pathogen mutation rate, but were low and driven mostly by heterozygote advantage when the pathogen mutation rate was low. We found that novel mutations at ABSs are strongly favored by the Red Queen process, but not by heterozygote advantage, regardless of the pathogen mutation rate. However, while the strong advantage of novel alleles increased the allele turnover rate, under a high pathogen mutation rate, allelic lineages persisted for a comparable length of time under Red Queen and under heterozygote advantage. Thus, when pathogens evolve quickly, the Red Queen is capable of explaining both positive selection and long coalescence times, but the tension between the novel allele advantage and persistence of alleles deserves further investigation.     

The Scent of Genetic Compatibility: Sexual Selection and the Major Histocompatibility Complex

Individuals in some species prefer mates carrying dissimilar genes at the major histocompatibility complex (MHC), which may function to increase the MHC or overall heterozygosity of progeny. Here I review the evidence for MHC-dependent mating preferences from recent studies, including studies on the underlying olfactory mechanisms and evolutionary functions. Many studies indicate that MHC genes influence odour, and some work is beginning to examine the potential role of MHC-linked olfactory receptor genes in mating preferences. MHC-dependent mating preference increases the MHC-heterozygosity of progeny, which is suspected to confer resistance to infectious diseases. In humans, heterozygosity at MHC loci is associated with increased resistance to hepatitis and HIV infections, but experimental evidence for the heterozygote advantage hypothesis has been lacking. Here I re-analyse data from previously published experimental infection studies with mice. I show that although overdominance is rare, resistance is often dominant, suggesting that heterozygotes are often protected. A second (nonmutually exclusive) possibility is that MHC-disassortative mating preferences promotes inbreeding avoidance. This hypothesis is supported by recent evidence that MHC genes play a role in kin recognition, and that mating with close kin has rather deleterious fitness consequences. In conclusion, I discuss other ways that MHC genes might influence sexual selection. The research on MHC-mediated mating preferences is integrating the study of animal behaviour with other seemingly disparate fields, including sensory biology and immunogenetics.     

Between‐year variation of MHC allele frequencies in great reed warblers: selection or drift?

The major histocompatibility complex (MHC) genes are extremely polymorphic and this variation is assumed to be maintained by balancing selection. Cyclic interactions between pathogens and their hosts could generate such selection, and specific MHC alleles or heterozygosity at certain MHC loci have been shown to confer resistance against particular pathogens. Here we compare the temporal variation in allele frequencies of 23 MHC class I alleles with that of 23 neutral microsatellite markers in adult great reed warblers (a passerine bird) in nine successive cohorts. Overall, the MHC alleles showed a significantly higher variation in allele frequencies between cohorts than the microsatellite alleles, using a multi-variate genetic analysis (amova). The frequency of two specific MHC alleles, A3e (P = 0.046) and B4b (P = 0.0018), varied more between cohorts than expected from random, whereas none of the microsatellite alleles showed fluctuations exceeding the expectation from stochastic variation. These results imply that the variation in MHC allele frequencies between cohorts is not a result of demographic events, but rather an effect of selection favouring different MHC alleles in different years.     

THE FITNESS CONSEQUENCES OF MULTIPLE-LOCUS HETEROZYGOSITY: THE RELATIONSHIP BETWEEN HETEROZYGOSITY AND GROWTH RATE IN PITCH PINE (PINUS RIGIDA MILL.)

Positive correlations between measures of “fitness” and the number of electrophoretic loci for which an individual is heterozygous have been observed in many species. Two major hypotheses have been proposed to explain this phenomenon: inbreeding depression and overdominance. Until recently, there has been no way to distinguish between these hypotheses. The overdominance model devised by Smouse (1986) is used here in a reanalysis of Ledig et al.‘s (1983) study of heterozygosity and growth rate in eight populations of pitch pine and is contrasted with an inbreeding-depression analysis. Ledig et al. (1983) regressed mean growth rate per heterozygosity class on the number of heterozygous loci, a method of analysis which, although it points to general trends in the data, does not differentiate between hypotheses. The correlations they obtained in four populations were significant only because regressing on the means eliminates most of the sum of squares for error and does not weight the unequally sized heterozygosity classes. Reanalysis of Ledig et al.‘s data using individuals, not means, showed no significant correlations between heterozygosity and fitness. A major assumption of Smouse's overdominance model is that genetic polymorphism is in part a reflection of selection for heterozygotes at genetic equlibrium. The homozygote for the most frequent allele at a locus should be more fit than a homozygote for a less frequent allele, with the heterozygote superior to both homozygotes. Smouse's model predicts a negative, linear relationship between fitness and “adaptive distance,” a variable that for a heterozygote is zero and for homozygotes is equal to the inverse of the frequency of the corresponding allele. The adaptive-distance model accounted for between 15% and 50% of the variation in growth rate within eight P. rigida population samples by accounting for genotypic differences at eight polymorphic loci. This is over twice as much of the variation in growth rate accounted for by Ledig et al.'s (1983) analysis using individuals. Significant correlations were found between adaptive distance and growth rate in four of the eight populations, but in only two of the populations were more of the partial coefficients negative than positive, as would be predicted by the overdominance hypothesis. The remaining two populations in which correlations were significant did not lend themselves to such clear-cut interpretation, as the majority of the partial coefficients were positive. Positive partial coefficients indicate that the growth rate of the heterozygote is inferior to that of at least one of the homozygotes. The adaptive-distance analysis provides evidence that specific genotypes do play a role in determining growth rate in pitch pine. The correlation between growth rate and adaptive distance increased significantly with the age of the population, possibly reflecting competition subsequent to crown closure.     

The dynamics of the rec-system in variable environments: haploid selection in a cyclical two-state environment

The dynamics of a 3-locus infinite population with non-overlapping generations and panmixia was studied. Loci are di-allelic: two loci affect fitness under cyclical symmetric haploid selection while the third one is a modifier of recombination (rec-modifier). Selection favors alternatively haplotypes AB and ab or Ab and aB. It has been proven that under alternating selection (when period of selection consists of two generations) a dominant suppressor of recombination is displaced and the allele for non-zero recombination becomes fixed within the population. For populations with inversion heterozygosity within the selective system (i.e. with zero recombination in heterozygote for rec-modifier and non-zero for homozygotes) fixation of one of the alleles (depending on the initial point) at the rec-modifier locus is predicted. For other values of recombination parameters, the behavior of the system was studied numerically. A full bifurcation picture of parameters was obtained. Many of the results related to the case of a two-generation period hold also in the case of longer period lengths.