Stability properties of underdominance in finite subdivided populations

IN ISOLATED populations underdominance leads to bistable evolutionary dynamics: below a certain mutant allele frequency the wildtype succeeds. Above this point, the potentially underdominant mutant allele fixes. In subdivided populations with gene flow there can be stable states with coexistence of wildtypes and mutants: polymorphism can be maintained because of a migration-selection equilibrium, i.e., selection against rare recent immigrant alleles that tend to be heterozygous. We focus on the stochastic evolutionary dynamics of systems where demographic fluctuations in the coupled populations are the main source of internal noise. We discuss the influence of fitness, migration rate, and the relative sizes of two interacting populations on the mean extinction times of a group of potentially underdominant mutant alleles. We classify realistic initial conditions according to their impact on the stochastic extinction process. Even in small populations, where demographic fluctuations are large, stability properties predicted from deterministic dynamics show remarkable robustness. Fixation of the mutant allele becomes unlikely but the time to its extinction can be long.     

The influence of local extinctions on the probability of fixation of chromosomal rearrangements

We investigated the influence of local extinctions in a subdivided population on the probability of fixation of an initially rare allele, for different migration rates. The selective regimes considered were strict underdominance, meiotic drive, and underdominance associated with meiotic drive. We show that local extinctions can increase the probability of fixation of initially rare alleles in underdominant loci for relatively high migration rates, even when both homozygotes have the same fitness. This increase is due to drift during founder events. On the contrary, local extinctions decrease the probability of fixation of alleles favoured by meiotic drive. For a locus where both meiotic drive and underdominance act, the effect of local extinctions depends on the relative strength of the two selective regimes and the initial frequency of the rare allele. For parameter values such that the rare allele is initially selected against, local extinctions decrease the probability of fixation for low migration rates while they cause an increase for moderate migration rates. When the parameter values are such that the rare allele should always be favoured by selection, local extinctions always decrease the probability of fixation. In this latter case, we show the existence of an optimal migration rate which maximizes the probability of fixation.     

Dispersal of introduced house sparrows Passer domesticus: an experiment

An important issue concerning the introduction of non-indigenous organisms into local populations is the potential of the introduced individuals to spread and interfere both demographically and genetically with the local population. Accordingly, the potential of spatial dispersal among introduced individuals compared with local individuals is a key parameter to understand the spatial and temporal dynamics of populations after an introduction event. In addition, if the variance in dispersal rate and distance is linked to individual characteristics, this may further affect the population dynamics. We conducted a large-scale experiment where we introduced 123 house sparrows from a distant population into 18 local populations without changing population density or sex ratio. Introduced individuals dispersed more frequently and over longer distances than residents. Furthermore, females had higher probability of dispersal than males. In females, there was also a positive relationship between the wing length and the probability of dispersal and dispersal distance. These results suggest that the distribution and frequency of introduced individuals may be predicted by their sex ratio as well as their phenotypic characteristics.     

Behavioral choices based on patch selection: a model using aggregation methods

The aim of this work is to study the influence of patch selection on the dynamics of a system describing the interactions between two populations, generically called 'population N' and 'population P'. Our model may be applied to prey-predator systems as well as to certain host-parasite or parasitoid systems. A situation in which population P affects the spatial distribution of population N is considered. We deal with a heterogeneous environment composed of two spatial patches: population P lives only in patch 1, while individuals belonging to population N migrate between patch 1 and patch 2, which may be a refuge. Therefore they are divided into two patch sub-populations and can migrate according to different migration laws. We make the assumption that the patch change is fast, whereas the growth and interaction processes are slower. We take advantage of the two time scales to perform aggregation methods in order to obtain a global model describing the time evolution of the total populations, at a slow time scale. At first, a migration law which is independent on population P density is considered. In this case the global model is equivalent to the local one, and under certain conditions, population P always gets extinct. Then, the same model, but in which individuals belonging to population N leave patch 1 proportionally to population P density, is studied. This particular behavioral choice leads to a dynamically richer global system, which favors stability and population coexistence. Finally, we study a third example corresponding to the addition of an aggregative behavior of population N on patch 1. This leads to a more complicated situation in which, according to initial conditions, the global system is described by two different aggregated models. Under certain conditions on parameters a stable limit cycle occurs, leading to periodic variations of the total population densities, as well as of the local densities on the spatial patches.     

Selection for intermediate mortality and reproduction rates in a spatially structured population

How local interactions influence both population and evolutionary dynamics is currently a key topic in theoretical ecology. We use a 'well-mixed' analytical model and spatially explicit individual-based models to investigate a system where a population is subject to rare disturbance events. The disturbance can only propagate through regions of the population where the density of individuals is sufficiently high and individuals affected by the disturbance die shortly after. We find that populations where individuals are sessile often exhibit very different dynamic behaviour when compared to populations where individuals are mobile and spatially well mixed. When mutations are allowed which affect either offspring birth rates or mortality rates, the well-mixed populations always evolve to a state where a single disturbance event leads to extinction. Populations often persist substantially longer if individuals are sessile and they disperse their offspring locally. We also find that for sessile populations selection may favour short-lived individuals with limited offspring production. Population dynamics are found to be strongly influenced by the host characters that are evolving and the rate at which host variation is introduced into the system.     

Evolution of dispersal in density regulated populations: A haploid model

The evolution of dispersal is explored in a density-dependent framework. Attention is restricted to haploid populations in which the genotypic fitnesses at a single diallelic locus are decreasing functions of the changing number of individuals in the population. It is shown that migration between two populations in which the genotypic response to density is reversed can maintain both alleles when the intermigration rates are constant or nondecreasing functions of the population densities. There is always a unique symmetric interior equilibrium with equal numbers but opposite gene frequencies in the two populations, provided the system is not degenerate. Numerical examples with exponential and hyperbolic fitnesses suggest that this is the only stable equilibrium state under constant positive migration rates (m) less than . Practically speaking, however, there is only convergence after a reasonable number of generations for relatively small migration rates ( ). A migration-modifying mutant at a second, neutral locus, can successfully enter two populations at a stable migration-selection balance if and only if it reduces the intermigration rates of its carriers at the original equilibrium population size. Moreover, migration modification will always result in a higher equilibrium population size, provided the system approaches another symmetric interior equilibrium. The new equilibrium migration rate will be lower than that at the original equilibrium, even when the modified migration rate is a nondecreasing function of the population sizes. Therefore, as in constant viability models, evolution will lead to reduced dispersal.     

Evolutionary dynamics of prey exploitation in a metapopulation of predators

In well-mixed populations of predators and prey, natural selection favors predators with high rates of prey consumption and population growth. When spatial structure prevents the populations from being well mixed, such predators may have a selective disadvantage because they do not make full use of the prey's growth capacity and hence produce fewer propagules. The best strategy then depends on the degree to which predators can monopolize the exploitation of local prey populations, which in turn depends on the spatial structure, the number of migrants, and, in particular, the stochastic nature of the colonization process. To analyze the evolutionary dynamics of predators in a spatially structured predator-prey system, we performed simulations with a metapopulation model that has explicit local dynamics of nonpersistent populations, keeps track of the number of emigrants entering the migration pool, assumes individuals within local populations as well as within the migration pool to be well mixed, and takes stochastic colonization into account. We investigated which of the predator's exploitation strategies are evolutionarily stable and whether these strategies minimize the overall density of prey, as is the case in Lotka-Volterra-type models of competitive exclusion. This was analyzed by pairwise invasibility plots based on short-term simulations and tested by long-term simulation experiments of competition between resident and mutant predator-types that differed in one of the following parameters: the prey-to-predator conversion efficiency, the per capita prey consumption rate, or the per capita emigration rate from local populations. In addition, we asked which of these three strategies are most likely to evolve. Our simulations showed that under selection for conversion efficiency the predator-prey system always goes globally extinct yet persists under selection for consumption or emigration rates and that the evolutionarily stable (ES) exploitation strategies do not maximize local population growth rates. The most successful exploitation strategy minimizes the overall density of prey but does not make it settle exactly at the minimum. The system did not settle at the point where the mean time to co-invasion (i.e., immigration of a second predator in a local prey population) equals the mean local interaction time (an idea borne out from studies on host exploitation strategies in host-pathogen systems) but rather where the mean time to co-invasion was larger. The ES exploitation strategies represent more prudent strategies than the ones that minimize prey density. Finally, we show that-compared to consumption-emigration is a more likely target for selection to achieve prudent exploitation and that prudent exploitation strategies can evolve only provided the prey-to-predator conversion efficiency is subject to constraints.     

The neutral effective migration rate in a mainland-island context

Genetic influx into a population often does not correspond to the real migration rate (m) of individuals, due to class structure within the population. The effective migration rate (m(e)) is a concept to measure gene flow in such a situation. The ratio of the effective migration rate to the real migration rate (m(e)/m) is called the gene flow factor, and represents the degree of gene flow modification. Prior authors proposed different definitions of the effective migration rate. These may be categorized into two groups: the neutral effective migration rate and the selective effective migration rate. In this article, we construct a general model of a class-structured population with a mainland-island structure. Using the model, we prove that the gene flow factor of the neutral effective migration rate converges to the mean reproductive value of immigrants if the limit is taken with the real migration rate converging to zero. This limit theorem provides a novel interpretation of gene flow and can be used to derive approximation formulae of the neutral effective migration rate. We illustrate this method analyzing two examples, sex ratio distortion due to extrinsic factors and hybrid zones with underdominance.     

On the stability of polymorphic host-pathogen populations

The stability of populations of hosts and micro-parasites is investigated where each consists of n varieties that are equal in every respect except that each strain of parasites can infect only one specific strain of hosts and none of the others. Collectively the host strains are limited by a carrying capacity and through this limitation the host populations interact with each other. Hosts are assumed to reproduce asexually or such that different strains do not mate or are not fertile if they do. When the excess death rate caused by the pathogenic parasites is sufficiently large, then the host population is regulated to an equilibrium below the carrying capacity of the environment. This polymorphic equilibrium is shown to be locally asymptotically stable. When one of the parasite strains is absent, then all the other strains die out asymptotically. However, if host resistance to all infectious strains of parasites is achieved at the cost of a lower birthrate of the resistant host strain, then, if a certain condition for the various parameters is satisfied, stable coexistence between infected and resistant hosts is possible. There are many examples where susceptibility and resistance of hosts depends upon the conformation of specific proteins that are involved in host-parasite interactions and hence upon alleles at genetic loci that code for these proteins. We propose that polymorphism in wildtype populations which has been the subject of much theorizing in mathematical genetics may be due to host-pathogen interactions. Our model suggests how a polymorphic population, once established, can remain polymorphic indefinitely.     

Stochasticity, complex spatial structure, and the feasibility of the shifting balance theory

Sewall Wright's shifting balance theory of evolution posits a mechanism by which a structured population may escape local fitness optima and find a global optimum. We examine a one-locus, two-allele model of underdominance in populations with differing spatial arrangements of demes, both analytically and with Monte Carlo simulations. We find that inclusion of variance in interpatch connectivities can significantly reduce the number of generations required for fixation of the more favorable allele relative to island and stepping-stone models. Although time to fixation increases with migration rate in all cases, the presence of one or two relatively isolated demes may reduce the number of generations by 80% or more. These results suggest that the shifting balance process may operate under less restrictive conditions than those found with a simple spatial arrangement of demes.     

The significance of over- and underdominance for the maintenance of genetic polymorphisms. II. Overdominance and instability with random mating

Part I of the present series demonstrates that globally stable polymorphic equilibria may show underdominance in Darwinian fitness. Hence, overdominance in fitness can no longer be conceived of as a necessary condition for the stability of a polymorphism. In the present paper, the question is posed as to whether overdominance is at least sufficient for this stability. A population of randomly mating individuals is considered, where selection operates uniquely through differential fecundities of particular mating types and may generate either a heterozygote excess or deficit relative to Hardy-Weinberg proportions. It turns out that both unstable central overdominance and stable central underdominance are possible and that their occurrence is strongly related to an excess or a deficiency of heterozygotes in the vicinity of the regions of instability or stability. As one consequence, the above suggested sufficiency of heterozygote superiority is not valid, even in random mating populations. Based on the results of both papers of this series, which demonstrate the inadequacy of over- and underdominance as indicators of stability or instability, a modified overdominance principle is discussed. This principle states that a biallelic polymorphism is maintained if the heterozygote is superior in its degree of "heterogamous self-replication" to the degrees of "autogamous self-replication" of the corresponding homozygotes. It is derived with the help of fractional fitnesses, and it is pointed out that certain ratios of these may be more useful for finding evolutionary constants which govern the maintenance of genetic polymorphisms than are ratios of total fitnesses.     

Rapid evolutionary responses in a translocated population of intertidal snail (<Emphasis Type="Italic">Bembicium vittatum</Emphasis>) utilise variation from different source populations

The artificial movement of individuals between populations (translocation) can be an effective way to increase genetic diversity within populations, but few studies have undertaken long term genetic monitoring to determine if variation introduced by translocation is maintained over many generations or whether it can be used to adapt to local conditions. Here, we report on the changes in morphological and molecular variation over a 12-year period in a population of an intertidal littorine snail (Bembicium vittatum) that was created by mixing individuals from three geographically disjunct populations. These source populations differ genetically in shell shape and in allele frequency at several allozyme loci. We found that the translocated population had higher allozyme diversity than any of the source populations and that this pattern was maintained over multiple generations. Variation in shell shape also increased, but this declined over time as shells became taller. Some allozyme loci also showed significant changes in frequency over time. These changes were not consistently towards the genetic makeup of a single source population, and in the case of shell shape, were towards a phenotype that was most suited to the local environment. Our results suggest that genetic variation introduced into a population by translocation can be rapidly incorporated and used to adapt to local conditions without domination by a single source population’s genome. However, more studies are needed before generalisations on the benefits of mixing individuals from disjunct populations can be made.     

Cytoplasmic feminizing elements in a two-population model: infection dynamics, gene flow modification, and the spread of autosomal suppressors

We investigate the dynamics of a cytoplasmic parasitic element with feminizing effect in a two-population model. We assume that the host species has a ZZ/ZW sex determination system. Our analysis reveals that the feminizer and the W chromosome can stably coexist by dominating different populations if the transmission rate differs significantly between the populations and migration is sufficiently weak. In the equilibrium of coexistence, genetic influx at any host autosomal locus is strongly enhanced in the population where infection is prevalent but not modified in the other population. We further explore conditions for the spread of autosomal suppressor genes that reduce transmission of feminizing elements to the cost of host viability, and compute their equilibrium frequencies. Our results confirm the hypothesis that selfish genetic elements convert infected host populations into genetic sinks, thereby restricting the spread of transmission suppressors.     

Genetic similarities within and between human populations

The proportion of human genetic variation due to differences between populations is modest, and individuals from different populations can be genetically more similar than individuals from the same population. Yet sufficient genetic data can permit accurate classification of individuals into populations. Both findings can be obtained from the same data set, using the same number of polymorphic loci. This article explains why. Our analysis focuses on the frequency, omega, with which a pair of random individuals from two different populations is genetically more similar than a pair of individuals randomly selected from any single population. We compare omega to the error rates of several classification methods, using data sets that vary in number of loci, average allele frequency, populations sampled, and polymorphism ascertainment strategy. We demonstrate that classification methods achieve higher discriminatory power than omega because of their use of aggregate properties of populations. The number of loci analyzed is the most critical variable: with 100 polymorphisms, accurate classification is possible, but omega remains sizable, even when using populations as distinct as sub-Saharan Africans and Europeans. Phenotypes controlled by a dozen or fewer loci can therefore be expected to show substantial overlap between human populations. This provides empirical justification for caution when using population labels in biomedical settings, with broad implications for personalized medicine, pharmacogenetics, and the meaning of race.     

Punishment of polygyny

We investigated the evolution of monogamy (one male, one female) and polygyny (one male, more than one female). In particular, we studied whether it is possible for a mutant polygynous mating strategy to invade a resident population of monogamous breeders and, alternatively, whether a mutant monogamy can invade resident polygyny. Our population obeys discrete-time Ricker dynamics. The role of males and females in the breeding system is incorporated via the harmonic birth function. The results of the invasability analysis are straightforward. Polygyny is an evolutionarily stable strategy mating system; this holds throughout the examined range of numbers of offspring produced per female. So that the two strategies can coexist, polygyny has to be punished. The coexistence of monogamy and polygyny is achieved by reducing the offspring number for polygyny relative to monogamy. This yields long-term persistence of the strategies for all offspring numbers studied. An alternative punishment is to increase the sensitivity of polygynous breeders to population density. The coexistence is possible only with a limited range of offspring produced. The third way to achieve coexistence of the two mating strategies is to assume that individuals live in a spatially structured population, where dispersal links population subunits to a network. Reducing the dispersal rate of polygynous breeders relative to that of monogamous individuals makes the coexistence feasible. However, for monogamy to persist, the number of offspring produced has to be relatively high.     

The spatial spread of altruism versus the evolutionary response of egoists

Several recent models have shown that altruism can spread in viscous populations, i.e. in spatially structured populations within which individuals interact only with their immediate neighbours and disperse only over short distances. I first confirm this result with an individual-based model of a viscous population, where an individual can vary its level of investment into a behaviour that is beneficial to its neighbours but costly to itself. Two distinct classes of individuals emerge: egoists with no or very little investment into altruism, and altruists with a high level of investment; intermediate levels of altruism are not maintained. I then extend the model to investigate the consequences of letting interaction and dispersal distances evolve along with altruism. Altruists maintain short distances, while the egoists respond to the spread of altruism by increasing their interaction and dispersal distances. This allows the egoistic individuals to be maintained in the population at a high frequency. Furthermore, the coevolution of investment into altruism and interaction distance can lead to a stable spatial pattern, where stripes of altruists (with local interactions) alternate with stripes of egoists (with far-reaching interactions). Perhaps most importantly, this approach shows that the ease with which altruism spreads in viscous populations is maintained despite countermeasures evolved by egoists.     

Costs of coexistence along a gradient of competitor densities: an experiment with arvicoline rodents

1. Costs of coexistence for species with indirect resource competition usually increase monotonically with competitor numbers. Very little is known though about the shape of the cost function for species with direct interference competition. 2. Here we report the results of an experiment with two vole species in artificial coexistence in large enclosures, where density of the dominant competitor species (Microtus agrestis) was manipulated. Experimental populations of the subordinate vole species (Clethrionomys glareolus) were composed of same aged individuals to study distribution of costs of coexistence with a dominant species within an age-cohort. 3. Survival and space use decreased gradually with increasing field vole numbers. Thus, responses to interference competition in our system appeared to be similar as expected from resource competition. The total number of breeders was stable. Reproductive characteristics such as the timing of breeding, and the litter size were not affected. In the single species enclosures a proportion of surviving individuals were not able to establish a breeding territory against stronger conspecifics. Under competition with heterospecifics such nonbreeders suffered high mortality, whereas the breeders survived. 4. Combined interference of dominant conspecifics and heterospecifics probably increased the frequency of aggressive interactions, social stress and mortality for the weaker individuals within a homogeneous age cohort of the subordinate competitor population. 5. Our results suggest, that in open systems where bank voles are outcompeted over the breeding season by faster reproducing field voles, animals able to establish a territory may be able to withstand competitor pressure, while nonbreeding bank vole individuals are forced to emigrate to suboptimal forest habitats.     

Examples of the effect of genetic variation on competing species

An ordinary differential equation model for two competing populations with genetic variation in one population is presented. The degree of frequency dependence needed to produce various configurations of stable equilibria is discussed. For example, if the fitnesses are frequency independent then there may exist stable polymorphism although the genetically varying population becomes extinct in each fixation plane. Stable polymorphism where the genetically invariant population becomes extinct in each fixation plane requires frequency dependence in the fitness of the genetically invariant population.     

ON THE EVOLUTION OF PSEUDOGAMY

A. density- and frequency-dependent model for the evolution and maintenance of pseudogamous females is developed and analyzed. Ecological as well as evolutionary aspects of pseudogamy are discussed. Criteria are described for the stable coexistence of sexual females and pseudogamous females under natural conditions. The conditions for invasion of a normal bisexual population by pseudogamous females are less stringent than the conditions for stable coexistence. Hence, we expect that some populations will be characterized by unstable sex ratios over time (with the resulting local extinction due to lack of males) while other populations will be characterized by stable sex ratios over time. If high population sex ratios (i.e., many females to few males) are to be stable, the net population growth rate must be large, and there can be no successful male preference for sexual females.     

Permanence of discrete-time Kolmogorov systems for two species and saturated fixed points

This paper considers the dynamics of a discrete-time Kolmogorov system for two-species populations. In particular, permanence of the system is considered. Permanence is one of the concepts to describe the species coexistence. By using the method of an average Liapunov function, we have found a simple sufficient condition for permanence of the system. That is, nonexistence of saturated boundary fixed points is enough for permanence of the system under some appropriate convexity or concavity properties for the population growth rate functions. Numerical investigations show that for the system with population growth rate functions without such properties, the nonexistence of saturated boundary fixed points is not sufficient for permanence, actually a boundary periodic orbit or a chaotic orbit can be attractive despite the existence of a stable coexistence fixed point. This result implies, in particular, that existence of a stable coexistence fixed point is not sufficient for permanence.