On a simple epidemic model with a heterogeneous infective population

This paper is concerned with a generalization of the simple epidemic model in which the infective population is partitioned intom classes, each of specific infectiousness. Attention is restricted, however, to the case where all the meeting rates between two individuals are equal to each other. Both deterministic and stochastic versions are examined. In either case the development in time of the epidemic process is investigated by exploiting a connection with the standard simple epidemic model. Finally, it is shown that the technique used also applies to a similar model for the spread of information.     

A simple model of a Gyrodactylus population

Experimental populations of 20 Gyrodactylus alexanderi Mizelle &; Kritsky, 1967, on 19 isolated Gasterosteus aculeatus at 15°C increased for 2 weeks to a mean of 61, then decreased in 2 further weeks to a mean of 9. Fish that lost their fluke infestations were refractory to further infestation for about 3 weeks.The chief factors affecting fluke abundance were measured, including reproduction and mortality rates of flukes on fish, rate of shedding by the fish, mortality rates of flukes while off fish, and the rate of reattachment of the flukes. Data on these individual factors were combined to form a simple deterministic model which simulated the population changes on isolated fish. This was later made more realistic by the introduction of a random variable. When the model was tested in a multiple-host situation it predicted results close to those observed experimentally.     

Traveling waves in a simple population model involving growth and death

The structure of solutions to a simple spatially dependent population model involving growth and death is investigated. Two forms of motility of the population are considered: (1) random motion only modeled by a Fickian law, and (2) a directed component of motion (chemotaxis), included in addition to the random motion. Under certain growth conditions a traveling wave of constant speed is approached. This speed can be increased by the addition of the chemotaxis with a corresponding increase in the asymptotic population. Development of initial conditions into a wave is illustrated numerically.     

Manifestation of multimodality in a simple ecological-genetic model of population evolution

An investigation of the nature of dynamics of the population size and genetic structure is carried out for a homogeneous ecologically limited population influenced by density-dependent r-K selection in a single diallelic genetic locus. The detailed study of the results of basic types of natural selection is carried out with additional consideration of the influence of initial conditions. It is shown that coexistence of several different asymptotic dynamic modes (with their own attraction basins) is possible in numerous enough parametric domains which are meaningful biologically.     

Parasite-mediated heterozygote advantage in an outbred songbird population

Coevolution with parasites is thought to maintain genetic diversity in host populations. However, while there are sound theoretical reasons to expect heterozygosity and parasite resistance to be related, this pattern has generally been shown only in inbred laboratory and island populations. This leaves doubt as to whether parasite-mediated selection for genetic diversity is in fact a general process. Here we show that haematozoan parasite load is linked to two complementary measures of microsatellite variability in an outbred population of mountain white-crowned sparrows (Zonotrichia leucophrys oriantha) for which we know that parasites reduce fitness. Moreover, each of the genetic measures predicts a subtly different aspect of parasitism. Microsatellite heterozygosity is related to an individual's risk of parasitism, and mean d2 (a broader, more long-term measure of parental relatedness) to the severity of infection among parasitized individuals.     

Postglacial population expansion drives the evolution of long-distance migration in a songbird

The evolution of long-distance migratory behavior from sedentary populations is a central problem in studies of animal migration. Three crucial issues that remain unresolved are: (1) the biotic and abiotic factors promoting evolution of migratory behavior, (2) the geographic origin of ancestral sedentary populations, and (3) the time scale over which migration evolves. We test the role of postglacial population expansions during the Quaternary in driving the evolution of songbird migration against prevailing views favoring the role of intraspecific competition. In contrast to previous attempts to investigate these questions using interspecific phylogenies, we adopt an intraspecific approach and examine the phylogeography of a North American songbird, the chipping sparrow (Spizella passerina), which exhibits both long-distance migratory behavior in temperate North America and sedentary behavior in Mexico and Central America. We show that migratory populations descend from sedentary populations in southern Mexico and that migration has evolved as a result of a northward population expansion into temperate North America since the last glacial maximum 18,000 years ago. Migration appears to have evolved rapidly in some species as populations colonized areas of high seasonality in the temperate zone. The phylogeography of the yellow-eyed junco (Junco phaeonotus), a strictly sedentary species, provides a null model supporting the view that northward range expansions were driven solely by environmental factors and not by a predisposition to evolve migratory behavior. These results provide the strongest evidence to date that historical climate patterns can drive the rapid evolution of avian migration in natural populations, and they suggest a general mechanism for the repeated evolution of migration within and across bird lineages.     

No population genetic structure in a widespread aquatic songbird from the Neotropics

Neotropical lowland organisms often show marked population genetic structure, suggesting restricted migration among populations. However, most phylogeographic studies have focused on species inhabiting humid forest interior. Little attention has been devoted to the study of species with ecologies conducive to dispersal, such as those of more open and variable environments associated with watercourses. Using mtDNA sequences, we examined patterns of genetic variation in a widely distributed Neotropical songbird of aquatic environments, the Yellow-hooded Blackbird (Icteridae, Chrysomus icterocephalus). In contrast to many forest species, Yellow-hooded Blackbirds showed no detectable genetic structure across their range, which includes lowland populations on both sides of the Andes, much of northeastern South America, Amazonia, as well as a phenotypically distinct highland population in Colombia. A coalescent-based analysis of the species indicated that its effective population size has increased considerably, suggesting a range expansion. Our results support the hypothesis that species occurring in open habitats and tracking temporally dynamic environments should show increased dispersal propensities (hence gene flow) relative to species from closed and more stable environments. The phenotypic and behavioral variation among populations of our study species appears to have arisen recently and perhaps in the face of gene flow.     

Implications of a simple mathematical model to cancer cell population dynamics

Recent research in cancer progression and treatment indicates that many forms of cancer arise from the development of a small subpopulation of abnormal cancer stem cells (CSCs) that promote cancer growth and spread. Many potential treatments preferentially interact with cells at certain stages of the cell cycle by either selective killing or halting the cell cycle, such as intense, nanosecond-duration pulsed electric fields (nsPEFs). Simple mathematical models of unfed cancer cell populations at the plateau of their growth characteristics may estimate the long-term consequences of these treatments on proliferating and quiescent cell populations. Applying such a model with no transition from the quiescent to proliferating state shows that it is possible for the proliferating cell population to fall below 1 if the quiescent cell population obtains a sufficient competitive advantage with respect to nutrient consumption and/or survival rate. Introducing small, realistic transition rates did not appreciably alter short-term or long-term population behaviour, indicating that the predicted small cell population behaviour (< 1 cell) is not an artefact of the simpler model. Experimental observations of nsPEF-induced effects on the cell cycle suggest that such a model may serve as a first step in assessing the viability of a given cancer treatment in vitro prior to clinical application.     

Maintenance of MHC Class IIB diversity in a recently established songbird population

We examined variation at MHC Class IIB genes in a recently established population of dark-eyed juncos (Junco hyemalis) in a coastal urban environment in southern California, USA relative to an ancestral-range population from a nearby species-typical montane environment. The founding population is estimated to have been quite small, but we predicted that variation at the major histocompatibility complex (MHC) among the founders would nevertheless be preserved owing to the high functional significance of MHC. Previous studies of MHC in songbirds have had varying degrees of success in isolating loci, as passerines show extensive MHC gene duplication. In order to compare diversity in the two populations, we employed two published approaches to sequencing MHC Class II exon 2: direct sequencing with exon-based primers, and traditional cloning and sequencing with intron-based primers. Results from both methods show that the colonist population has maintained high levels of variation. Our results also indicate varying numbers of alleles across individuals, corroborating evidence for gene duplication in songbird MHC. While future studies in songbirds may need to take a genomic approach to fully understand the structure of MHC in this lineage, our results show that it is possible to use traditional methods to reveal functional variation across populations.     

Neighbour recognition in two sister songbird species with a simple and complex repertoire – a playback study

The ability of individuals to recognize others based on vocalizations has been proven in many species of birds. However, we are still far from understanding important aspects of the discrimination process. For example, it is still not fully understood whether, and why, repertoire size hinders discrimination between individuals. Further, the strategies and vocal cues used for discrimination between individuals are largely unexplored. In this study, we tested the ability of chiffchaffs Phylloscopus collybita and willow warblers Phylloscopus trochilus, two closely related species with different repertoire sizes and song organization, to differentiate between their neighbours on the basis of a single song of a neighbouring male. We did playback experiments within the ‘dear enemy’ paradigm in which we tested resident males with a single song of a neighbour broadcast from the correct and opposite, incorrect territory border. Both species displayed a strong ability to discriminate between their neighbours representing further evidence that repertoire size per se has no negative impact on individual recognition in songbirds. Using a single song for playback allowed us to speculate about the nature of the possible cues used by males for recognition. Individual recognition in both species is most likely based on the modulation of syllables or on general voice characteristics. We suggest that specific changes in song organization, for example the tendency of individuals to insert a distinct phrase at the beginning of each song, may facilitate individual recognition and compensate for increased repertoire size in willow warblers.     

Density dependence and stochastic variation in a newly established population of a small songbird

Models describing fluctuations in population size should include both density dependence and stochastic effects. We examine the relative contribution of variation in parameters of the expected dynamics as well as demographic and environmental stochasticity to fluctuations in a population of a small passerine bird, the pied flycatcher, that was newly established in a Dutch study area. Using the theta-logistic model of density regulation, we demonstrate that the estimated quasi-stationary distribution including demographic stochasticity is close to the stationary distribution ignoring demographic stochasticity, indicating a long expected time to extinction. We also show that the variance in the estimated quasi-stationary distribution is especially sensitive to variation in the density regulation function. Reliable population projections must therefore account for uncertainties in parameter estimates which we do by using the population prediction interval (PPI). After 2 years the width of the 90% PPI was already larger than the corresponding estimated range of variation in the quasi-stationary distribution. More precise prediction of future population size than can be derived from the quasi-stationary distribution could only be made for a time span less than about five years.     

Consequential classes of resources: Subtle global bifurcation with dramatic ecological consequences in a simple population model

Numerous situations exist in which a consumer uses two different kinds of resources, one fixed, the other renewable, e.g., nesting resources and food resources. With an elementary modification of the basic Lotka-Volterra consumer resource equations, we investigate the population dynamics of a consumer dependent on two resources, one fixed, the other renewable. Emerging from this structure is a situation of alternative attractors that remain qualitatively robust over a significant range of parameter values. However, a dramatic change in basins of attraction is induced by very small changes in parameters due to a global bifurcation. Noteworthy is the fact that the qualitative nature of the alternative equilibria remains constant but the dramatic change in the basins does not arise from subtle differences in initial conditions. Rather, there is a major restructuring of the vector field such that a permanent change involving large sets of initial conditions results from very small changes in parameters.     

A simple model of plankton population dynamics coupled with a LES of the surface mixed layer

The concentration of phytoplankton in the sea is affected by biological processes, such as growth/mortality rates, predatory zooplankton concentrations and nutrient levels. Phytoplankton concentrations are also influenced by physical processes, in particular the mixing properties of the local fluid environment. On planktonic scales (approximately 10-1000 microm) one can assume the local turbulent flow is isotropic, with no distinction between horizontal and vertical mixing. However, agglomerations of phytoplankton into patches are observed on larger scales of up to hundreds of metres, whose formation will be influenced by the anisotropic advection/mixing properties and large-eddy structures prevalent in the surface mixed layer. This paper presents the results of the coupling of a large-eddy simulation (LES) model of the mixed layer with an advection-diffusion system of coupled equations for nitrate-phytoplankton-zooplankton (NPZ) concentration, incorporating sub-grid parameterizations of the biological processes. Typically these include phytoplankton growth due to light levels and ambient nitrate concentration, offset by grazing losses due to the presence of zooplankton. The primary goal of this work is to investigate how the characteristics of the mixed layer turbulence influence the observed distribution of phytoplankton. One novel feature is the incorporation of a 'vortex-force' term in the LES code in order to generate Langmuir circulations. It has been speculated that the enhanced mixing rates associated with 'Langmuir turbulence' play a significant role in regulating planktonic activity. Results derived from the coupled LES-NPZ model, run with and without the presence of Langmuir circulations, are presented in order to investigate these ideas.     

Modelling population growth in stage-grouped organisms: a simple extension to the Leslie model

A simple method for decomposing a population's stage grouping into the underlying age structure is described. The population's dynamics can then be predicted using standard age-structured models, such as Leslie's matrix model. The method overcomes objections to previous attempts to use Leslie's procedure for modelling population growth in stage-grouped organisms. A hypothetical example is used to illustrate the new technique.     

The hydra effect, bubbles, and chaos in a simple discrete population model with constant effort harvesting

We analyze the effects of a strategy of constant effort harvesting in the global dynamics of a one-dimensional discrete population model that includes density-independent survivorship of adults and overcompensating density dependence. We discuss the phenomenon of bubbling (which indicates that harvesting can magnify fluctuations in population abundance) and the hydra effect, which means that the stock size gets larger as harvesting rate increases. Moreover, we show that the system displays chaotic behaviour under the combination of high per capita recruitment and small survivorship rates.     

Environmental stochasticity and extinction risk in a population of a small songbird, the great tit

Using a long-term demographic data set, we estimated the separate effects of demographic and environmental stochasticity in the growth rate of the great tit population in Wytham Wood, United Kingdom. Assuming logistic density regulation, both the demographic (sigma2d = 0.569) and environmental (sigma2e = 0.0793) variance, with interactions included, were significantly greater than zero. The estimates of the demographic variance seemed to be relatively insensitive to the length of the study period, whereas reliable estimates of the environmental variance required long time series (at least 15 yr of data). The demographic variance decreased significantly with increasing population density. These estimates are used in a quantitative analysis of the demographic factors affecting the risk of extinction of this population. The very long expected time to extinction of this population (approximately 10(19) yr) was related to a relatively large population size (>/=120 pairs during the study period). However, for a given population size, the expected time to extinction was sensitive to both variation in population growth rate and environmental stochasticity. Furthermore, the form of the density regulation strongly affected the expected time to extinction. Time to extinction decreased when the maximum density regulation approached K. This suggests that estimates of viability of small populations should be given both with and without inclusion of density dependence.     

Heterozygosity–fitness correlations in a declining seabird population

Loss of genetic diversity is thought to lead to increased risk of extinction in endangered populations due to decreasing fitness of homozygous individuals. Here, we evaluated the presence of inbreeding depression in a long‐lived seabird, the European shag (Phalacrocorax aristotelis), after a severe decline in population size by nearly 70%. During three reproductive seasons, 85 breeders were captured and genotyped at seven microsatellite loci. Nest sites were monitored during the breeding season to estimate reproductive success as the number of chicks surviving to full‐size‐grown per nest. Captured birds were tagged with a ring with an individual code, and resighting data were collected during 7‐year period. We found a strong effect of multilocus heterozygosity on female reproductive performance, and a significant, although weaker, effect on breeder survival. However, our matrix population model suggests that this relatively small effect of genetic diversity on breeder survival may have a profound effect on fitness. This highlights the importance of integrating life history consequences in HFC studies. Importantly, heterozygosity was correlated across loci, suggesting that genomewide effects, rather than single loci, are responsible for the observed HFCs. Overall, the HFCs are a worrying symptom of genetic erosion in this declining population. Many long‐lived species are prone to extinction, and future studies should evaluate the magnitude of fitness impact of genetic deterioration on key population parameters, such as survival of breeders.     

A bayesian random-effects markov model for tumor progression in women with a family history of breast cancer

SUMMARY: It makes intuitive sense to model the natural history of breast cancer, tumor progression from preclinical screen-detectable phase (PCDP) to clinical disease, as a multistate process, but the multilevel structure of the available data, which generally comes from cluster (family)-based service screening programs, makes the required parameter estimation intractable because there is a correlation between screening rounds in the same individual, and between subjects within clusters (families). There is also residual heterogeneity after adjusting for covariates. We therefore proposed a Bayesian hierarchical multistate Markov model with fixed and random effects and applied it to data from a high-risk group (women with a family history of breast cancer) participating in a family-based screening program for breast cancer. A total of 4867 women attended (representing 4464 families) and by the end of 2002, a total of 130 breast cancer cases were identified. Parameter estimation was carried out using Markov chain Monte Carlo (MCMC) simulation and the Bayesian software package WinBUGS. Models with and without random effects were considered. Our preferred model included a random-effect term for the transition rate from preclinical to clinical disease (sigma(2)(2f)), which was estimated to be 0.50 (95% credible interval = 0.22-1.49). Failure to account for this random effect was shown to lead to bias. The incorporation of covariates into multistate models with random effect not only reduced residual heterogeneity but also improved the convergence of stationary distribution. Our proposed Bayesian hierarchical multistate model is a valuable tool for estimating the rate of transitions between disease states in the natural history of breast cancer (and possibly other conditions). Unlike existing models, it can cope with the correlation structure of multilevel screening data, covariates, and residual (unexplained) variation.