Dynamical behavior of differential equation models of frequency and density dependent populations

This paper studies the dynamics of a mathematical model of a continuously reproducing diploid population with two alleles at one locus. The dependent variables are allele frequency and population density. If the genotype fitnesses are frequency and density dependent, the stability of equilibria is related to the geometry of the zero allele fitness curves. The asymptotic behavior of solutions where fitness is only density dependent is contrasted to the asymptotic behavior where fitness is frequency and density dependent. A parameterized family of fitness functions giving a Hopf bifurcation and limit cycles is investigated analytically and numerically.     

Studies in the Dynamics of Genetically Variable Populations. I. Frequency- and Density-Dependent Selection in Experimental Populations of DROSOPHILA MELANOGASTER

Changes in genetic composition and in fitness of karyotypes in synthetic populations of Drosophila melanogaster carrying fourth chromosomes marked with recessive mutants in repulsion were measured as functions of initial population composition and density. There was no detectable influence of initial population density on the magnitude of chromosome frequency change or on adaptive values of karyotypes, but both measures proved sensitive to population composition. That fitness be measured zygote-to-zygote is argued to be of dubious necessity in all contexts except demonstration of frequency-dependent selection. Some prior reports cited as evidence of the operation of density-dependent selection, like the present study, contain no evidence that the rate of natural selection is influenced by population crowding. Reports that do demonstrate density-dependent selection suggest that the relationship between fitness and crowding differs from that assumed in most theoretical treatments. Further work is needed to elucidate the interaction between fitness, population crowding, and population composition.     

Density-Dependent Selection Incorporating Intraspecific Competition. II. a Diploid Model

A diploid model is introduced and analyzed in which intraspecific competition is incorporated within the context of density-regulated selection. It is assumed that each genotype has a unique carrying capacity corresponding to the equilibrium population size when only that type is present. Each genotypic fitness at a single diallelic autosomal locus is a decreasing function of a distinctive effective population size perceived as a result of intraspecific competition. The resulting fitnesses are both density and frequency dependent with selective advantage determined by a balance between genotypic carrying capacity and sensitivity to intraspecific competition. A major finding is that intergenotypic interactions may allow genetic variation to be more easily maintained than in the corresponding model of purely density-dependent selection. In addition, numerical study confirms the possible existence of multiple interior equilibria and that neither overdominance in fitness nor carrying capacity is necessary for stability. The magnitude of the equilibrium population size and optimization principles are also discussed.     

Natural selection and density-dependent population growth

Natural selection was studied in the context of density-dependent population growth using a single locus, continuous time model for the rates of change of population size and allele frequency. The maximization principle of density-dependent selection was applied to a class of fitness expressions with explicit recruitment and mortality terms. Three general results were obtained: First, at low population densities, the genetic basis of selection is the difference between the mean recruitment rate and the mean mortality rate. Second, at densities much higher than the equilibrium population size, selection is expected to act to minimize the mean mortality rate. Third, as the population approaches its equilibrium density, selection is predicted to maximize the ratio of the mean recruitment rate to the mean mortality rate.     

Natural selection and fitness entropy in a density-regulated population

The entropy H(p_o,p*) of a population with the initial allele frequency p_o given the equilibrium polymorphic frequency p* has been proposed as a measure of natural selection. In the present paper, we have extended this concept to include a particular aspect of density-dependent selection. We compared size trajectory of a population initially at genetic equilibrium, N(t), with the size trajectories of populations not initially at p*,N(t), but which do eventually converge to a common equilibrium allele frequency and equilibrium density, N*. The following experimentally testable hyopthesis was established. The total area defined by the difference between the trajectories of N(t) and N(t) as they converge to N* is directly proportional to the fitness entropy when population size is transformed using the density-dependent fitness value. Two properties of this relationship were noted. First, it is independent of the magnitude of natural selection and, secondly, it does not depend upon the initial population density as long as the equilibrium and nonequilibrium populations have the same initial numbers. This hypothesis was evaluated with experimental data on the flour beetle Tribolium castaneum.     

Stable periodic solutions for two species,density dependent coevolution

This paper studies a four dimensional system of time-autonomous ordinary differential equations which models the interaction of two diploid, diallelic populations with overlapping generations. The variables are two population densities and an allele frequency in each of the populations. For single species models, the existence of periodic solutions requires that the genotype fitness functions be both frequency and density dependent. But, for two species exhibiting a predator-prey interaction, two examples are presented where there exists asymptotically stable cycles with fitness functions only density dependent. In the first example, the Hopf bifurcation theorem is used on a two parameter, polynomial vector field. The second example has a Michaelis-Menten or Holling term for the interaction between predator and prey; and, for this example, the existence and uniqueness of limit cycles for a wide range of parameter values has been established in the literature.     

Density-dependent selection migration model with non-monotone fitness functions

This paper studies the classical single locus, diallelic selection model with diffusion for a continuously reproducing population. The phase variables are population density and allele frequency (or allele density). The genotype fitness depend only on population density but include one-hump functions of the density variable. With mild assumptions on genotype fitnesses, we study the geometry of the nullclines and the asymptotic behavior of solutions of the selection model without diffusion. For the diffusion model with zero Neumann boundary conditions, we use this geometric information to show that if the initial data satisfy certain conditions then the corresponding solution to the reaction-diffusion equation converges to the spatially constant stable equilibrium which is closest to the initial data.Research partially supported by NSF grant DMS-8920597Research supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Pioneering (Population Genetics of Forest Trees) Research Unit, Raleigh, North Carolina     

The frequency-dependent Wright–Fisher model: diffusive and non-diffusive approximations

We study a class of processes that are akin to the Wright–Fisher model, with transition probabilities weighted in terms of the frequency-dependent fitness of the population types. By considering an approximate weak formulation of the discrete problem, we are able to derive a corresponding continuous weak formulation for the probability density. Therefore, we obtain a family of partial differential equations for the evolution of the probability density, and which will be an approximation of the discrete process in the joint large population, small time-steps and weak selection limit. If the fitness functions are sufficiently regular, we can recast the weak formulation in a more standard formulation, without any boundary conditions, but supplemented by a number of conservation laws. The equations in this family can be purely diffusive, purely hyperbolic or of convection–diffusion type, with frequency dependent convection. The particular outcome will depend on the assumed scalings. The diffusive equations are of the degenerate type; using a duality approach, we also obtain a frequency dependent version of the Kimura equation without any further assumptions. We also show that the convective approximation is related to the replicator dynamics and provide some estimate of how accurate is the convective approximation, with respect to the convective-diffusion approximation. In particular, we show that the mode, but not the expected value, of the probability distribution is modelled by the replicator dynamics. Some numerical simulations that illustrate the results are also presented.     

Density dependence and the spread of anthelmintic resistance

Variation in the strength of selection pressures acting upon different subpopulations may cause density-dependent regulatory processes to act differentially on particular genotypes and may influence the rate of selection of adaptive traits. Using host-helminth parasite systems as examples, we investigate the impact of different positive and negative density dependence on the potential spread of anthelmintic resistance. Following chemotherapy, the negative density-dependent processes restricting parasite population growth will be relaxed, increasing the genetic contribution of resistant parasites to the next generation. Simple deterministic models of directly transmitted nematodes that merge population dynamics and genetics show that the frequency of drug-resistant alleles may increase faster in species whose population size is down-regulated by density-dependent parasite fecundity than in species with density-dependent establishment or parasite mortality. A genetically structured population dynamics model of an indirectly transmitted nematode is used to highlight how population regulation will influence the resistance allele frequency in different parasite lifestages. Results indicate that surveys aimed at monitoring the evolution of drug resistance should consider carefully which life stage to sample, and the time following treatment samples should be collected. Anthelmintic resistance offers a good opportunity to apply fundamental evolutionary and ecological principles to the management of a potentially crucial public health problem.     

Population dynamic models of the spread of Wolbachia

Wolbachia are endosymbionts that are found in many insect species and can spread rapidly when introduced into a naive host population. Most Wolbachia spread when their infection frequency exceeds a threshold normally calculated using purely population genetic models. However, spread may also depend on the population dynamics of the insect host. We develop models to explore interactions between host population dynamics and Wolbachia infection frequency for an age-structured insect population regulated by larval density dependence. We first derive a new expression for the threshold frequency that extends existing theory to incorporate important details of the insect's life history. In the presence of immigration and emigration, the threshold also depends on the form of density-dependent regulation. We show how the type of immigration (constant or pulsed) and the temporal dynamics of the host population can strongly affect the spread of Wolbachia. The results help understand the natural dynamics of Wolbachia infections and aid the design of programs to introduce Wolbachia to control insects that are disease vectors or pests.     

Dynamical behavior for population genetics models of differential and difference equations with nonmonotone fitnesses

This paper studies the dynamical behavior of classical 2-dimensional models of continuously and discretely reproducing diploid populations with two alleles at one locus. The phase variables are allele frequency and population density. The genotype fitnesses are not assumed to be monotonically decreasing functions of density. Hence the mean fitness curve is more complicated than in the monotonic case. If genotype fitnesses are only density dependent, results concerning equilibrium stability are obtained similar to those for the monotonic case, and periodic solutions are precluded in the differential equation model. An example with one-hump genotype fitnesses is presented and analyzed.Research supported by funds provided by the USDA-Forest Service, Southeastern Forest Experiment Station, Pioneering (Population Genetics of Forest Trees) Research Unit, Raleigh, North Carolina     

Convergence to constant equilibrium for a density-dependent selection model with diffusion

We consider the classical single locus two alleles selection model with diffusion where the fitnesses of the genotypes are density dependent. Using a theorem of Peter Brown, we show that in a bounded domain with homogeneous Neumann boundary conditions, the allele frequency and population density converge to a constant equilibrium lying on the zero population mean fitness curve. The results agree with the case without diffusion obtained by Selgrade and Namkoong. Frequency and density dependent selection is also considered.Research partially supported by NSF grant DMS-8601585     

Hamilton's rule confronts ideal free habitat selection

If individuals occupy habitats in a way that maximizes their fitness, if they are free to occupy the habitats they choose and if fitness declines with population density, then their abundance across habitats should follow an ideal free distribution. But, if individuals are genetically related, this simple fitness-maximization mechanism breaks down. Habitat occupation should obey Hamilton's rule (natural selection favours traits causing a loss in individual fitness as long as they result in an equal or greater gain in inclusive fitness) and depends more on inclusive fitness than it does on individual fitness. We demonstrate that the resulting inclusive-fitness distribution inflates the population density in habitats of poorer inherent quality, creating pronounced source sink dynamics. We also show that density-dependent habitat selection among relatives reinforces behaviours such as group defence and interspecific territoriality, and that it explains many anomalies in dispersal and foraging.     

Density-dependent immune responses against the gastrointestinal nematode Strongyloides ratti

Negative density-dependent effects on the fitness of parasite populations are an important force in their population dynamics. For the parasitic nematode Strongyloides ratti, density-dependent fitness effects require the rat host immune response. By analysis of both measurements of components of parasite fitness and of the host immune response to different doses of S. ratti infection, we have identified specific parts of the host immune response underlying the negative density-dependent effects on the fitness of S. ratti. The host immune response changes both qualitatively from an inflammatory Th1- to a Th2-type immune profile and the Th2-type response increases quantitatively, as the density of S. ratti infection increases. Parasite survivorship was significantly negatively related to the concentration of parasite-specific IgG(1) and IgA, whereas parasite fecundity was significantly negatively related to the concentration of IgA only.     

Incorporating density dependence into the oviposition preference-offspring performance hypothesis

1. Although theory predicts a positive relationship between oviposition preferences and the developmental performance of offspring, the strength of this relationship may depend not only on breeding site quality, but also on the complex interactions between environmental heterogeneity and density-dependent processes. Environmental heterogeneity may not only alter the strength of density dependence, but may also fundamentally alter density-dependent relationships and the preference-performance relationship. 2. Here I present results from a series of field experiments testing the effects of environmental heterogeneity and density-dependent feedback on offspring performance in tree-hole mosquitoes. Specifically, I asked: (i) how do oviposition activity, patterns of colonization and larval density differ among habitats and among oviposition sites with different resources; and (ii) how is performance influenced by the density of conspecifics, the type of resource in the oviposition site, and the type of habitat in which the oviposition site is located? 3. Performance did not differ among habitats at low offspring densities, but was higher in deciduous forest habitats than in evergreen forest habitats at high densities. Oviposition activity and larval densities were also higher in deciduous forests, suggesting a weak preference for these habitats. 4. The observed divergence of fitness among habitats with increasing density may select for consistent but weak preferences for deciduous habitats if regional abundances vary temporally. This would generate a negative preference-performance relationship when population densities are low, but a positive relationship when population densities are high. 5. This study demonstrates that failure to recognize that fitness differences among habitats may themselves be density-dependent may bias our assumptions about the ecological and evolutionary processes determining oviposition preferences in natural systems.     

Density-dependent mortality and the evolution of social interactions

Existing models of kin selection implicitly assume that the intensity of density-dependent mortality depends on the size or density of the entire deme. If individuals compete for locally limited resources, then mortality may also depend on local population density. In this paper, I derive conditions for evolutionary equilibrium when the fitness of an individual depends on (1) the phenotypes of a local group of conspecifics with which it interacts and (2) the size of the local group. In general, altruistic behaviours which reduce the individual fitness of the actor but increase that of the recipient are favoured towards individuals whose coefficient of relationship exceeds a threshold value. ‘Harming’ behaviours which reduce the individual fitness of both the actor and the recipient are favoured towards individuals below this threshold value. The magnitude of this threshold depends on the nature of the processes that cause density-dependent mortality and the average coefficient of relationship in the group.     

Influence of polyphenols on bacterial biofilm formation and quorum-sensing

Many bacteria utilize sophisticated regulatory systems to ensure that some functions are only expressed when a particular population density has been reached. The term 'quorum-sensing' has been coined to describe this form of density-dependent gene regulation which relies on the production and perception of small signal molecules by bacterial cells. As in many pathogenic bacteria the production of virulence factors is quorum-sensing regulated, it has been suggested that this form of gene regulation allows the bacteria to remain invisible to the defence systems of the host until the population is sufficiently large to successfully establish the infection. Here we present first evidence that polyphenolic compounds can interfere with bacterial quorum-sensing. Since polyphenols are widely distributed in the plant kingdom, they may be important for promoting plant fitness.     

Density-dependent habitat selection: Testing the theory with fitness data

Summary According to density-dependent habitat selection theory, reproductive success should decline with increased density. Fitness should be similar between habitats if habitat selection follows an ideal free distribution; fitness should be dissimilar between habitats if habitat selection is modified by territorial behavior. I tested these assumptions by examining a variety of fitness estimates obtained from white-footed mice living in nest boxes in forest, forest edge and fencerow habitats in southwestern Ontario. As expected, mean litter size declined with increased population density. Litter sizes, adult longevity and the proportion of adult animals in breeding condition were not significantly different among the three habitats. The success at recruiting at least one offspring to the adult population and the number of recruits per litter were much greater in the forest than in either of the other two habitats. Fitness was thus unequal among habitats and the results confirm both assumptions of density-dependent habitat selection theory for territorial white-footed mice.     

Fluctuations in numbers in box populations of Drosophila and selective genetic mechanism of their regulation

Box populations of Drosophila melanogaster are characterized by two types of periodical fluctuations of numbers: with low and high frequency. High frequency fluctuations are determined by existence of preimago and imago stages and subsequent delay in density-dependent limitation of imago reproduction, duration of which is determined by time of preimago stage. The period of these fluctuations should be limited within two generation, that is confirmed by experimental data. Low frequency fluctuations with the period of 13-15 generations are the result of ecological density-dependent effect. In this case during pick density one can observe continuous degradation of population (i.e. decrease in fecundity and life time of imago) and following decrease in numbers. Temporary changes in fecundity of females and their offspring of the second generation are positively correlated with low frequency fluctuations in numbers. Such relationships show the possibility of density-dependent, cyclic, genetic changes in fecundity connected with fluctuations in numbers. It means that at the phase of growth in numbers when the density is still low, the selection is directed to the individuals with high fecundity sensible to overpopulation. The phase of decline in numbers is connected with high density and selection directed to the individuals with low fecundity in low density populations. The changes in genetic structure of fluctuating population lead to the weakening of this fluctuations and to the maintaining of population under such conditions.     

Fixation in haploid populations exhibiting density dependence I: The non-neutral case

We extend the one-locus two allele Moran model of fixation in a haploid population to the case where the total size of the population is not fixed. The model is defined as a two-dimensional birth-and-death process for allele number. Changes in allele number occur through density-independent death events and birth events whose per capita rate decreases linearly with the total population density. Uniquely for models of this type, the latter is determined by these same birth-and-death events. This provides a framework for investigating both the effects of fluctuation in total population number through demographic stochasticity, and deterministic density-dependent changes in mean density, on allele fixation. We analyze this model using a combination of asymptotic analytic approximations supported by numerics. We find that for advantageous mutants demographic stochasticity of the resident population does not affect the fixation probability, but that deterministic changes in total density do. In contrast, for deleterious mutants, the fixation probability increases with increasing resident population fluctuation size, but is relatively insensitive to initial density. These phenomena cannot be described by simply using a harmonic mean effective population size.