Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.     

Language dynamics in finite populations

Any mechanism of language acquisition can only learn a restricted set of grammars. The human brain contains a mechanism for language acquisition which can learn a restricted set of grammars. The theory of this restricted set is universal grammar (UG). UG has to be sufficiently specific to induce linguistic coherence in a population. This phenomenon is known as "coherence threshold". Previously, we have calculated the coherence threshold for deterministic dynamics and infinitely large populations. Here, we extend the framework to stochastic processes and finite populations. If there is selection for communicative function (selective language dynamics), then the analytic results for infinite populations are excellent approximations for finite populations; as expected, finite populations need a slightly higher accuracy of language acquisition to maintain coherence. If there is no selection for communicative function (neutral language dynamics), then linguistic coherence is only possible for finite populations.     

The effect of linkage and population size on inbreeding depression due to mutational load.

Using a stochastic model of a finite population in which there is mutation to partially recessive detrimental alleles at many loci, we study the effects of population size and linkage between the loci on the population mean fitness and inbreeding depression values. Although linkage between the selected loci decreases the amount of inbreeding depression, neither population size nor recombination rate have strong effects on these quantities, unless extremely small values are assumed. We also investigate how partial linkage between the loci that determine fitness affects the invasion of populations by alleles at a modifier locus that controls the selfing rate. In most of the cases studied, the direction of selection on modifiers was consistent with that found in our previous deterministic calculations. However, there was some evidence that linkage between the modifier locus and the selected loci makes outcrossing less likely to evolve; more losses of alleles promoting outcrossing occurred in runs with linkage than in runs with free recombination. We also studied the fate of neutral alleles introduced into populations carrying detrimental mutations. The times to loss of neutral alleles introduced at low frequency were shorter than those predicted for alleles in the absence of selected loci, taking into account the reduction of the effective population size due to inbreeding. Previous studies have been confined to outbreeding populations, and to alleles at frequencies close to one-half, and have found an effect in the opposite direction. It therefore appears that associations between neutral and selected loci may produce effects that differ according to the initial frequencies of the neutral alleles.     

Comparison of deterministic and stochastic SIS and SIR models in discrete time

The dynamics of deterministic and stochastic discrete-time epidemic models are analyzed and compared. The discrete-time stochastic models are Markov chains, approximations to the continuous-time models. Models of SIS and SIR type with constant population size and general force of infection are analyzed, then a more general SIS model with variable population size is analyzed. In the deterministic models, the value of the basic reproductive number R0 determines persistence or extinction of the disease. If R0 < 1, the disease is eliminated, whereas if R0 > 1, the disease persists in the population. Since all stochastic models considered in this paper have finite state spaces with at least one absorbing state, ultimate disease extinction is certain regardless of the value of R0. However, in some cases, the time until disease extinction may be very long. In these cases, if the probability distribution is conditioned on non-extinction, then when R0 > 1, there exists a quasi-stationary probability distribution whose mean agrees with deterministic endemic equilibrium. The expected duration of the epidemic is investigated numerically.     

Stochastic models of kleptoparasitism

In this paper, we consider a model of kleptoparasitism amongst a small group of individuals, where the state of the population is described by the distribution of its individuals over three specific types of behaviour (handling, searching for or fighting over, food). The model used is based upon earlier work which considered an equivalent deterministic model relating to large, effectively infinite, populations. We find explicit equations for the probability of the population being in each state. For any reasonably sized population, the number of possible states, and hence the number of equations, is large. These equations are used to find a set of equations for the means, variances, covariances and higher moments for the number of individuals performing each type of behaviour. Given the fixed population size, there are five moments of order one or two (two means, two variances and a covariance). A normal approximation is used to find a set of equations for these five principal moments. The results of our model are then analysed numerically, with the exact solutions, the normal approximation and the deterministic infinite population model compared. It is found that the original deterministic models approximate the stochastic model well in most situations, but that the normal approximations are better, proving to be good approximations to the exact distribution, which can greatly reduce computing time.     

Transition between Stochastic Evolution and Deterministic Evolution in the Presence of Selection: General Theory and Application to Virology

We present here a self-contained analytic review of the role of stochastic factors acting on a virus population. We develop a simple one-locus, two-allele model of a haploid population of constant size including the factors of random drift, purifying selection, and random mutation. We consider different virological experiments: accumulation and reversion of deleterious mutations, competition between mutant and wild-type viruses, gene fixation, mutation frequencies at the steady state, divergence of two populations split from one population, and genetic turnover within a single population. In the first part of the review, we present all principal results in qualitative terms and illustrate them with examples obtained by computer simulation. In the second part, we derive the results formally from a diffusion equation of the Wright-Fisher type and boundary conditions, all derived from the first principles for the virus population model. We show that the leading factors and observable behavior of evolution differ significantly in three broad intervals of population size, N. The “neutral limit” is reached when N is smaller than the inverse selection coefficient. When N is larger than the inverse mutation rate per base, selection dominates and evolution is “almost” deterministic. If the selection coefficient is much larger than the mutation rate, there exists a broad interval of population sizes, in which weakly diverse populations are almost neutral while highly diverse populations are controlled by selection pressure. We discuss in detail the application of our results to human immunodeficiency virus population in vivo, sampling effects, and limitations of the model.     

Rate of Adaptation in Sexuals and Asexuals: A Solvable Model of the Fisher–Muller Effect

The adaptation of large asexual populations is hampered by the competition between independently arising beneficial mutations in different individuals, which is known as clonal interference. In classic work, Fisher and Muller proposed that recombination provides an evolutionary advantage in large populations by alleviating this competition. Based on recent progress in quantifying the speed of adaptation in asexual populations undergoing clonal interference, we present a detailed analysis of the Fisher–Muller mechanism for a model genome consisting of two loci with an infinite number of beneficial alleles each and multiplicative (nonepistatic) fitness effects. We solve the deterministic, infinite population dynamics exactly and show that, for a particular, natural mutation scheme, the speed of adaptation in sexuals is twice as large as in asexuals. This result is argued to hold for any nonzero value of the rate of recombination. Guided by the infinite population result and by previous work on asexual adaptation, we postulate an expression for the speed of adaptation in finite sexual populations that agrees with numerical simulations over a wide range of population sizes and recombination rates. The ratio of the sexual to asexual adaptation speed is a function of population size that increases in the clonal interference regime and approaches 2 for extremely large populations. The simulations also show that the imbalance between the numbers of accumulated mutations at the two loci is strongly suppressed even by a small amount of recombination. The generalization of the model to an arbitrary number L of loci is briefly discussed. If each offspring samples the alleles at each locus from the gene pool of the whole population rather than from two parents, the ratio of the sexual to asexual adaptation speed is approximately equal to L in large populations. A possible realization of this scenario is the reassortment of genetic material in RNA viruses with L genomic segments.     

The Effect of Bacterial Recombination on Adaptation on Fitness Landscapes with Limited Peak Accessibility

There is ample empirical evidence revealing that fitness landscapes are often complex: the fitness effect of a newly arisen mutation can depend strongly on the allelic state at other loci. However, little is known about the effects of recombination on adaptation on such fitness landscapes. Here, we investigate how recombination influences the rate of adaptation on a special type of complex fitness landscapes. On these landscapes, the mutational trajectories from the least to the most fit genotype are interrupted by genotypes with low relative fitness. We study the dynamics of adapting populations on landscapes with different compositions and numbers of low fitness genotypes, with and without recombination. Our results of the deterministic model (assuming an infinite population size) show that recombination generally decelerates adaptation on these landscapes. However, in finite populations, this deceleration is outweighed by the accelerating Fisher-Muller effect under certain conditions. We conclude that recombination has complex effects on adaptation that are highly dependent on the particular fitness landscape, population size and recombination rate.     

Transition between stochastic evolution and deterministic evolution in the presence of selection: general theory and application to virology.

We present here a self-contained analytic review of the role of stochastic factors acting on a virus population. We develop a simple one-locus, two-allele model of a haploid population of constant size including the factors of random drift, purifying selection, and random mutation. We consider different virological experiments: accumulation and reversion of deleterious mutations, competition between mutant and wild-type viruses, gene fixation, mutation frequencies at the steady state, divergence of two populations split from one population, and genetic turnover within a single population. In the first part of the review, we present all principal results in qualitative terms and illustrate them with examples obtained by computer simulation. In the second part, we derive the results formally from a diffusion equation of the Wright-Fisher type and boundary conditions, all derived from the first principles for the virus population model. We show that the leading factors and observable behavior of evolution differ significantly in three broad intervals of population size, N. The "neutral limit" is reached when N is smaller than the inverse selection coefficient. When N is larger than the inverse mutation rate per base, selection dominates and evolution is "almost" deterministic. If the selection coefficient is much larger than the mutation rate, there exists a broad interval of population sizes, in which weakly diverse populations are almost neutral while highly diverse populations are controlled by selection pressure. We discuss in detail the application of our results to human immunodeficiency virus population in vivo, sampling effects, and limitations of the model.     

Genetic load in sexual and asexual diploids: segregation, dominance and genetic drift

In diploid organisms, sexual reproduction rearranges allelic combinations between loci (recombination) as well as within loci (segregation). Several studies have analyzed the effect of segregation on the genetic load due to recurrent deleterious mutations, but considered infinite populations, thus neglecting the effects of genetic drift. Here, we use single-locus models to explore the combined effects of segregation, selection, and drift. We find that, for partly recessive deleterious alleles, segregation affects both the deterministic component of the change in allele frequencies and the stochastic component due to drift. As a result, we find that the mutation load may be far greater in asexuals than in sexuals in finite and/or subdivided populations. In finite populations, this effect arises primarily because, in the absence of segregation, heterozygotes may reach high frequencies due to drift, while homozygotes are still efficiently selected against; this is not possible with segregation, as matings between heterozygotes constantly produce new homozygotes. If deleterious alleles are partly, but not fully recessive, this causes an excess load in asexuals at intermediate population sizes. In subdivided populations without extinction, drift mostly occurs locally, which reduces the efficiency of selection in both sexuals and asexuals, but does not lead to global fixation. Yet, local drift is stronger in asexuals than in sexuals, leading to a higher mutation load in asexuals. In metapopulations with turnover, global drift becomes again important, leading to similar results as in finite, unstructured populations. Overall, the mutation load that arises through the absence of segregation in asexuals may greatly exceed previous predictions that ignored genetic drift.     

Local Replicator Dynamics: A Simple Link Between Deterministic and Stochastic Models of Evolutionary Game Theory

Classical replicator dynamics assumes that individuals play their games and adopt new strategies on a global level: Each player interacts with a representative sample of the population and if a strategy yields a payoff above the average, then it is expected to spread. In this article, we connect evolutionary models for infinite and finite populations: While the population itself is infinite, interactions and reproduction occurs in random groups of size N. Surprisingly, the resulting dynamics simplifies to the traditional replicator system with a slightly modified payoff matrix. The qualitative results, however, mirror the findings for finite populations, in which strategies are selected according to a probabilistic Moran process. In particular, we derive a one-third law that holds for any population size. In this way, we show that the deterministic replicator equation in an infinite population can be used to study the Moran process in a finite population and vice versa. We apply the results to three examples to shed light on the evolution of cooperation in the iterated prisoner’s dilemma, on risk aversion in coordination games and on the maintenance of dominated strategies.     

Allee effects in stochastic populations

The Allee effect, or inverse density dependence at low population sizes, could seriously impact preservation and management of biological populations. The mounting evidence for widespread Allee effects has lately inspired theoretical studies of how Allee effects alter population dynamics. However, the recent mathematical models of Allee effects have been missing another important force prevalent at low population sizes: stochasticity. In this paper, the combination of Allee effects and stochasticity is studied using diffusion processes, a type of general stochastic population model that accommodates both demographic and environmental stochastic fluctuations. Including an Allee effect in a conventional deterministic population model typically produces an unstable equilibrium at a low population size, a critical population level below which extinction is certain. In a stochastic version of such a model, the probability of reaching a lower size a before reaching an upper size b , when considered as a function of initial population size, has an inflection point at the underlying deterministic unstable equilibrium. The inflection point represents a threshold in the probabilistic prospects for the population and is independent of the type of stochastic fluctuations in the model. In particular, models containing demographic noise alone (absent Allee effects) do not display this threshold behavior, even though demographic noise is considered an "extinction vortex". The results in this paper provide a new understanding of the interplay of stochastic and deterministic forces in ecological populations.     

Evolution of mating types in finite populations: The precarious advantage of being rare

Sexually reproducing populations with self‐incompatibility bear the cost of limiting potential mates to individuals of a different type. Rare mating types escape this cost since they are unlikely to encounter incompatible partners, leading to the deterministic prediction of continuous invasion by new mutants and an ever‐increasing number of types. However, rare types are also at an increased risk of being lost by random drift. Calculating the number of mating types that a population can maintain requires consideration of both the deterministic advantages and the stochastic risks. By comparing the relative importance of selection and drift, we show that a population of size N can maintain a maximum of approximately N^1/3 mating types for intermediate population sizes, whereas for large N, we derive a formal estimate. Although the number of mating types in a population is quite stable, the rare‐type advantage promotes turnover of types. We derive explicit formulas for both the invasion and turnover probabilities in finite populations.     

Marital mores as a mechanism for the maintenance of ethnic variations of lethal gene frequencies.

A computer simulation model was developed to study the effects of various marital mores on the incidence of lethal autosomal recessive genes in populations that are subdivided into small isolates. The problem was studied in isolates where initial generation size was 30, 40, and 50 individuals. In each of these, the mean fertility rate was varied from 2.3 to 2.7 surviving (to adulthood) children per couple whose marriage had been contracted in accordance with the prevailing convention: marriage between first cousins and siblings prohibited; marriage between siblings prohibited; marriage allowed between any individuals; marriage prohibited between siblings but encouraged between cousins; and marriage encouraged between siblings. In all cases, the mean gene frequency in generation 20 was lower than that predicted by the deterministic model with random mating in an unsubdivided population of infinite size, due to gene loss through random drift (to zero) in many of the isolates. The mores that encouraged consanguineous marriages had the lowest final lethal-gene frequencies. Random mating produced intermediate values, and the restrictive mores, the highest final frequencies. The deterministic model (assuming infinite population size and random mating) predictions of the final gene frequency were exceeded only if there was reproductive compensation. It is concluded that restrictive marital mores significantly reduce the selective pressures on lethal recessive genes in small isolates, but that this is counteracted by the increased rate of gene loss through random drift.     

The effect of intragenic recombination on the number of alleles in a finite population

A two-site infinite allele model is constructed to study the effect of intragenic recombination on the number of neutral alleles and the distribution of their frequencies in a finite population. The results of theory and Monte Carlo simulation of the two-site model demonstrate that intragenic recombination significantly increases the mean and variance of the number of alleles when the rates of mutation and recombination are as large as the reciprocal of the population size. Data from natural populations indicate that this may be a significant process in generating variation and determining its distribution.     

Stochastic interplay between mutation and recombination during the acquisition of drug resistance mutations in human immunodeficiency virus type 1

The emergence of drug resistance mutations in human immunodeficiency virus (HIV) has been a major setback in the treatment of infected patients. Besides the high mutation rate, recombination has been conjectured to have an important impact on the emergence of drug resistance. Population genetic theory suggests that in populations limited in size recombination may facilitate the acquisition of beneficial mutations. The viral population in an infected patient may indeed represent such a population limited in size, since current estimates of the effective population size range from 500 to 10(5). To address the effects of limited population size, we therefore expand a previously described deterministic population genetic model of HIV replication by incorporating the stochastic processes that occur in finite populations of infected cells. Using parameter estimates from the literature, we simulate the evolution of drug-resistant viral strains. The simulations show that recombination has only a minor effect on the rate of acquisition of drug resistance mutations in populations with effective population sizes as small as 1,000, since in these populations, viral strains typically fix beneficial mutations sequentially. However, for intermediate effective population sizes (10(4) to 10(5)), recombination can accelerate the evolution of drug resistance by up to 25%. Furthermore, a reduction in population size caused by drug therapy can be overcome by a higher viral mutation rate, leading to a faster evolution of drug resistance.     

MUTATIONS DO NOT ACCUMULATE IN ASEXUAL ISOLATES CAPABLE OF GROWTH AND EXTINCTION–MULLER'S RATCHET RE-EXAMINED

The rarity of exclusively asexual species is often attributed to Muller's Ratchet. This supposes that because asexual populations cannot recreate individuals with fewer mutations than the currently least-loaded line, mutations will accumulate in such isolates. However, because the computer models that corroborate this theory have assumed isolate immortality, it is possible that mutations will accumulate only if there is “soft” selection acting on relative, rather than absolute, fitness. Here we, therefore, describe several models in which 200 asexual organisms randomly selected from an infinite population in genetic equilibrium under “hard” selection (acting through absolute fitness), were followed for 100 generations. When there were no limits to the fluctuations in population size, the deterministic distribution of mutations per individual was maintained for 100 (as well as for 200) generations. If population growth was limited by a proportional decrease in fertility of the whole isolate, then the isolates tended to become extinct. The rate of extinction was inversely related to maximum isolate size. When resource limitation at maximum population size had an extra deleterious effect on mutants, then isolates shed the mutant classes. Mutations accumulated (ad inifinitum) in immortal isolates whose population numbers were kept constant by proportionately increasing or decreasing each class's size whenever isolate size ≠ 200. Muller's Ratchet, therefore, operates only when mutations affect the outcome of intraspecific contests, but not the organisms' intrinsic ability to survive in the ecosystem.     

Monte Carlo simulation of biospecific interactions

Numerical simulations of the stochastic time evolution of biospecific interactions are described and show that when molecular populations are large, time course predictions match those obtained using a deterministic expression. When population size is decreased the effects of stochastic noise become apparent. The significance of stochastic noise in sensitive binding-based assay systems suggests an immediate need for models of this type.     

Population dependent Fourier decomposition of fitness landscapes over recombination spaces: Evolvability of complex characters

The effect of recombination on genotypes can be represented in the form of P-structures, i.e., a map from the set of pairs of genotypes to the power set of genotypes. The interpretation is that the P-structure maps the pair of parental genotypes to the set of recombinant genotypes which result from the recombination of the parental genotypes. A recombination fitness landscape is then a function from the genotypes in a P-structure to the real numbers. In previous papers we have shown that the eigenfunctions of (a matrix associated with) the P-structure provide a basis for the Fourier decomposition of arbitrary recombination landscapes. Here we generalize this framework to include the effect of genotype frequencies, assuming linkage equilibrium. We find that the autocorrelation of the eigenfunctions of the population-weighted P-structure is independent of the population composition. As a consequence we can directly compare the performance of mutation and recombination operators by comparing the autocorrelations on the finite set of elementary landscapes. This comparison suggests that point mutation is a superior search strategy on landscapes with a low order and a moderate order of interaction p < n/3 (n is the number of loci). For more complex landscapes 1-point recombination is superior to both mutation and uniform recombination, but only if the distance among the interacting loci (defining length) is minimal. Furthermore we find that the autocorrelation on any landscape is increasing as the distribution of genotypes becomes more extreme, i.e., if the population occupies a location close to the boundary of the frequency simplex. Landscapes are smoother the more biased the distribution of genotype frequencies is. We suggest that this result explains the paradox that there is little epistatic interaction for quantitative traits detected in natural populations if one uses variance decomposition methods while there is evidence for strong interactions in molecular mapping studies for quantitative trait loci.     

Persistence of structured populations in random environments

Environmental fluctuations often have different impacts on individuals that differ in size, age, or spatial location. To understand how population structure, environmental fluctuations, and density-dependent interactions influence population dynamics, we provide a general theory for persistence for density-dependent matrix models in random environments. For populations with compensating density dependence, exhibiting “bounded” dynamics, and living in a stationary environment, we show that persistence is determined by the stochastic growth rate (alternatively, dominant Lyapunov exponent) when the population is rare. If this stochastic growth rate is negative, then the total population abundance goes to zero with probability one. If this stochastic growth rate is positive, there is a unique positive stationary distribution. Provided there are initially some individuals in the population, the population converges in distribution to this stationary distribution and the empirical measures almost surely converge to the distribution of the stationary distribution. For models with overcompensating density-dependence, weaker results are proven. Methods to estimate stochastic growth rates are presented. To illustrate the utility of these results, applications to unstructured, spatially structured, and stage-structured population models are given. For instance, we show that diffusively coupled sink populations can persist provided that within patch fitness is sufficiently variable in time but not strongly correlated across space.