Continuous-discrete models of the dynamics of an isolated population and of 2 competing species

The paper presents the analysis of various mathematical models for dynamics of isolated population and for competition between two species. It is assumed that mortality is continuous and birth of individuals of new generations takes place in certain fixed moments. Influence of winter upon the population dynamics and conditions of classic discrete model "deduction" of population dynamics (in particular, Moran-Ricker and Hassel's models) are investigated. Dynamic regimes of models under various assumptions about the birth and death rates upon the population states are also examined. Analysis of models of isolated population dynamics with nonoverlapping generations showed the density changes regularly if the birth rate is constant. Moreover, there exists a unique global stable level and population size stabilizes asymptotically at this equilibrium, i.e. cycle and chaotic regimes in various discrete models depend on correlation between individual productivity and population state in previous time. When the correlation is exponential upon mean population size the discrete Hassel model is realized. Modification of basis model, based on the assumption that during winter survival/death changes are constant, showed that population size at global level is stable. Generally, the dependence of population rate upon "winter parameters" has nonlinear character. Nonparametric models of competition between two species does not vary if the individual productivity is constant. In a phase space there are several stable stationary states and population stabilizes at one or other level asymptotically. So, in discrete models of competition between two species oscillation can be explained by dependence of population growth rate on the population size at previous times.     

Demographic heterogeneity impacts density-dependent population dynamics

Among-individual variation in vital parameters such as birth and death rates that is unrelated to age, stage, sex, or environmental fluctuations is referred to as demographic heterogeneity. This kind of heterogeneity is prevalent in ecological populations, but is almost always left out of models. Demographic heterogeneity has been shown to affect demographic stochasticity in small populations and to increase growth rates for density-independent populations. The latter is due to ??cohort selection,?? where the most frail individuals die out first, lowering the cohort??s average mortality as it ages. The importance of cohort selection to population dynamics has only recently been recognized. We use a continuous-time model with density dependence, based on the logistic equation, to study the effects of demographic heterogeneity in mortality and reproduction. Reproductive heterogeneity is introduced in three ways: parent fertility, offspring viability, and parent?Coffspring correlation. We find that both the low-density growth rate and the equilibrium population size increase as the magnitude of mortality heterogeneity increases or as parent?Coffspring phenotypic correlation increases. Population dynamics are affected by complex interactions among the different types of heterogeneity, and trade-off scenarios are examined which can sometimes reverse the effect of increased heterogeneity. We show that there are a number of different homogeneous approximations to heterogeneous models, but all fail to capture important parts of the dynamics of the full model.     

LOCAL ADAPTATION WHEN COMPETITION DEPENDS ON PHENOTYPIC SIMILARITY

Recent work incorporating demographic–genetic interactions indicates the importance of population size, gene flow, and selection in influencing local adaptation. This work typically assumes that density‐dependent survival affects individuals equally, but individuals in natural population rarely compete equally. Among‐individual differences in resource use generate stronger competition between more similar phenotypes (frequency‐dependent competition) but it remains unclear how this additional form of selection changes the interactions between population size, gene flow, and local stabilizing selection. Here, we integrate migration–selection dynamics with frequency‐dependent competition. We developed a coupled demographic‐quantitative genetic model consisting of two patches connected by dispersal and subject to local stabilizing selection and competition. Our model shows that frequency‐dependent competition slightly increases local adaptation, greatly increases genetic variance within patches, and reduces the amount that migration depresses population size, despite the increased genetic variance load. The effects of frequency‐dependence depend on the strength of divergent selection, trait heritability, and when mortality occurs in the life cycle in relation to migration and reproduction. Essentially, frequency‐dependent competition reduces the density‐dependent interactions between migrants and residents, the extent to which depends on how different and common immigrants are compared to residents. Our results add new dynamics that illustrate how competition can alter the effects of gene flow and divergent selection on local adaptation and population carrying capacities.     

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.     

Stochastic sampling of interaction partners versus deterministic payoff assignment

Evolutionary game dynamics describes how successful strategies spread in a population. In well-mixed populations, the usual assumption, e.g. underlying the replicator dynamics, is that individuals obtain a payoff from interactions with a representative sample of the population. This determines their fitness. Here, we analyze a situation in which payoffs are obtained through a single interaction, so that individuals of the same type can have different payoffs. We show analytically that for weak selection, this scenario is identical to the usual approach in which an individual interacts with the whole population. For strong selection, however, differences arise that are reflected in the fixation probabilities and lead to deviating evolutionary dynamics.     

Natural selection, fitness entropy, and the dynamics of coevolution

The coevolutionary dynamics of interacting populations were studied by combining continuous time Lotka-Volterra models of population growth with single-locus genetic models of weak selection. The effects of natural selection on population growth were evaluated using Ginzburg's fitness entropy function as a measure of the deviation of a population's initial allele frequencies from their polymorphic equilibrium values. This entropy measure was used to relate the dynamics of a community composed of evolving populations to the dynamics of a "reference community" whose populations are initially in genetic equilibrium. Specifically, a quantity called the "selective difference area" was defined as the total difference between the population size trajectories of a reference and evolving population. The selective difference area represents the amount of extra life a species would realize if the entire community were at genetic equilibrium. It was shown that this selective difference area is a simple linear function of the initial fitness entropies of each species. This prediction is independent of the strength of selection and holds for any arbitrary set of initial population densities. Numerical examples were presented to illustrate the results. Under the assumption of weak selection, a generalization for arbitrary population growth models was outlined.     

Effects of releasing maladapted individuals: a demographic-evolutionary model

A model of the joint dynamics of change in population size N and evolution in a quantitative trait z, as a result of a general form of density dependence, local stabilizing selection, and immigration of individuals deviating from the local optimum, is analyzed. For weak selection and migration, a reduction in total equilibrium population size below the initial level without immigration, K, is shown to occur if the immigrants deviates more than square root of 8 = 2.83 genetic standard deviations from the optimum and if the rate of migration m is sufficiently large relative to the strength of stabilizing selection s. For the Lotka-Volterra form of density dependence, two additional equilibria are shown to exist below K, provided that the strength of selection is large relative to the strength of density dependence. Reintroduction of an initially extinct population is possible if the immigrants are not too maladapted and if the genetic variance is sufficiently large. For a simplified version of the model corresponding to competition between similar species or different haplotypes, the equilibrium population size is always exactly at K if m < Ksz1(2) and is above K otherwise, which shows the importance of including recombination in the model.     

Equilibrium selection in evolutionary games with random matching of players

We discuss stochastic dynamics of populations of individuals playing games. Our models possess two evolutionarily stable strategies: an efficient one, where a population is in a state with the maximal payoff (fitness) and a risk-dominant one, where players are averse to risks. We assume that individuals play with randomly chosen opponents (they do not play against average strategies as in the standard replicator dynamics). We show that the long-run behavior of a population depends on its size and the mutation level.     

Density dependence in an island population of silvereyes

The population of silvereyes Zosterops lateralis chlorocephalus , on Heron Island, Great Barrier Reef has been monitored accurately since 1965. Between 1979 and 1993, the breeding success of all birds was determined by monitoring nests. The population fluctuated between 225 and 483 individuals. Four cyclones led to substantial mortality. As this data set is long-term, has little observation error, and is from an effectively closed population, it provides an unusual opportunity to examine density dependence in reproduction or mortality. Using a stochastic logistic model, we found clear evidence of density dependence in adult population size. Logistic regression suggested that fledgling survival decreased with the numbers of birds attempting to breed. There was also some suggestion that adult survival might be density dependent. The fitted stochastic logistic model predicts negligible risks of extinction for this population, in contrast to the predictions of a published population viability analysis. Whilst our statistical model including density dependence may provide better predictions of the "usual" behaviour of a population than a population viability analysis, we suggest that caution should be exercised when statistically fitted models are used to predict the behaviour of the population at extremes, such as near extinction.     

The Allee-type ideal free distribution

The ideal free distribution (IFD) in a two-patch environment where individual fitness is positively density dependent at low population densities is studied. The IFD is defined as an evolutionarily stable strategy of the habitat selection game. It is shown that for low and high population densities only one IFD exists, but for intermediate population densities there are up to three IFDs. Population and distributional dynamics described by the replicator dynamics are studied. It is shown that distributional stability (i.e., IFD) does not imply local stability of a population equilibrium. Thus distributional stability is not sufficient for population stability. Results of this article demonstrate that the Allee effect can strongly influence not only population dynamics, but also population distribution in space.     

Two SIS epidemiologic models with delays

 The SIS epidemiologic models have a delay corresponding to the infectious period, and disease-related deaths, so that the population size is variable. The population dynamics structures are either logistic or recruitment with natural deaths. Here the thresholds and equilibria are determined, and stabilities are examined. In a similar SIS model with exponential population dynamics, the delay destabilized the endemic equilibrium and led to periodic solutions. In the model with logistic dynamics, periodic solutions in the infectious fraction can occur as the population approaches extinction for a small set of parameter values. Received: 10 January 1997 / 18 November 1997     

The influence of size-dependent life-history traits on the structure and dynamics of populations and communities

Individual organisms often show pronounced changes in body size throughout life with concomitant changes in ecological performance. We synthesize recent insight into the relationship between size dependence in individual life history and population dynamics. Most studies have focused on size‐dependent life‐history traits and population size‐structure in the highest trophic level, which generally leads to population cycles with a period equal to the juvenile delay. These cycles are driven by differences in competitiveness of differently sized individuals. In multi‐trophic systems, size dependence in life‐history traits at lower trophic levels may have consequences for both the dynamics and structure of communities, as size‐selective predation may lead to the occurrence of emergent Allee effects and the stabilization of predator–prey cycles. These consequences are linked to that individual development is density dependent. We conjecture that especially this population feedback on individual development may lead to new theoretical insight compared to theory based on unstructured or age‐dependent models. Density‐dependent individual development may also cause individuals to realize radically different life histories, dependent on the state and dynamics of the population during their life and may therefore have consequences for individual behaviour or the evolution of life‐history traits as well.     

Long-term growth dynamics of natural forests in Hokkaido,northern Japan

Abstract. The long-term growth dynamics of natural forest stands on the island of Hokkaido were described on the basis of an analysis of data from 38 permanent plots spanning 15–22 yr. Stand structure was characterized by basal area, stem density and tree size variability. To detect trends in stand structure, regression models for recruitment rate (per ha per yr), mortality rate and the rate of change in stem density and tree size variability were developed by a stepwise method using initial basal area, stem density, tree size variability, species composition summarized by LNMDS ordination, altitude, annual mean temperature, annual precipitation, type of understorey vegetation, topography and slope aspect as candidates for predictor variables. The same analyses were conducted for basal area increment (net growth) and its components: survivor growth = basal area gain by growth of surviving individuals and mortality = basal area loss by death of individuals. Stem density remained generally unchanged; recruitment was relatively low even in very sparse stands. Stand basal area generally increased as survivor growth was approximately double the mortality. Recruitment rate was strongly affected by the presence of dwarf bamboo (Sasa spp.) vegetation on the forest floor which inhibited tree regeneration. Mortality rate was density-dependent; dense stands had higher mortality than sparse stands. Density change rate (recruitment rate - mortality rate) was, therefore, determined by both the type of understorey vegetation and stem density. Survivor growth was high in stands with high stem density and basal area. Mortality was dependent on basal area and altitude. Net basal area increment (net growth) was dependent only on stem density with other factors that influenced survivor growth and mortality omitted. Tree size variability decreased in stands with high tree size variability whereas it increased in stands with low size variability. Based on the obtained models for density change rate and net basal area increment, trajectories of stands were illustrated on a log-log diagram of stem density and basal area. The predicted differences in trajectories as affected by the understorey vegetation type indicated the importance of dwarf bamboo vegetation for forest dynamics on Hokkaido.     

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.     

Mutation-selection equilibrium in games with multiple strategies

In evolutionary games the fitness of individuals is not constant but depends on the relative abundance of the various strategies in the population. Here we study general games among n strategies in populations of large but finite size. We explore stochastic evolutionary dynamics under weak selection, but for any mutation rate. We analyze the frequency dependent Moran process in well-mixed populations, but almost identical results are found for the Wright-Fisher and Pairwise Comparison processes. Surprisingly simple conditions specify whether a strategy is more abundant on average than 1/n, or than another strategy, in the mutation-selection equilibrium. We find one condition that holds for low mutation rate and another condition that holds for high mutation rate. A linear combination of these two conditions holds for any mutation rate. Our results allow a complete characterization of n×n games in the limit of weak selection.     

Behavioral states help translate dispersal movements into spatial distribution patterns of floaters

Within the field of spatial ecology, it is important to study animal movements in order to better understand population dynamics. Dispersal is a nonlinear process through which different behavioral mechanisms could affect movement patterns. One of the most common approaches to analyzing the trajectories of organisms within patches is to use random-walk models to describe movement features. These models express individual movements within a specific area in terms of random-walk parameters in an effort to relate movement patterns to the distributions of organisms in space. However, only using the movement trajectories of individuals to predict the spatial spread of animal populations may not fit the complex distribution of individuals across heterogeneous environments. When we empirically tested the results from a random-walk model (a residence index) used to predict the spatial equilibrium distribution of individuals, we found that the index severely underestimated the spatial spread of dispersing individuals. We believe this is because random-walk models only account for the effects of environmental conditions on individual movements, completely overlooking the crucial influence of behavior changes over time. In the future, both aspects should be accounted for when predicting general rules of (meta)population abundance, distribution, and dynamics from patterns of animal movements.     

Patterns and processes of population dynamics with fluctuating habitat size: a case study of a marine copepod inhabiting tide pools

The logistic model is a fundamental population model often used as the basis for analyzing wildlife population dynamics. In the classic logistic model, however, population dynamics may be difficult to characterize if habitat size is temporally variable because population density can vary at a constant abundance, which results in variable strength of density‐dependent feedback for a given population size. To incorporate habitat size variability, we developed a general population model in which changes in population abundance, density, and habitat size are taken into account. From this model, we deduced several predictions for patterns and processes of population dynamics: 1) patterns of fluctuation in population abundance and density can diverge, with respect of their correlation and relative variability; and 2) along with density dependence, habitat size fluctuation can affect population growth with a time lag because changes in habitat size result in changes in population density. In order to test these predictions, we applied our model to population dynamics data of 36 populations of Tigriopus japonicus, a marine copepod inhabiting tide pools of variable sizes caused by weather processes. As expected, we found a significant difference in the fluctuation patterns of population abundance and density of T. japonicus populations with respect to the correlation between abundance and density and their relative variability, which correlates positively with the variability of habitat size. In addition, we found direct and lagged‐indirect effects of weather processes on population growth, which were associated with density dependence and impose regulatory forces on local and regional population dynamics. These results illustrate how changes in habitat size can have an impact on patterns and processes of wildlife population dynamics. We suggest that without knowledge of habitat size fluctuation, measures of population size and its variability as well as inferences about the processes of population dynamics may be misleading.     

DOES INCREASED MORTALITY FAVOR THE EVOLUTION OF MORE RAPID SENESCENCE?

It is widely believed (following the 1957 hypothesis of G. C. Williams) that greater rates of “extrinsic” (age- and condition-independent) mortality favor more rapid senescence. However, a recent analysis of mammalian life tables failed to find a significant correlation between minimum adult mortality rate and the rate of senescence. This article presents a simple theoretical analysis of how extrinsic mortality should affect the rate of senescence (i.e., the rate at which probability of mortality increases with age) under different evolutionary and population dynamical assumptions. If population dynamics are density independent, extrinsic mortality should not alter the senescence rate favored by natural selection. If population growth is density dependent and populations are stable, the effect of extrinsic mortality depends on the age specificity of the density dependence and on whether survival or reproduction (or both) are functions of density. It is possible that higher extrinsic mortality will increase the rate of senescence at all ages, decrease the rate at all ages, or increase it at some ages while decreasing it at others. Williams's hypothesis is most likely to be supported when density dependence acts primarily on fertility and does not differentially decrease the fertilities of older individuals. Patterns contrary to Williams's prediction are possible when density dependence acts primarily on the survival or fertility of later ages or when most variation in mortality rates is due to variation in nonextrinsic mortality.     

The role of sexually abstained groups in two-sex demographic and epidemic logistic models with non-linear mortality

We describe several gender structured population models governed by logistic growth with non-linear death rate. We extend these models to include groups of people isolated from sexual activity and individuals exposed to a mild and long-lasting sexually transmitted disease, i.e. without disease-induced mortality and recovery. The transmission of the disease is modeled through formation/separation of heterosexual couples assuming that one infected individual automatically infects his/her partner. We are interested in how the non-reproductive class may change the demographic tendencies in the general population and whether they can curb the growth of the infected group while keeping the healthy one at acceptable levels. A comparison of the equilibrium total population size in the presence and the absence of the isolated class is also provided.     

Mutation Rate Evolution in Replicator Dynamics

The mutation rate of an organism is itself evolvable. In stable environments, if faithful replication is costless, theory predicts that mutation rates will evolve to zero. However, positive mutation rates can evolve in novel or fluctuating environments, as analytical and empirical studies have shown. Previous work on this question has focused on environments that fluctuate independently of the evolving population. Here we consider fluctuations that arise from frequency-dependent selection in the evolving population itself. We investigate how the dynamics of competing traits can induce selective pressure on the rates of mutation between these traits. To address this question, we introduce a theoretical framework combining replicator dynamics and adaptive dynamics. We suppose that changes in mutation rates are rare, compared to changes in the traits under direct selection, so that the expected evolutionary trajectories of mutation rates can be obtained from analysis of pairwise competition between strains of different rates. Depending on the nature of frequency-dependent trait dynamics, we demonstrate three possible outcomes of this competition. First, if trait frequencies are at a mutation–selection equilibrium, lower mutation rates can displace higher ones. Second, if trait dynamics converge to a heteroclinic cycle—arising, for example, from “rock-paper-scissors” interactions—mutator strains succeed against non-mutators. Third, in cases where selection alone maintains all traits at positive frequencies, zero and nonzero mutation rates can coexist indefinitely. Our second result suggests that relatively high mutation rates may be observed for traits subject to cyclical frequency-dependent dynamics.