Survival in continuous structured populations models

The extended McKendrick-von Foerster structured population model is employed to derive a nonautonomous ordinary differential equation model of a population. The derivation assumes that the individual life history can be delineated into several physiological stages. We study the persistence of the population when the model is autonomous and base the nonautonomous survival analysis on the autonomous case and a comparison principle. A brief excursion into alternate life history strategies is presented.This work was supported in part by the U.S. Environmental Protection Agency under cooperative agreement CR 813353010     

Equilibria in structured populations

The existence of a stable positive equilibrium state for the density of a population which is internally structured by means of a single scalar such as age, size, etc. is studied as a bifurcation problem. Using an inherent birth modulus n as a bifurcation parameter it is shown for very general nonlinear model equations, in which vital birth and growth processes depend on population density, that a global unbounded continuum of nontrivial equilibrium pairs (n, ) bifurcates from the unique (normalized) critical point (1, 0). The pairs are locally positive and conditions are given under which the continuum is globally positive. Local stability is shown to depend on the direction of bifurcation. For the important case when density dependence is a nonlinear expression involving a linear functional of density (such as total population size) it is shown how a detailed global bifurcation diagram is easily constructed in applications from the graph of a certain real valued function obtained from an invariant on the continuum. Uniqueness and nonuniqueness of positive equilibrium states are studied. The results are illustrated by several applications to models appearing in the literature.This research was done while the author was on leave at the Lehrstuhl für Biomathematik, Universität Tübingen, Auf der Morgenstelle 10, 7400 Tübingen 1, Federal Republic of Germany     

Persistence in fluctuating environments for interacting structured populations

Individuals within any species exhibit differences in size, developmental state, or spatial location. These differences coupled with environmental fluctuations in demographic rates can have subtle effects on population persistence and species coexistence. To understand these effects, we provide a general theory for coexistence of structured, interacting species living in a stochastic environment. The theory is applicable to nonlinear, multi species matrix models with stochastically varying parameters. The theory relies on long-term growth rates of species corresponding to the dominant Lyapunov exponents of random matrix products. Our coexistence criterion requires that a convex combination of these long-term growth rates is positive with probability one whenever one or more species are at low density. When this condition holds, the community is stochastically persistent: the fraction of time that a species density goes below \(\delta >0\) approaches zero as \(\delta \) approaches zero. Applications to predator-prey interactions in an autocorrelated environment, a stochastic LPA model, and spatial lottery models are provided. These applications demonstrate that positive autocorrelations in temporal fluctuations can disrupt predator-prey coexistence, fluctuations in log-fecundity can facilitate persistence in structured populations, and long-lived, relatively sedentary competing populations are likely to coexist in spatially and temporally heterogenous environments.     

Coalescent approximation for structured populations in a stationary random environment

We establish convergence to the Kingman coalescent for the genealogy of a geographically-or otherwise-structured version of the Wright-Fisher population model with fast migration. The new feature is that migration probabilities may change in a random fashion. This brings a novel formula for the coalescent effective population size (EPS). We call it a quenched EPS to emphasize the key feature of our model — random environment. The quenched EPS is compared with an annealed (mean-field) EPS which describes the case of constant migration probabilities obtained by averaging the random migration probabilities over possible environments.     

Numerical solution of structured population models

Numerical methods are presented for a general age-structured population model with demographic rates depending on age and the total population size. The accuracy of these methods is established by solving problems for which alternate solution techniques are available and are used for comparison. The methods reliably solve test problems with a variety of dynamic behavior. Simulations of a blowfly population exhibit cyclic fluctuations, whereas a simulated squirrel population reaches a stable age distribution and stable equilibrium population size. Life-history attributes are easily studied from the computed solutions, and are discussed for these examples. Recovery of a stressed population back to equilibrium is examined by computing the transition in age structure, and the transient behavior of other properties of the population such as the per capita growth rate, the average age, and the generation length.     

Epidemic outbreaks on structured populations

Our chances to halt epidemic outbreaks rely on how accurately we represent the population structure underlying the disease spread. When analysing global epidemics this force us to consider metapopulation models taking into account intra- and inter-community interactions. Here I introduce and analyze a metapopulation model which accounts for several features observed in real outbreaks. First, I demonstrate that depending on the intra-community expected outbreak size and the fraction of social bridges the epidemic outbreaks die out or there is a finite probability to observe a global epidemics. Second, I show that the global scenario is characterized by resurgent epidemics, their number increasing with increasing the intra-community average distance between individuals. Finally, I present empirical data for the AIDS epidemics supporting the model predictions.     

Strategy selection in structured populations

Evolutionary game theory studies frequency dependent selection. The fitness of a strategy is not constant, but depends on the relative frequencies of strategies in the population. This type of evolutionary dynamics occurs in many settings of ecology, infectious disease dynamics, animal behavior and social interactions of humans. Traditionally evolutionary game dynamics are studied in well-mixed populations, where the interaction between any two individuals is equally likely. There have also been several approaches to study evolutionary games in structured populations. In this paper we present a simple result that holds for a large variety of population structures. We consider the game between two strategies, A and B, described by the payoff matrix . We study a mutation and selection process. For weak selection strategy A is favored over B if and only if σa+b>c+σd. This means the effect of population structure on strategy selection can be described by a single parameter, σ. We present the values of σ for various examples including the well-mixed population, games on graphs, games in phenotype space and games on sets. We give a proof for the existence of such a σ, which holds for all population structures and update rules that have certain (natural) properties. We assume weak selection, but allow any mutation rate. We discuss the relationship between σ and the critical benefit to cost ratio for the evolution of cooperation. The single parameter, σ, allows us to quantify the ability of a population structure to promote the evolution of cooperation or to choose efficient equilibria in coordination games.     

A spatially structured metapopulation model with patch dynamics

Metapopulation models that incorporate both spatial and temporal structure are studied in this paper. The existence and stability of equilibria are provided, and an extinction threshold condition is derived which depends on patch dynamics (patch destruction and creation) and metapopulation dynamics (patch colonization and extinction). These results refine threshold conditions given by previous metapopulation models. By comparing landscapes with different spatial heterogeneities with respect to weighted long-term patch occupancies, we conclude that the pattern of a landscape is of overwhelming importance in determining metapopulation persistence and patch occupancy. We show that the same conclusion holds when a rescue effect is considered. We also derive a stochastic differential equations (SDE) model of the It? type based on our deterministic model. Our simulations reveal good agreement between the deterministic model and the SDE model.     

Persistence despite perturbations for interacting populations

Two definitions of persistence despite perturbations in deterministic models are presented. The first definition, persistence despite frequent small perturbations, is shown to be equivalent to the existence of a positive attractor i.e. an attractor bounded away from extinction. The second definition, persistence despite rare large perturbations, is shown to be equivalent to permanence i.e. a positive attractor whose basin of attraction includes all positive states. Both definitions set up a natural dichotomy for classifying models of interacting populations. Namely, a model is either persistent despite perturbations or not. When it is not persistent, it follows that all initial conditions are prone to extinction due to perturbations of the appropriate type. For frequent small perturbations, this method of classification is shown to be generically robust: there is a dense set of models for which persistent (respectively, extinction prone) models lies within an open set of persistent (resp. extinction prone) models. For rare large perturbations, this method of classification is shown not to be generically robust. Namely, work of Josef Hofbauer and the author have shown there are open sets of ecological models containing a dense sets of permanent models and a dense set of extinction prone models. The merits and drawbacks of these different definitions are discussed.     

Numerical solution of structured population models

Numerical methods are presented for a general mass-structured population model with demographic rates that depend on individual mass, time, and total population mass. Several types of numerical methods are described, Eulerian methods, implicit methods, and the method of characteristics. These methods are compared for a sample problem with an exact solution. The preferred numerical technique combines the method of characteristics with an adaptive grid. Numerical solution of model equations developed for mosquitofish illustrates this method and demonstrates how seasons can play a dominant role in shaping population development.     

Synchronous dynamics and rates of extinction in spatially structured populations

We explore extinction rates using a spatially arranged set of subpopulations obeying Ricker dynamics. The population system is subjected to dispersal of individuals among the subpopulations as well as to local and global disturbances. We observe a tight positive correlation between global extinction rate and the level of synchrony in dynamics among thesubpopulations. Global disturbances and to a lesser extent, migration, are capable of synchronizing the temporal dynamics of the subpopulations over a rather wide span of the population growth rate r. Local noise decreases synchrony, as does increasing distance among the subpopulations. Synchrony also levels off with increasing r: in the chaotic region, subpopulations almost invariably behave asynchronously. We conclude that it is asynchrony that reduces the probability of global extinctions, not chaos as such: chaos is a special case only. The relationship between global extinction rate, synchronous dynamics and population growth rate is robust to changes in dispersal rates and ranges.     

Dispersal patterns of endemic alpine butterflies with contrasting population structures:<Emphasis Type="Italic"> Erebia epiphron</Emphasis> and<Emphasis Type="Italic"> E. sudetica</Emphasis>

We studied population sizes and mobility of Erebia epiphron and Erebia sudetica, two high mountain butterflies forming endemic subspecies in the Hrubý Jeseník Mountains, Czech Republic. E. epiphron formed two continuous populations containing 100,000 and 4,500 individuals on alpine grasslands. The butterflies moved freely within their habitats, but movements between the two populations were highly unlikely. E. sudetica formed a system of colonies at timberline sites on valley headwalls and in forest clearings. Two such colonies studied in detail contained 4,500 and 450 adults and were interconnected by limited dispersal. The negative exponential function and the sigmoid function (this assumes flat decrease of movements over short distances) were superior to the inverse power function in fitting mobility data for both species. For E. sudetica, the functions describing movements within a habitat differed significantly from total movements, suggesting different behaviours of dispersing individuals. The habitats of E. epiphron are uniform and highly isolated, favouring free within-habitat mobility but prohibiting leaving their boundaries. The habitats of E. sudetica are diverse and disturbance-dependent; leaving such habitats is less risky, and a source-sink model may explain the persistence of the species in the mountains.     

Invasion speeds for structured populations in fluctuating environments

We live in a time where climate models predict future increases in environmental variability and biological invasions are becoming increasingly frequent. A key to developing effective responses to biological invasions in increasingly variable environments will be estimates of their rates of spatial spread and the associated uncertainty of these estimates. Using stochastic, stage-structured, integrodifference equation models, we show analytically that invasion speeds are asymptotically normally distributed with a variance that decreases in time. We apply our methods to a simple juvenile–adult model with stochastic variation in reproduction and an illustrative example with published data for the perennial herb, Calathea ovandensis. These examples buttressed by additional analysis reveal that increased variability in vital rates simultaneously slow down invasions yet generate greater uncertainty about rates of spatial spread. Moreover, while temporal autocorrelations in vital rates inflate variability in invasion speeds, the effect of these autocorrelations on the average invasion speed can be positive or negative depending on life history traits and how well vital rates “remember” the past.     

Dynamics of age- and size-structured populations in fluctuating environments: Applications of stochastic matrix models to natural populations

Recent developments of the theory of stochastic matrix modeling have made it possible to estimate general properties of age- and size-structured populations in fluctuating environments. However, applications of the theory to natural populations are still few. The empirical studies which have used stochastic matrix models are reviewed here to examine whether predictions made by the theory can be generally found in wild populations. The organisms studied include terrestrial grasses and herbs, a seaweed, a fish, a reptile, a deer and some marine invertebrates. In all the studies, the stochastic population growth rate (ln λ _s ) was no greater than the deterministic population growth rate determined using average vital rates, suggesting that the model based only on average vital rates may overestimate growth rates of populations in fluctuating environments. Factors affecting ln λ _s include the magnitude of variation in vital rates, probability distribution of random environments, fluctuation in different types of vital rates, covariances between vital rates, and autocorrelation between successive environments. However, comprehensive rules were hardly found through the comparisons of the empirical studies. Based on shortcomings of previous studies, I address some important subjects which should be examined in future studies.     

Frequency responses of age-structured populations: Pacific salmon as an example

Increasing evidence of the effects of changing climate on physical ocean conditions and long-term changes in fish populations adds to the need to understand the effects of stochastic forcing on marine populations. Cohort resonance is of particular interest because it involves selective sensitivity to specific time scales of environmental variability, including that of mean age of reproduction, and, more importantly, very low frequencies (i.e., trends). We present an age-structured model for two Pacific salmon species with environmental variability in survival rate and in individual growth rate, hence spawning age distribution. We use computed frequency response curves and analysis of the linearized dynamics to obtain two main results. First, the frequency response of the population is affected by the life history stage at which variability affects the population; varying growth rate tends to excite periodic resonance in age structure, while varying survival tends to excite low frequency fluctuation with more effect on total population size. Second, decreasing adult survival strengthens the cohort resonance effect at all frequencies, a finding that addresses the question of how fishing and climate change will interact.     

Equilibria in systems of interacting structured populations

The existence of a stable positive equilibrium density for a community of k interacting structured species is studied as a bifurcation problem. Under the assumption that a subcommunity of k–1 species has a positive equilibrium and under only very mild restrictions on the density dependent vital growth rates, it is shown that a global continuum of equilibria for the full community bifurcates from the subcommunity equilibrium at a unique critical value of a certain inherent birth modulus for the kth species. Local stability is shown to depend upon the direction of bifurcation. The direction of bifurcation is studied in more detail for the case when vital per unity birth and death rates depend on population density through positive linear functionals of density and for the important case of two interacting species. Some examples involving competition, predation and epidemics are given.     

Persistence of invading gypsy moth populations in the United States

Exotic invasive species are a mounting threat to native biodiversity, and their effects are gaining more public attention as each new species is detected. Equally important are the dynamics of exotic invasives that are previously well established. While the literature reports many examples of the ability of a newly arrived exotic invader to persist prior to detection and population growth, we focused on the persistence dynamics of an established invader, the European gypsy moth (Lymantria dispar) in the United States. The spread of gypsy moth is largely thought to be the result of the growth and coalescence of isolated colonies in a transition zone ahead of the generally infested area. One important question is thus the ability of these isolated colonies to persist when subject to Allee effects and inimical stochastic events. We analyzed the US gypsy moth survey data and identified isolated colonies of gypsy moth using the local indicator of spatial autocorrelation. We then determined region-specific probabilities of colony persistence given the population abundance in the previous year and its relationship to a suite of ecological factors. We observed that colonies in Wisconsin, US, were significantly more likely to persist in the following year than in other geographic regions of the transition zone, and in all regions, the abundance of preferred host tree species and land use category did not appear to influence persistence. We propose that differences in region-specific rates of persistence may be attributed to Allee effects that are differentially expressed in space, and that the inclusion of geographically varying Allee effects into colony-invasion models may provide an improved paradigm for addressing the establishment and spread of gypsy moth and other invasive exotic species.     

Evolution of fitnesses in structured populations with correlated environments

The outcome of selection in structured populations with spatially varying selection pressures depends on the interaction of two factors: the level of gene flow and the amount of heterogeneity among the demes. Here we investigate the effect of three different levels of spatial heterogeneity on the levels of genetic polymorphisms for different levels of gene flow, using a construction approach in which a population is constantly bombarded with new mutations. We further compare the relative importance of two kinds of balancing selection (heterozygote advantage and selection arising from spatial heterogeneity), the level of adaptation and the stability of the resulting polymorphic equilibria. The different levels of environmental heterogeneity and gene flow have a large influence on the final level of polymorphism. Both factors also influence the relative importance of the two kinds of balancing selection in the maintenance of variation. In particular, selection arising from spatial heterogeneity does not appear to be an important form of balancing selection for the most homogeneous scenario. The level of adaptation is highest for low levels of gene flow and, at those levels, remarkably similar for the different levels of spatial heterogeneity, whereas for higher levels of gene flow the level of adaptation is substantially reduced.     

The strong-migration limit in geographically structured populations

Summary Some strong-migration limits are established for geographically structured populations. A diploid monoecious population is subdivided into a finite number of colonies, which exchange migrants. The migration pattern is fixed and ergodic, but otherwise arbitrary. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus. In all the limiting results, an effective population number N _e ( N _T ) appears instead of the actual total population number N _T . 1. If there is no selection, every allele mutates at rate u to types not preexisting in the population, and the (finite) subpopulation numbers N _i are very large, then the ultimate rate and pattern of convergence of the probabilities of allelic identity are approximately the same as for panmixia. If, in addition, the N _i are proportional to 1/u, as N _T 8, the equilibrium probabilities of identity converge to the panmictic value. 2. With a finite number of alleles, any mutation pattern, an arbitrary selection scheme for each colony, and the mutation rates and selection coefficients proportional to 1/N _T , let P _j be the frequency of the allele A _j in the entire population, averaged with respect to the stationary distribution of the backward migration matrix M. As N _T 8, the deviations of the allelic frequencies in each of the subpopulations from P _j converge to zero; the usual panmictic mutation-selection diffusion is obtained for P _j , with the selection intensities averaged with respect to the stationary distribution of M. In both models, N _e = N _T and all effects of population subdivision disappear in the limit if, and only if, migration does not alter the subpopulation numbers.Supported by the National Science Foundation (Grant No. DEB77-21494)