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.     

Persistence, spread and the drift paradox

We derive conditions for persistence and spread of a population where individuals are either immobile or dispersing by advection and diffusion through a one-dimensional medium with a unidirectional flow. Reproduction occurs only in the stationary phase. Examples of such systems are found in rivers and streams, marine currents, and areas with prevalent wind direction. In streams, a long-standing question, dubbed 'the drift paradox', asks why aquatic insects faced with downstream drift are able to persist in upper stream reaches. For our two-phase model, persistence of the population is guaranteed if, at low population densities, the local growth rate of the stationary component of the population exceeds the rate of entry of individuals into the drift. Otherwise the persistence condition involves all the model parameters, and persistence requires a critical (minimum) domain size. We calculate the rate at which invasion fronts propagate up- and downstream, and show that persistence and ability to spread are closely connected: if the population cannot advance upstream against the flow, it also cannot persist on any finite spatial domain. By studying two limiting cases of our model, we show that residence in the immobile state always enhances population persistence. We use our findings to evaluate a number of mechanisms previously proposed in the ecological literature as resolutions of the drift paradox.     

Building the Foundation for International Conservation Planning for Breeding Ducks across the U.S. and Canadian Border

We used publically available data on duck breeding distribution and recently compiled geospatial data on upland habitat and environmental conditions to develop a spatially explicit model of breeding duck populations across the entire Prairie Pothole Region (PPR). Our spatial population models were able to identify key areas for duck conservation across the PPR and predict between 62.1 – 79.1% (68.4% avg.) of the variation in duck counts by year from 2002 – 2010. The median difference in observed vs. predicted duck counts at a transect segment level was 4.6 ducks. Our models are the first seamless spatially explicit models of waterfowl abundance across the entire PPR and represent an initial step toward joint conservation planning between Prairie Pothole and Prairie Habitat Joint Ventures. Our work demonstrates that when spatial and temporal variation for highly mobile birds is incorporated into conservation planning it will likely increase the habitat area required to support defined population goals. A major goal of the current North American Waterfowl Management Plan and subsequent action plan is the linking of harvest and habitat management. We contend incorporation of spatial aspects will increase the likelihood of coherent joint harvest and habitat management decisions. Our results show at a minimum, it is possible to produce spatially explicit waterfowl abundance models that when summed across survey strata will produce similar strata level population estimates as the design-based Waterfowl Breeding Pair and Habitat Survey (r² = 0.977). This is important because these design-based population estimates are currently used to set duck harvest regulations and to set duck population and habitat goals for the North American Waterfowl Management Plan. We hope this effort generates discussion on the important linkages between spatial and temporal variation in population size, and distribution relative to habitat quantity and quality when linking habitat and population goals across this important region.     

Morphologically-structured models of growing budding yeast populations

It has been well recognized that many key aspects of cell cycle regulation are encoded into the size distributions of growing budding yeast populations due to the tight coupling between cell growth and cell division present in this organism. Several attempts have been made to model the cell size distribution of growing yeast populations in order to obtain insight on the underlying control mechanisms, but most were based on the age structure of asymmetrically dividing populations. Here we propose a new framework that couples a morphologically-structured representation of the population with population balance theory to formulate a dynamic model for the size distribution of growing yeast populations. An advantage of the presented framework is that it allows derivation of simpler models that are directly identifiable from experiments. We show how such models can be derived from the general framework and demonstrate their utility in analyzing yeast population data. Finally, by employing a recently proposed numerical scheme, we proceed to integrate numerically the full distributed model to provide predictions of dynamics of the cell size structure of growing yeast populations.     

Spatial distribution of dispersing animals

Summary A mathematical model for the dispersal of an animal population is presented for a system in which animals are initially released in the central region of a uniform field and migrate randomly, exerting mutually repulsive influences (population pressure) until they eventually become sedentary. The effect of the population pressure, which acts to enhance the dispersal of animals as their density becomes high, is modeled in terms of a nonlinear-diffusion equation. From this model, the density distribution of animals is obtained as a function of time and the initial number of released animals. The analysis of this function shows that the population ultimately reaches a nonzero stationary distribution which is confined to a finite region if both the sedentary effect and the population pressure are present. Our results are in good agreement with the experimental data on ant lions reported by Morisita, and we can also interpret some general features known for the spatial distribution of dispersing insects.     

Beyond diffusion: Modelling local and long-distance dispersal for organisms exhibiting intensive and extensive search modes

The distribution of foragers on the landscape has important consequences to, for example, the spread rate of an invasive species or the outcrossing levels between neighbouring crops. Since forager distribution can be difficult to measure directly, mathematical models are often used to predict the population density of dispersing foragers on the landscape. We model organism movement using a diffusion framework in which the foraging population is divided into two subpopulations engaged in intensive and extensive search modes respectively. Movement in the intensive search mode (ISM) is modeled by diffusion, and movement in the extensive search mode (ESM) is modeled by advection. We show that our model provides a superior fit to organism movement data than more traditional diffusion or diffusion-advection models in which the forager population is considered homogeneous. Our results have implications for the understanding of dispersal in a wide variety of applications.     

Do small animals have a biogeography?

It has been stated that small organisms do not have barriers for distribution and will not show biogeographic discreteness. General models for size-mediated biogeographies establish a transition region between ubiquitous dispersal and restricted biogeography at about 1–10 mm. We tested patterns of distribution versus size with water mites, a group of freshwater organisms with sizes between 300 μm and 10 mm.We compiled a list of all known water mite species for Sierra del Guadarrama (a mountain range in the centre of the Iberian Peninsula) from different authors and our own studies in the area. Recorded habitats include lotic, lentic and interstitial environments. Species body size and world distribution were drawn from our work and published specialized taxonomic literature. The null hypothesis was that distribution is size-independent. The relationship between distribution and size was approached via analysis of variance and between size and habitat via logistic regression. Contrary to expectations, there is no special relationship between water mite size and area size distribution. On the other hand, water mite size is differentially distributed among habitats, although this ecological sorting is very weak. Larger water mites are more common in lentic habitats and smaller water mites in lotic habitats. Size-dependent distribution in which small organisms tend to be cosmopolitan breaks down when the particular biology comes into play. Water mites do not fit a previously proposed size-dependent biogeographical distribution, and are in accordance with similar data published on Tardigrada, Rotifera, Gastrotricha and the like.     

THE EVOLUTION OF STRATEGY VARIATION: WILL AN ESS EVOLVE?

Evolutionarily stable strategy (ESS) models are widely viewed as predicting the strategy of an individual that when monomorphic or nearly so prevents a mutant with any other strategy from entering the population. In fact, the prediction of some of these models is ambiguous when the predicted strategy is "mixed", as in the case of a sex ratio, which may be regarded as a mixture of the subtraits "produce a daughter" and "produce a son." Some models predict only that such a mixture be manifested by the population as a whole, that is, as an "evolutionarily stable state"; consequently, strategy monomorphism or polymorphism is consistent with the prediction. The hawk-dove game and the sex-ratio game in a panmictic population are models that make such a "degenerate" prediction. We show here that the incorporation of population finiteness into degenerate models has effects for and against the evolution of a monomorphism (an ESS) that are of equal order in the population size, so that no one effect can be said to predominate. Therefore, we used Monte Carlo simulations to determine the probability that a finite population evolves to an ESS as opposed to a polymorphism. We show that the probability that an ESS will evolve is generally much less than has been reported and that this probability depends on the population size, the type of competition among individuals, and the number of and distribution of strategies in the initial population. We also demonstrate how the strength of natural selection on strategies can increase as population size decreases. This inverse dependency underscores the incorrectness of Fisher's and Wright's assumption that there is just one qualitative relationship between population size and the intensity of natural selection.     

Basing population genetic inferences and models of molecular evolution upon desired stationary distributions of DNA or protein sequences

Models of molecular evolution tend to be overly simplistic caricatures of biology that are prone to assigning high probabilities to biologically implausible DNA or protein sequences. Here, we explore how to construct time-reversible evolutionary models that yield stationary distributions of sequences that match given target distributions. By adopting comparatively realistic target distributions,evolutionary models can be improved. Instead of focusing on estimating parameters, we concentrate on the population genetic implications of these models. Specifically, we obtain estimates of the product of effective population size and relative fitness difference of alleles. The approach is illustrated with two applications to protein-coding DNA. In the first, a codon-based evolutionary model yields a stationary distribution of sequences, which, when the sequences are translated,matches a variable-length Markov model trained on human proteins. In the second, we introduce an insertion-deletion model that describes selectively neutral evolutionary changes to DNA. We then show how to modify the neutral model so that its stationary distribution at the amino acid level can match a profile hidden Markov model, such as the one associated with the Pfam database.     

Pairwise Comparisons of Mitochondrial DNA Sequences in Stable and Exponentially Growing Populations

We consider the distribution of pairwise sequence differences of mitochondrial DNA or of other nonrecombining portions of the genome in a population that has been of constant size and in a population that has been growing in size exponentially for a long time. We show that, in a population of constant size, the sample distribution of pairwise differences will typically deviate substantially from the geometric distribution expected, because the history of coalescent events in a single sample of genes imposes a substantial correlation on pairwise differences. Consequently, a goodness-of-fit test of observed pairwise differences to the geometric distribution, which assumes that each pairwise comparison is independent, is not a valid test of the hypothesis that the genes were sampled from a panmictic population of constant size. In an exponentially growing population in which the product of the current population size and the growth rate is substantially larger than one, our analytical and simulation results show that most coalescent events occur relatively early and in a restricted range of times. Hence, the "gene tree" will be nearly a "star phylogeny" and the distribution of pairwise differences will be nearly a Poisson distribution. In that case, it is possible to estimate r, the population growth rate, if the mutation rate, mu, and current population size, N0, are assumed known. The estimate of r is the solution to ri/mu = ln(N0r) - gamma, where i is the average pairwise difference and gamma approximately 0.577 is Euler's constant.     

Distance measures in terms of substitution processes.

Phylogenetic reconstruction from DNA or amino acid sequences relies heavily on suitable distance measures. A number of new distance measures (asynchronous, LogDet, and paralinear distances) which possess the desired property of tree additivity under fairly general models of sequence evolution have been proposed recently, but they are not well understood from a mechanistic point of view. We review them here in a unifying framework, which is the substitution process in continuous time. The emerging interpretation will also clarify the relationship among these distance measures. We also tackle situations with site-to-site variation of substitution rates which is well known to cause non-additive distances and inconsistent branch lengths. For homogeneous, stationary, time-reversible models, this may be repaired provided that the distribution of rates is known. In contrast, we will show that, for non-stationary models, different tree topologies may produce identical joint distributions of letters in pairs of sequences, given the same distribution of rates. This precludes the existence of any tree-additive pairwise distance measure.     

Stationary distribution of population size inTribolium

We propose the use of a stationary probability distribution for the analysis of data on population size. Predicting this long term population property from short term individual events is accomplished by the use of the asymptotic theory of stochastic processes. A WKB approximation to the stationary density is obtained and then applied to observations on the flour beetleTribolium.     

Spatial distribution of competing populations

We consider spatial distributions of two competing and diffusing populations whose habitats are partly overlapping. As a model, certain reaction—diffusion equations are used in the finite and in the infinite regions with Dirichlet boundary conditions. On the assumption of extremely different diffusive rates of the two species, it is verified, by the use of singular perturbation techniques, that the slowly diffusing species can survive in some subregions, although the species with the greater diffusive rate rapidly occupies the region at the initial stage, and that coexistence of two populations is realized, reducing the effect of interspecific competition by spatial segregation. It will be also shown that the size of the region where the slowly moving population can survive exhibits a markedly qualitative change, depending on the values of some parameters, and that the population can extends the distribution infinitely, when the parameters satisfy a certain condition.     

Experimental analysis and modelling of in vitro HUVECs proliferation in the presence of various types of drugs

Objectives: This study focuses on experimental analysis and corresponding mathematical simulation of in vitro HUVECs (human umbilical vein endothelial cells) proliferation in the presence of various types of drugs. Materials and methods: HUVECs, once seeded in Petri dishes, were expanded to confluence. Temporal profiles of total count obtained by classic haemocytometry and cell size distribution measured using an electronic Coulter counter, are quantitatively simulated by a suitable model based on the population balance approach. Influence of drugs on cell proliferation is also properly simulated by accounting for suitable kinetic equations. Results and discussion: The models’ parameters have been determined by comparison with experimental data related to cell population expansion and cell size distribution in the absence of drugs. Inhibition constant for each type of drug has been estimated by comparing the experimental data with model results concerning temporal profiles of total cell count. The reliability of the model and its predictive capability have been tested by simulating cell size distribution for experiments performed in the presence of drugs. The proposed model will be useful in interpreting effects of selected drugs on expansion of readily available human cells.     

The stationary distribution of allele frequencies when selection acts at unlinked loci

We consider population genetics models where selection acts at a set of unlinked loci. It is known that if the fitness of an individual is multiplicative across loci, then these loci are independent. We consider general selection models, but assume parent-independent mutation at each locus. For such a model, the joint stationary distribution of allele frequencies is proportional to the stationary distribution under neutrality multiplied by a known function of the mean fitness of the population. We further show how knowledge of this stationary distribution enables direct simulation of the genealogy of a sample at a single-locus. For a specific selection model appropriate for complex disease genes, we use simulation to determine what features of the genealogy differ between our general selection model and a multiplicative model.     

Convergence time to the Ewens sampling formula in the infinite alleles Moran model

In this paper, we establish an upper bound for time to convergence to stationarity for the discrete time infinite alleles Moran model. If M is the population size and μ is the mutation rate, this bound gives a cutoff time of log(M μ)/μ generations. The stationary distribution for this process in the case of sampling without replacement is the Ewens sampling formula. We show that the bound for the total variation distance from the generation t distribution to the Ewens sampling formula is well approximated by one of the extreme value distributions, namely, a standard Gumbel distribution. Beginning with the card shuffling examples of Aldous and Diaconis and extending the ideas of Donnelly and Rodrigues for the two allele model, this model adds to the list of Markov chains that show evidence for the cutoff phenomenon. Because of the broad use of infinite alleles models, this cutoff sets the time scale of applicability for statistical tests based on the Ewens sampling formula and other tests of neutrality in a number of population genetic studies.     

On the linear birth and death processes of biology as Markoff chains

Stochastic Markoff models for the linear birth and death population growth processes of biology are constructed using the Q-matrix method of Doob. The relationship of the stochastic theory to the classical deterministic foundations of these processes is stressed by showing in detail how the classical postulates are mathematically transformed via the Q-matrix elements into the basis for a stationary Markoff process with continuous time parameter and denumerably many “populations states.” It is shown that the resulting stochastic models predict that the population size will fluctuate about the deterministic time curve, the extent of fluctuation being measured by the variance functions. General formulas covering all possible transitions from one population size to another are derived.     

The effect of static and dynamic spatially structured disturbances on a locally dispersing population

Previous models of locally dispersing populations have shown that in the presence of spatially structured fixed habitat heterogeneity, increasing local spatial autocorrelation in habitat generally has a beneficial effect on such populations, increasing equilibrium population density. It has also been shown that with large-scale disturbance events which simultaneously affect contiguous blocks of sites, increasing spatial autocorrelation in the disturbances has a harmful effect, decreasing equilibrium population density. Here, spatial population models are developed which include both of these spatially structured exogenous influences, to determine how they interact with each other and with the endogenously generated spatial structure produced by the population dynamics. The models show that when habitat is fragmented and disturbance occurs at large spatial scales, the population cannot persist no matter how large its birth rate, an effect not seen in previous simpler models of this type. The behavior of the model is also explored when the local autocorrelation of habitat heterogeneity and disturbance events are equal, i.e. the two effects occur at the same spatial scale. When this scale parameter is very small, habitat fragmentation prevents the population from persisting because sites attempting to reproduce will drop most of their offspring on unsuitable sites; when the parameter is very large, large-scale disturbance events drive the population to extinction. Population levels reach their maximum at intermediate values of the scale parameter, and the critical values in the model show that the population will persist most easily at these intermediate scales of spatial influences. The models are investigated via spatially explicit stochastic simulations, traditional (infinite-dispersal) and improved (local-dispersal) mean-field approximations, and pair approximations.     

The Effect of an Upper Limit to Population Size on Persistence Time

For a population with density-independent vital rates in a randomly varying environment, previous authors have calculated the probability that population size will first drop to some specified (arbitrary) low level at a given time (the first passage time distribution (FPTD), which may be interpreted as a distribution of extinction times). In this paper, we study the FPTD For a stochastic model of density-independent population growth which includes a hard upper limit to population size. We discuss the conditions under which this distribution may be approximated by the FPTD of a Wiener process with a reflecting boundary condition, for which an exact calculation is presented in an appendix. We compare the FPTD of the new model with its counterpart in the model without an upper limit. The most important effects of introducing the upper limit are: (a) ultimate extinction becomes certain; (b) if the long run growth rate in the absence of the upper boundary was small but positive, extinction within ecologically significant times is likely; (c) for larger values of the long run growth rate, persistence over ecologically significant times is almost certain. We discuss the implications of result (b) for conservation. Result (c) establishes that "density-vague" regulation can produce persistent, but bounded, populations.