Integral projection models for finite populations in a stochastic environment

Continuous types of population structure occur when continuous variables such as body size or habitat quality affect the vital parameters of individuals. These structures can give rise to complex population dynamics and interact with environmental conditions. Here we present a model for continuously structured populations with finite size, including both demographic and environmental stochasticity in the dynamics. Using recent methods developed for discrete age-structured models we derive the demographic and environmental variance of the population growth as functions of a continuous state variable. These two parameters, together with the expected population growth rate, are used to define a one-dimensional diffusion approximation of the population dynamics. Thus, a substantial reduction in complexity is achieved as the dynamics of the complex structured model can be described by only three population parameters. We provide methods for numerical calculation of the model parameters and demonstrate the accuracy of the diffusion approximation by computer simulation of specific examples. The general modeling framework makes it possible to analyze and predict future dynamics and extinction risk of populations with various types of structure, and to explore consequences of changes in demography caused by, e.g., climate change or different management decisions. Our results are especially relevant for small populations that are often of conservation concern.     

Collective decisions in ants when foraging under crowded conditions

In this paper we examine the effect of crowding on the selection of a path in the mass-recruiting ant Lasius niger. In our experiment, ants had to go from their nest to a food source by crossing a diamond-shaped bridge, giving the choice between two paths. Two types of bridges were used: the first had two branches of equal length but different width while the second had two branches of different length and width. Experiments at high traffic volume always ended up with the selection of the wider branch, even if it was longer. This result shows that overcrowding on the narrow branch plays an essential role in the mechanism underlying the choice of route in ants. A mathematical model was developed to evaluate the importance of two mechanisms that could account for this result. The first is based on the difference in travel duration between the two paths. The second is based on the repulsive interactions between workers making head-on encounters. The model shows that travel duration per se is not sufficient to explain path choice. Rather, it is the interplay between trail following behaviour and repulsive interactions that allows ants to choose the path that minimizes their travel time. When choosing a path ants thus prefer to trade time against energy. Our results demonstrate that any environmental constraint that alters the dynamics of trail recruitment can lead to the emergence of adaptive foraging decisions without any explicit coding of information by the foragers at the individual level.     

Zum Stichprobenfehler im Human-Biomonitoring

Goal and Scope

Human biomonitoring determines the concentration of xenobiotics in populations by means of smaller samples, thus necessarily arising sampling errors. These are determined.

Methods

For a fictitious population of 200,000 persons, differently broad xenobiotic concentration distributions were simulated. Samples of varying size were randomly drawn and the sampling error, defined as the proportional difference between the geometric means of sample and population, was determined.

Results and Conclusions

The sampling error depends on the sample size and the width of the concentration distribution; its estimation is possible for any xenobiotic, given it has lognormal distribution, and the sample size is between 10 and 50,000. For its estimation an equation was derived.

Outlook

When presenting and interpreting results of human biomonitoring, the sampling error must be considered, together with the uncertainty of the measurement.     

Efficient implementation of the Metropolis-Hastings algorithm,with application to the Cormack–Jolly–Seber model

Judicious choice of candidate generating distributions improves efficiency of the Metropolis-Hastings algorithm. In Bayesian applications, it is sometimes possible to identify an approximation to the target posterior distribution; this approximate posterior distribution is a good choice for candidate generation. These observations are applied to analysis of the Cormack–Jolly–Seber model and its extensions.     

Predicting functional response and size selectivity of juvenile Notonecta maculata foraging on Daphnia magna

A mechanistic model was developed to assess the impact of predation of juvenile Notonecta maculata on size structured Daphnia magna populations and to provide a framework for quantifying the backswimmers uptake of food. Results of experiments and model predictions clearly demonstrate selective predation of backswimmers when fed with a choice of daphnid size classes, with patterns of selectivity differing across N. maculata instars. The model describes the foraging process empirically on the base of a general predation cycle including four conditional events instead of using classic functional response curves. For model parameterisation components of predation, namely probability of encounter, attack and success as well as time spent on handling prey was directly observed by means of video tracking experiments. Since attack rate, capture success and handling time appeared to be a function of prey size differing between Notonecta instars, the model takes into account ontogenic changes in both predator and prey characteristics. Independent data of functional response and size selectivity experiments were used for model validation and proved the model outcome to be consistent with observations.     

Predicting reproduction rate of varroa

A single equation is derived to predict population-density effects on the reproduction rate of the honey bee parasite Varroa destructor Anderson and Trueman. This equation provides a simpler alternative to the approach currently used in the literature, and additionally corrects an anomaly in that approach. The method is then extended to the case of co-existing haplotypes of Varroa. It thus derives an equation used without proof for modelling biocontrol of Varroa, and examines the error caused by an approximation necessary for a closed form solution. Additionally a varroa population model incorporating the derived equation is described.     

Nonlinear relationships between vital rates and state variables in demographic models

To accurately estimate population dynamics and viability, structured population models account for among-individual differences in demographic parameters that are related to individual state. In the widely used matrix models, such differences are incorporated in terms of discrete state categories, whereas integral projection models (IPMs) use continuous state variables to avoid artificial classes. In IPMs, and sometimes also in matrix models, parameterization is based on regressions that do not always model nonlinear relationships between demographic parameters and state variables. We stress the importance of testing for nonlinearity and propose using restricted cubic splines in order to allow for a wide variety of relationships in regressions and demographic models. For the plant Borderea pyrenaica, we found that vital rate relationships with size and age were nonlinear and that the parameterization method had large effects on predicted population growth rates, X (linear IPM, 0.95; nonlinear IPMs, 1.00; matrix model, 0.96). Our results suggest that restricted cubic spline models are more reliable than linear or polynomial models. Because even weak nonlinearity in relationships between vital rates and state variables can have large effects on model predictions, we suggest that restricted cubic regression splines should be considered for parameterizing models of population dynamics whenever linearity cannot be assumed.     

Simulating regeneration and vegetation dynamics in Mediterranean coniferous forests

This study aims to provide a quantitative framework to model the dynamics of Mediterranean coniferous forests by integrating existing ecological data within a generic mathematical simulator. We developed an individual-based vegetation dynamics model, constrained on long-term field regeneration data, analyses of tree-rings and seed germination experiments. The simulator implements an asymmetric competition algorithm which is based on the location and size of each individual. Growth is parameterized through the analysis of tree-rings from more than thirty individuals of each of the three species of interest. A super-individual approach is implemented to simulate regeneration dynamics, constrained with available regeneration data across time-since-disturbance and light-availability gradients. The study concerns an insular population of an endemic to Greece Mediterranean fir (Abies cephalonica Loudon) on the island of Cephalonia (Ionian Sea) and two interacting populations of a Mediterranean pine (Pinus brutia Ten.) and a more temperate-oriented pine (Pinus nigra Arn. ssp. pallasiana) on the island of Lesbos (NE Aegean Sea), Greece. The model was validated against plot-level observations in terms of species standing biomass and regeneration vigour and adequately captured regeneration patterns and overall vegetation dynamics in both study sites. The potential effects of changing climatic patterns on the regeneration dynamics of the three species of interest were subsequently explored. With the assumption that a warmer future would probably cause changes in the duration of cold days, we tested how this change would affect the overall dynamics of the study sites, by focusing on the process of cold stratification upon seed germination. Following scenarios of a warmer future and under the current model parameterization, changes in the overall regeneration vigour controlled by a reduction in the amount of cold days, did not alter the overall dynamics in all plant populations studied. No changes were identified in the relative dominance of the interacting pine populations on Lesbos, while the observed reduction in the amount of emerging seedlings of A. cephalonica on Cephalonia did not affect biomass yield at later stages of stand development.     

Effectiveness of Corridors Relative to Enlargement of Habitat Patches

Abstract:  The establishment of biological corridors between two otherwise isolated habitat patches is a common yet contentious strategy for conserving populations in fragmented landscapes. We compared the effectiveness of corridors with the effectiveness of an alternate conservation strategy, the enlargement of existing habitat patches. We used a spatially explicit population model that simulated population size in two kinds of patches. One patch had a corridor that connected it to a larger "source" patch and the other patch was unconnected and enlarged at the periphery by an area the same size as the corridor. Patch isolation, corridor width, patch size, and the probability that individuals would cross the border from habitat to matrix were varied independently. In general, population size was greater in enlarged patches than in connected patches when patches were relatively large and isolated. Corridor width and the probability of crossing the border from habitat to matrix did not affect the relative benefit of corridors versus patch enlargement. Although biological corridors may mitigate potential effects of inbreeding depression at long time scales, our results suggest that they are not always the best method of conserving fragmented populations.     

Adsorption and partition of benzo(a)pyrene on sediments with different particle sizes from the Yellow River

Experiments have been carried out to study the sorption of Benzo(a)pyrene(Bap) on sediment particles from the Yellow River using a batch equilibration technique. Effects of particle size on the adsorption and partition of Bap were investigated with the particle content of 3 g/L. Several significant results were obtained from the study. (1) Isotherms of Bap could be fitted with the dual adsorption-partition model under different particle sizes, and the measured value of the adsorption and partition was in agreement with the theoretical value of the dual adsorption-partition model. (2) When the particle diameter was d ? 0.025 mm, the adsorption was predominant in the sorption of Bap, which accounted for 68.7%–82.4% of the sorption. For the particles with the size of 0.007 mm?d<0.025 mm, the adsorption was predominant when the equilibrium concentration of Bap was 0–8.87 μg/L in the water phase; and the partition was predominant when the equilibrium concentration of Bap was higher than 8.87 μg/L in the water phase. When the particle diameter was d<0.007 mm, the partition was predominant. (3) On the point of particle size, the contribution of adsorption to sorption followed the order: “d?0.025 mm”>“0.007 mm ?d<0.025 mm” >“d<0.007 mm”. (4) The partition coefficients of Bap in solids with different particle sizes were linearly correlated with the organic content, and the K _oc of Bap was about 1.26 × 10⁵ (L/kg).     

Assessing the parameterisation of the settling flux in a depth-integrated model of the fate of decaying and sinking particles, with application to fecal bacteria in the Scheldt Estuary

The fate of reactive tracers is often modelled by depth-averaged equations. When integrating the depth-resolved equations, it appears that the term describing the settling of particles is dependent on the concentration just above the bottom. Because in a depth-averaged framework this quantity is not available, the settling term needs to be parameterised. The most natural choice is to make the settling flux dependent on the average concentration. This approximation is acceptable if the water column is well mixed, but these conditions are not necessarily met in real applications. Therefore, this study aims at assessing and understanding the error made by using a depth-averaged model in a range of realistic conditions. For the definition of these conditions, typical values for the Scheldt Estuary and the Dutch-Belgian coast were taken. The realistic inspiration for the reactive tracer in this study is the fecal bacterium Escherichia coli, whose own dynamics are characterised by settling and gradual decay by mortality. In an attempt to understand the relative importance of several factors like settling, mortality, mixing and stratification on the error made by a depth-averaged approach, a number of simplified test cases were investigated. It follows that, as expected, the error is acceptable if the situation is mixing-dominated. However, the effect of mortality and stratification was less obvious in advance. For instance, it appeared that errors can also be significant if settling and mortality have the same characteristic timescales. Stratification often has the effect to increase the error made by the depth-averaged model.     

Predicting [corrected] extinction risk in spite of predator-prey oscillations.

Most population viability analyses (PVA) assume that the effects of species interactions are subsumed by population-level parameters. We examine how robust five commonly used PVA models are to violations of this assumption. We develop a stochastic, stage-structured predator-prey model and simulate prey population vital rates and abundance. We then use simulated data to parameterize and estimate risk for three demographic models (static projection matrix, stochastic projection matrix, stochastic vital rate matrix) and two time series models (diffusion approximation [DA], corrupted diffusion approximation [CDA]). Model bias is measured as the absolute deviation between estimated and observed quasi-extinction risk. Our results highlight three generalities about the application of single-species models to multi-species conservation problems. First, our collective model results suggest that most single-species PVA models overestimate extinction risk when species interactions cause periodic variation in abundance. Second, the DA model produces the most (conservatively) biased risk forecasts. Finally, the CDA model is the most robust PVA to population cycles caused by species interactions. CDA models produce virtually unbiased and relatively precise risk estimates even when populations cycle strongly. High performance of simple time series models like the CDA owes to their ability to effectively partition stochastic and deterministic sources of variation in population abundance.     

Increasing accuracy of dispersal kernels in grid-based population models

Dispersal kernels in grid-based population models specify the proportion, distance and direction of movements within the model landscape. Spatial errors in dispersal kernels can have large compounding effects on model accuracy. Circular Gaussian and Laplacian dispersal kernels at a range of spatial resolutions were investigated, and methods for minimizing errors caused by the discretizing process were explored. Kernels of progressively smaller sizes relative to the landscape grid size were calculated using cell-integration and cell-center methods. These kernels were convolved repeatedly, and the final distribution was compared with a reference analytical solution. For large Gaussian kernels (σ > 10 cells), the total kernel error was <10⁻¹¹ compared to analytical results. Using an invasion model that tracked the time a population took to reach a defined goal, the discrete model results were comparable to the analytical reference. With Gaussian kernels that had σ ≤ 0.12 using the cell integration method, or σ ≤ 0.22 using the cell center method, the kernel error was greater than 10%, which resulted in invasion times that were orders of magnitude different than theoretical results. A goal-seeking routine was developed to adjust the kernels to minimize overall error. With this, corrections for small kernels were found that decreased overall kernel error to <10⁻¹¹ and invasion time error to <5%.     

Application of a Random Forest algorithm to predict spatial distribution of the potential yield of Ruditapes philippinarum in the Venice lagoon, Italy

We present a modelling framework that combines machine learning techniques and Geographic Information Systems to support the management of an important aquaculture species, Manila clam (Ruditapes philippinarum). We use the Venice lagoon (Italy), the first site in Europe for the production of R. philippinarum, to illustrate the potential of this modelling approach. To investigate the relationship between the yield of R. philippinarum and a set of environmental factors, we used a Random Forest (RF) algorithm. The RF model was tuned with a large data set (n = 1698) and validated by an independent data set (n = 841). Overall, the model provided good predictions of site-specific yields and the analysis of marginal effect of predictors showed substantial agreement among the modelled responses and available ecological knowledge for R. philippinarum. The most influent environmental factors for yield estimation were percentage of sand in the sediment, salinity, and water depth. Our results agree with findings from other North Adriatic lagoons. The application of the fitted RF model to continuous maps of all the environmental variables allowed estimates of the potential yield for the whole basin. Such a spatial representation enabled site-specific estimates of yield in different farming areas within the lagoon. We present a possible management application of our model by estimating the potential yield under the current farming distribution and comparing it to a proposed re-organization of the farming areas. Our analysis suggests a reduction of total yield is likely to result from the proposed re-organization.     

An EM algorithm for capture-recapture estimation

Analysis of capture—recapture data often involves maximizing a complex likelihood function with many unknown parameters. Statistical inference based on selection of a proper model depends on successful attainment of this maximum. An EM algorithm is developed for obtaining maximum likelihood estimates of capture and survival probabilities conditional on first capture from standard capture—recapture data. The algorithm does not require the use of numerical derivatives which may improve precision and stability relative to other estimation schemes. The asymptotic covariance matrix of the estimated parameters can be obtained using the supplemented EM algorithm. The EM algorithm is compared to a more traditional Newton-Raphson algorithm with both a simulated and a real dataset. The two algorithms result in the same parameter estimates, but Newton-Raphson variance estimates depend on a numerically estimated Hessian matrix that is sensitive to step size choice.     

Projecting population trends of endangered amphibian species in the face of uncertainty: A pattern-oriented approach

Amphibian populations have been declining worldwide for the last three decades. Determining the risk of extinction is one of the major goals of amphibian conservation, yet few quantitative models have been developed for amphibian populations. Like most rare or threatened populations, there is a paucity of life history data available for most amphibian populations. Data on the critical juvenile life stage are particularly lacking. Pattern oriented modeling (POM) has been used successfully to estimate life history parameters indirectly when critical data lacking, but has not been applied to amphibian populations. We describe a spatially explicit, individual-based, stochastic simulation model developed to project population dynamics of pond-breeding amphibian populations. We parameterized the model with life history and habitat data collected for the endangered Houston toad (Bufohoustonensis), a species for which there is a high degree of uncertainty for juvenile and adult male survival. During model evaluation, we focused on explicitly reducing this uncertainty, evaluating 16 different versions of the model that represented the range of parametric uncertainty for juvenile and adult male survival. Following POM protocol, we compared simulation results to four population-level patterns observed in the field: population size, adult sex ratio, proportion of toads returning to their natal pond, and mean maximum distance moved. Based on these comparisons, we rejected 11 of the 16 model versions. Results of the remaining versions confirmed that population persistence depends heavily on juvenile survival, and further suggested that probability of juvenile survival is likely between 0.0075 and 0.015 (previous estimates ranged from 0.003 to 0.02), and that annual male survival is near 0.15 (previous estimates ranged up to 0.43).     

On the use of growth rate parameters for projecting population sizes: Application to aphids

This paper extends the application of the cumulative size based mechanistic model, which has previously been shown to describe diverse aphid population size data well. The mechanistic model is reviewed with a focus on the explanatory role of the birth and death rate formulation. An analysis of two data sets, one on the mustard aphid and the other on the pecan aphid, indicates that multiple linear regression equations based on the estimated birth and death rate parameters alone account for nearly all (R² > 0.95) of the variability in two key population attributes, namely the peak count and the cumulative density. This indicates that population size variables may be projected directly from the growth rate parameters using linear equations. Such linear relationships based on the birth and death rate parameters are shown to hold also for certain generalized mechanistic models for which the analytical solution is not available. The birth and death rate coefficients, therefore, constitute a new succinct set of variables that could be included in the predictive modeling of aphid populations, as well as other insect and animal populations with local collapse which follow similar growth dynamics.     

Simulating population variation and movement within fragmented landscapes: An application to the gopher tortoise (Gopherus polyphemus)

As the human activity footprint grows, land-use decisions play an increasing role in determining the future of plant and animal species. Studies have shown that urban and agricultural development cannot only harm species populations directly through habitat destruction, but also by destroying the corridors that connect habitat patches and populations within a metapopulation. Without these pathways, populations can encounter inbreeding depression and degeneration, which can increase death rates and lower rates of reproduction. This article describes the development and application of the FRAGGLE model, a spatial system dynamics model designed to calculate connectivity indices among populations. FRAGGLE can help planners and managers identify the relative contribution of populations associated with habitat patches to future populations in those patches, taking into account the importance of interstitial land to migration success. The model is applied to the gopher tortoise (Gopherus polyphemus), a threatened species whose southeastern U.S. distribution has diminished significantly within its native range due to agricultural and urban development over the last several decades. This model is parameterized with life history and movement traits of the gopher tortoise in order to simulate population demographics and spatial distribution within an area in west-central Georgia that supports a significant tortoise population. The implications of this simulation modeling effort are demonstrated using simple landscape representations and a hypothetical on land-use management scenario. Our findings show that development resulting in even limited habitat losses (10%) may lead to significant increases in fragmentation as measured by a loss in the rate of dispersions (31%) among area subpopulations.     

Integrating farming techniques in an ecological matrix model: Implementation on the primrose (Primula vulgaris)

Several studies have proven the importance of field margins in sustaining biodiversity and other work has been done on the effect of field management on field margin flora. However few models have been built to predict the effects of field management on the flora. Our project addresses this need for a model capable of predicting the effect of cropping techniques and their timing on the flora of field margins. Primula vulgaris is a biodiversity indicator, characteristic of undisturbed flora and found in field margins and woodlands: its population has been declining for several years. We created a temporal matrix model of P. vulgaris populations on field margins, taking into account the effects of field, field margin and roadside management based on literature and expert knowledge. We then analysed its sensitivity to demographic parameters by comparing lambda (growth rate) sensitivity and elasticity. We compared the management parameter effect using the relative growth rate of the population after 6 years of simulation. Sensitivity analysis to biological parameters showed the importance of adult survival and seed production and germination. Results show that P. vulgaris is particularly sensitive to broad-spectrum herbicides and that other management techniques like early mowing, scything and scrub-killer (diluted broad-spectrum herbicide or specific herbicide) are less aggressive. Our simulations show that management of cash crops in Brittany is too aggressive for P. vulgaris populations and that 4-5 years of grassland in the adjacent field are necessary to maintain populations.     

Mathematical modelling for conservation and management of gorgonians corals: youngs and olds,could they coexist?

Gorgonian corals are long-lived, slow-growing marine species dominating Mediterranean rocky bottoms. Endowed with complex morphologies they give a structure to the whole community, moreover, being efficient suspension feeders, they play a key role in plankton-benthos energy flow and CO₂${\text{CO}}_2$ storage. Thus, the structure and the development of benthic, hard bottom communities are linked to gorgonian survival. The red coral Corallium rubrum (L. 1758) is a precious gorgonian endemic to the Mediterranean Sea. Harvested and traded world-wide since ancient times red coral is a clear example of overexploited marine resource. This species is structured into self-seeding, genetically differentiated populations, some of which, living in the shallower part of the species bathymetric distribution, was recently affected by anomalous mortality events linked to global climate change. The co-occurrence of overharvesting and mass mortality could dramatically affect such populations. Demographic population models, widely applied by conservation biologists to check population viability and to project population trends over time are fundamental to foster survival of such populations matching harvesting to population growth rates. Therefore we set out a dynamic model of a genetically differentiated red coral population living in shallow waters. This population is characterised by small/young, crowded colonies and high recruitment rate. On the basis of the size–age structure determined for this population, a static life-history table, in which survival and reproduction coefficients of the different size–age classes were reported, has been set out. Demographic data were included in a non-linear, discrete, age-structured dynamic model, based on a Leslie-Lewis transition matrix. Our field data indicate that the recruits-to-larvae ratio is actually density-dependent. Such dependence, positive for low and negative for high density values, was included into the model and the effect of colonies of different size–age classes on recruits-to-larvae ratio was considered to be proportional to the number of polyps they have. We applied such model to simulate the trends of the studied population under different increases of survival and life-span. As some populations of gorgonians actually show the dominance of sparse, big/old colonies and low recruitment rate, while others are characterised by crowded, small/young colonies and high recruitment rate, we simulated the shift from the former to the latter structure increasing survival and life-span. Our results suggest that a dramatic mortality increase of bigger–older colonies (due, in the case of red coral to overfishing) could have determined the population structure we found.