Combining state and transition models with dynamic Bayesian networks

Bashari et al. (2009) propose combining state and transition models (STMs) with Bayesian networks for decision support tools where the focus is on modelling the system dynamics. There is already an extension of Bayesian networks - so-called dynamic Bayesian networks (DBNs) - for explicitly modelling systems that change over time, that has also been applied in ecological modelling. In this paper we propose a combination of STMs and DBNs that overcome some of the limitations of Bashari et al.’s approach including providing an explicit representation of the next state, while retaining its advantages, such an the explicit representation of transitions. We then show that the new model can be applied iteratively to predict into the future consistently with different time frames. We use Bashari et al.’s rangeland management problem as an illustrative case study. We present a comparative complexity analysis of the different approaches, based on the structure inherent in the problem being modelled. This analysis showed that any models that explicitly represent all the transitions only remain tractable when there are natural constraints in the domain. Thus we recommend modellers should analyse these aspects of their problem before deciding whether to use the framework.     

Modelling the influence of demographic parameters on group structure in social species with dispersal asymmetry and group fission

Female philopatry characterizes many mammal populations subdivided into social groups. Fission of these social groups is a relatively discrete event in the life of groups or of individuals, leading to the distribution of females among several newly formed groups. Fission is an important event because it can be a way for females to disperse. Group fissions have rarely been observed and their modalities generally remain poorly known, the best-documented species being primates. Most group fissions occur along lines of maternal relatedness, but the death of a matriarch may disrupt the cohesion within a matriline, inducing separation of sisters, accompanied by their descendants, when a group splits. Our model shows that the numbers and sizes of matrilines within groups depend on the precise demographic parameters and age structure of a population and not only on its rate of increase. For comparable population-growth periods, high survival rates of adult females induce an increase in the sizes of matrilines, whereas high survival rates of immature individuals induce an increase in the numbers of matrilines. Following fission, groups of a given size included, in the first case, only a few large matrilines, whereas in the second case, they consisted mainly of many small matrilines. The present study constitutes a preliminary stage, before modelling consequences of demographic structure of groups or populations on their genetic structure.     

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.     

Persistence of Different-sized Populations: An Empirical Assessment of Rapid Extinctions in Bighorn Sheep

Abstract: Theory and simulation models suggest that small populations are more susceptible to extinction than large populations, yet assessment of this idea has been hampered by lack of an empirical base. I address the problem by asking how long different-sized populations persist and present demographic and weather data spanning up to 70 years for 122 bighorn sheep ( Ovis canadensis ) populations in southwestern North America Analyses reveal that: (1) 100 percent of the populations with fewer than 50 individuals went extinct within 50 years; (2) populations with greater than 100 individuals persisted for up to 70 years; and (3) the rapid loss of populations was not likely to be caused by food shortages, severe weather, predation, or interspecific competition These data suggest that population size is a marker of persistence trajectories and they indicate that local extinction cannot be overcome because 50 individuals, even in the short term, are not a minimum viable population size for bighorn sheep.     

Identifying fishing trip behaviour and estimating fishing effort from VMS data using Bayesian Hidden Markov Models

Recent advances in technologies have lead to a vast influx of data on movements, based on discrete recorded position of animals or fishing boats, opening new horizons for future analyses. However, most of the potential interest of tracking data depends on the ability to develop suitable modelling strategies to analyze trajectories from discrete recorded positions. A serious modelling challenge is to infer the evolution of the true position and the associated spatio-temporal distribution of behavioural states using discrete, error-prone and incomplete observations. In this paper, a Bayesian Hierarchical Model (HBM) using Hidden Markov Process (HMP) is proposed as a template for analyzing fishing boats trajectories based on data available from satellite-based vessel monitoring systems (VMS). The analysis seeks to enhance the definition of the fishing pressure exerted on fish stocks, by discriminating between the different behavioural states of a fishing trip, and also by quantifying the relative importance of each of these states during a fishing trip. The HBM approach is tested to analyse the behaviour of pelagic trawlers in the Bay of Biscay. A hidden Markov chain with a regular discrete time step is used to model transitions between successive behavioural states (e.g., fishing, steaming, stopping (at Port or at sea)) of each vessel. The parameters of the movement process (speed and turning angles) are defined conditionally upon the behavioural states. Bayesian methods are used to integrate the available data (typically VMS position recorded at discrete time) and to draw inferences on any unknown parameters of the model. The model is first tested on simulated data with different parameters structures. Results provide insights on the potential of HBM with HMP to analyze VMS data. They show that if VMS positions are recorded synchronously with the instants at which the process switch from one behavioural state to another, the estimation method provides unbiased and precise inferences on behavioural states and on associated movement parameters. However, if the observations are not gathered with a sufficiently high frequency, the performance of the estimation method could be drastically impacted when the discrete observations are not synchronous with the switching instants. The model is then applied to real pathways to estimate variables of interest such as the number of operations per trip, time and distance spent fishing or travelling.     

Evaluation d'une population planctonique

In order to evaluate quantitative aspects of dynamics in planetonic populations, a method is presented permitting the more accurate estimation of the number of individuals within, each size-group. The samples analysed may have been caught with any type of collecting gear. Correcting coefficients are determined only once for each species studied and gear employed from suitable material, and may then be used for any other sample concerned with the same species and gear. The method is based on the fact that, in monospecific populations with all generations present, the number of individuals is always lowest in the oldest age group. A model calculation is presented for euphausiid crustaceans caught with a 10 ft Isaacs-Kidd midwater trawl.     

Chaos and pattern formation in a spatial tritrophic food chain

The model of Hastings and Powell describes a tritrophic food chain that exhibits chaotic dynamics. The model assumes that the populations are homogeneously mixed, so that the probability that any two individuals interact is uniform and space can be ignored. In this paper we propose a spatial version of the Hastings and Powell model in which predators seek their preys only in a finite neighborhood of their home location, breaking the mixing hypothesis. Treating both space and time as discrete variables we derive a set of coupled equations that describe the evolution of the populations at each site of the spatial domain. We show that the introduction of local predator–prey interactions result in qualitatively distinct dynamics of predator and prey populations. The evolution equations for the predators involve averages over the local density of preys, whereas the equations for the preys involve double averages, where the local density of both preys and predators appear. Our numerical simulations show that local predation also leads to spontaneous pattern formation and to qualitative changes in the global dynamics of the system. In particular, depending on the size of the predation neighborhoods, the chaotic strange attractor present in the original model of Hastings and Powell can be replaced by a stable fixed point or by an attractor of simpler topology.     

Who should wear mask against airborne infections? Altering the contact network for controlling the spread of contagious diseases

There are several ways of controlling the propagation of a contagious disease. For instance, to reduce the spreading of an airborne infection, individuals can be encouraged to remain in their homes and/or to wear face masks outside their domiciles. However, when a limited amount of masks is available, who should use them: the susceptible subjects, the infective persons or both populations? Here we employ susceptible-infective-recovered (SIR) models described in terms of ordinary differential equations and probabilistic cellular automata in order to investigate how the deletion of links in the random complex network representing the social contacts among individuals affects the dynamics of a contagious disease. The inspiration for this study comes from recent discussions about the impact of measures usually recommended by health public organizations for preventing the propagation of the swine influenza A (H1N1) virus. Our answer to this question can be valid for other eco-epidemiological systems.     

Extinction Rate Estimates for Plant Populations in Revisitation Studies: Importance of Detectability

Abstract:  Many researchers have obtained extinction-rate estimates for plant populations by comparing historical and current records of occurrence. A population that is no longer found is assumed to have gone extinct. Extinction can then be related to characteristics of these populations, such as habitat type, size, or species, to test ideas about what factors may affect extinction. Such studies neglect the fact that a population may be overlooked, however, which may bias estimates of extinction rates upward. In addition, if populations are unequally detectable across groups to be compared, such as habitat type or population size, comparisons become distorted to an unknown degree. To illustrate the problem, I simulated two data sets, assuming a constant extinction rate, in which populations occurred in different habitats or habitats of different size and these factors affected their detectability. The conventional analysis implicitly assumed that detectability equalled 1 and used logistic regression to estimate extinction rates. It wrongly identified habitat and population size as factors affecting extinction risk. In contrast, with capture-recapture methods, unbiased estimates of extinction rates were recovered. I argue that capture-recapture methods should be considered more often in estimations of demographic parameters in plant populations and communities.     

Super-individuals a simple solution for modelling large populations on an individual basis

Modelling populations on an individual-by-individual basis has proven to be a fruitful approach. Many complex patterns that are observed on the population level have been shown to arise from simple interactions between individuals. However, a major problem with these models is that the typically large number of individuals needed requires impractically large computation times. The common solution, reduction of the number of individuals in the model, can lead to loss of variation, irregular dynamics, and large sensitivity to the value of random generator seeds. As a solution to these problems, we propose to add an extra variable feature to each model individual, namely the number of real individuals it actually represents. This approach allows zooming from a real individual-by-individual model to a cohort representation or ultimately an all-animals-are-equal view without changing the model formulation. Therefore, the super-individual concept offers easy possibilities to check whether the observed behaviour is an artifact of following a limited number of individuals or of lumping individuals, and also to verify whether individual variability is indeed an essential ingredient for the observed behaviour. In addition the approach offers arbitrarily large computational advantages. As an example the super-individual approach is applied to a generic model of the dynamics of a size-distributed consumer cohort as well as to an elaborate applied simulation model of the recruitment of striped bass.     

Area-Based Refinement for Selection of Reserve Sites with the Benefit-Function Approach

Abstract:  Optimization of resource use is necessary for efficient conservation planning. Many reserve-selection algorithms aim to identify representative but inexpensive networks, which may lead to selecting small sites due to their lower costs and collectively higher species richness. Nevertheless, larger sites would be preferable regarding species' long-term persistence. An area-based refinement can be used to overcome this problem. We used a reserve-planning framework in which continuous benefit functions valued representation (numbers of populations), and differential species weights were based on a species' local rarity and threatened status. We introduced a refinement based on the species-area relationship that provides relatively higher values for larger sites. We applied the proposed method to rich fen vegetation in southern Finland. The species-area refinement resulted in a network of significantly larger sites with minor trade-offs with representation (numbers of populations). Giving endangered species higher weights ensured that the trade-off occurred mostly between site size and representation of low-priority species. We recommend using a species-area refinement for practical, maximum-coverage conservation planning.     

Modeling the Effects of Habitat Fragmentation on Source and Sink Demography of Neotropical Migrant Birds

Many songbird populations in the midwestem United States are structured as a network of sources and sinks that are linked by dispersal. We used a modeling approach to examine explicitly how populations respond to incremental fragmentation of source habitat and how this response may vary depending upon two life-history attributes: fidelity to natal habitat type and reproductive strength of the source. Fragmentation of source habitat led to a predictable decline in population for both attributes examined, but the manner in which populations declined varied depending upon the reproductive strength of the source and the level of fidelity. When the source was weak and produced few excess individuals, fragmentation of source habitats resulted in a predictable and parallel population decline of adults in both the source and the sink. In this situation high fidelity to natal habitats was important for maintenance of population size and structure. Low fidelity to weak sources resulted in population extinction; populations experienced a demographic cost by dispersing from high quality source habitat to low quality sink habitat. In contrast, when the source was strong and produced many excess individuals, fragmentation of the source led to population declines in both the source and the sink, but this decline was more abrupt in sink habitats. When the source was strong and produced a large excess of individuals, nonfidelity to natal habitats had little effect on metapopulation size and structure.     

Importance of Accounting for Detection Heterogeneity When Estimating Abundance: the Case of French Wolves

Abstract: Assessing conservation strategies requires reliable estimates of abundance. Because detecting all individuals is most often impossible in free‐ranging populations, estimation procedures have to account for a <1 detection probability. Capture–recapture methods allow biologists to cope with this issue of detectability. Nevertheless, capture–recapture models for open populations are built on the assumption that all individuals share the same detection probability, although detection heterogeneity among individuals has led to underestimating abundance of closed populations. We developed multievent capture–recapture models for an open population and proposed an associated estimator of population size that both account for individual detection heterogeneity (IDH). We considered a two‐class mixture model with weakly and highly detectable individuals to account for IDH. In a noninvasive capture–recapture study of wolves we based on genotypes identified in feces and hairs, we found a large underestimation of population size (27% on average) occurred when IDH was ignored.     

Variation in size at sex-change among natural populations of the protandrous hermaphrodite, <Emphasis Type="Italic">Crepidula fornicata</Emphasis> (Gastropoda,Calyptraeidae)

Protandrous hermaphrodites are predicted to change sex from male to female when relative reproductive fitness of females surpasses that of males. How size at sex transition varies with population, mating group and individual parameters was investigated for five populations of the protandrous hermaphrodite slipper snail, Crepidula fornicata. The populations varied for density, size distribution, average mating group size and sex ratio. Size at sex-change was correlated with the population sex ratio. Comparisons of multiple hypotheses revealed that variables predicting the sex of a snail vary among positions in the mating group. The variables included body size, the relative size of the snail sitting atop the focal snail and population density. Our data support the conclusions that size at sex-change (and by inference, the size at which one sex has relatively greater fitness) is not fixed for these hermaphrodites and that individual size, social conditions and population differences all influence variation in relative fitness.     

A simple spatially explicit ideal-free distribution: a model and an experiment

This investigation presents a simple spatially explicit analysis of the ideal-free distribution. The traditional ideal-free distribution assumes discrete sites with definite boundaries, and predicts how many individuals should occupy each site. In contrast, the present analysis assumes that a forager’s gains gradually decline with distance from a site, and asks where in space individuals ought to be. Although many interesting situations may arise, the analysis asks how individuals should position themselves as the distance between two identical sources increases. Nash equilibrium positions should follow a pitchfork pattern as the distance between sites is increased; that is, an individual should maintain a position between two sources when they are close together but should move nearer one of the sources when they are far apart. In addition, the text describes an experimental study that parallels the theoretical analysis. The experiment supports the predicted pitchfork pattern, and provides somewhat weaker support for the predicted differences in ”individual” and ”paired” pitchforks. Received: 14 June 2000 / Revised: 20 September 2000 / Accepted: 7 October 2000     

Nocturnal feeding migrations of Nerita plicata,N. undata and N. textilis (Prosobranchia: Neritacea) on the rocky shores at Mkomani,Mombasa, Kenya

The vertical zonation of the three common rocky shore neritids at Mkomani, Mombasa, Kenya, Nerita plicata Linnaeus, N. undata Linnaeus, and N. textilis Dillwyn, as a function of feeding migrations and of size, was studied from 28 February to 24 March 1983. These snails perform feeding migrations at night starting at around mid-ebb tide and return to their resting positions with the flood tide. They remain in their resting positions throughout the day until the next nocturnal ebb tide. The direction of migration is sizerelated, with the larger snails of each species moving in the opposite vertical direction to the smaller ones, so that the populations as a whole exhibit no statistically significant net vertical displacement. The larger individuals of two of the species, N. plicata and N. undata, invariably move downwards to their feeding levels, while the smaller individuals move upwards; the larger individuals of N. textilis display a different pattern of migration, moving downwards on and around spring-tide days and upwards on and around neap tide days, while the smaller individuals move in the opposite directions. N. textilis rest above their feeding level around spring tides, and below that level around neap tides. It is demonstrated how these nocturnal migratory feeding rhythms are integrated into the spring-neap and seasonal cycles of the snails' daytime resting positions. The adaptive significance of these migrations is also discussed.     

A resource-based approach to modelling the dynamics of interacting populations

The procedure for modelling the growth of single-species populations [Sakanoue, S., 2007. Extended logistic model for growth of single-species populations. Ecol. Model. 205, 159–168] is improved to be applicable to the study of the dynamics of interacting populations. The improved procedure is based on three assumptions: resource availability changes with population size as a variable, resource supply to populations and population demand for resources are defined as functions of resource availability and population size, and the variables of resource availability and population size shift in the supply function attracted to the demand function. These assumptions are organized into three equations. The equations can generate the dynamics models of plant, herbivore, and detritivore populations, and their own resources. The models can be used to describe prey–predator dynamics. They naturally contain nonlinear terms for the predator’s numerical and functional responses. Depending on the terms, the fluctuations in resource availability and population size stabilize. The three equations can also generate the dynamics models of different populations consuming the same resources. The analysis of zero isoclines of the models shows that a superior population can be simply defined as one with a higher intrinsic rate of natural increase, that a stable coexistence may be realized with the intraspecific interference or the interspecific facilitation of superiors, and that the interspecific interference or the intraspecific facilitation of inferiors may make the coexistence unstable and the inferiors winners depending on their initial population size.     

Long-term demographic and genetic effects of releasing captive-born individuals into the wild

Because of continued habitat destruction and species extirpations, the need to use captive breeding for conservation purposes has been increasing steadily. However, the long-term demographic and genetic effects associated with releasing captive-born individuals with varied life histories into the wild remain largely unknown. To address this question, we developed forward-time, agent-based models for 4 species with long-running captive-breeding and release programs: coho salmon (Oncorhynchus kisutch), golden lion tamarin (Leontopithecus rosalia), western toad (Anaxyrus boreas), and Whooping Crane (Grus americana). We measured the effects of supplementation by comparing population size and neutral genetic diversity in supplemented populations to the same characteristics in unaltered populations 100 years after supplementation ended. Releasing even slightly less fit captive-born individuals to supplement wild populations typically resulted in reductions in population sizes and genetic diversity over the long term when the fitness reductions were heritable (i.e., due to genetic adaptation to captivity) and populations continued to be regulated by density-dependent mechanisms over time. Negative effects for species with longer life spans and lower rates of population replacement were smaller than for species with shorter life spans and higher rates of population replacement. Programs that released captive-born individuals over fewer years or that avoided breeding individuals with captive ancestry had smaller reductions in population size and genetic diversity over the long term. Relying on selection in the wild to remove individuals with reduced fitness mitigated some negative demographic effects, but at a substantial cost to neutral genetic diversity. Our results suggest that conservation-focused captive-breeding programs should take measures to prevent even small amounts of genetic adaptation to captivity, quantitatively determine the minimum number of captive-born individuals to release each year, and fully account for the interactions among genetic adaptation to captivity, population regulation, and life-history variation.     

Incorporating spatial autocorrelation into cellular automata model: An application to the dynamics of Chinese tamarisk (Tamarix chinensis Lour.)

Spatial autocorrelation (SAC) is frequently encountered in most spatial data in ecology. Cellular automata (CA) models have been widely used to simulate complex spatial phenomena. However, little has been done to examine the impact of incorporating SAC into CA models. Using image-derived maps of Chinese tamarisk (Tamarix chinensis Lour.), CA models based on ordinary logistic regression (OLCA model) and autologistic regression (ALCA model) were developed to simulate landscape dynamics of T. chinensis. In this study, significant positive SAC was detected in residuals of ordinary logistic models, whereas non-significant SAC was found in autologistic models. All autologistic models obtained lower Akaike's information criterion corrected for small sample size (AICc) values than the best ordinary logistic models. Although the performance of ALCA models only satisfied the minimum requirement, ALCA models showed considerable improvement upon OLCA models. Our results suggested that the incorporation of the autocovariate term not only accounted for SAC in model residuals but also provided more accurate estimates of regression coefficients. The study also found that the neglect of SAC might affect the statistical inference on underlying mechanisms driving landscape changes and obtain false ecological conclusions and management recommendations. The ALCA model is statistically sound when coping with spatially structured data, and the adoption of the ALCA model in future landscape transition simulations may provide more precise probability maps on landscape transition, better model performance and more reasonable mechanisms that are responsible for landscape changes.     

Choosing classes for size projection matrix models

Projection matrix models are intensely used in ecology to model the dynamics of structured populations. When dealing with size-structured populations, there is no satisfactory algorithm to partition size into discrete classes. We show that the Vandermeer-Moloney algorithm for choosing classes is inconsistent with the Usher model, and systematically selects the finest classes. Considering that the matrix model is a discrete approximation of a continuous model, we define an approximation error as the sum of a distribution error (the difference between the discrete distribution and its continuous counterpart), and a sample error. The optimal partition of size into classes is the one that minimizes the approximation error. This method for choosing classes also shows that the choice of the class width cannot be disconnected from the choice of the time step. When applied to 520 trees of Dicorynia guianensis in French Guiana, this algorithm identified 8 classes of 11.4 cm in width, which is in agreement with the empirical choice of foresters.