Estimating demographic models for the range dynamics of plant species

Aims To better understand how demographic processes shape the range dynamics of woody plants (in this case, Proteaceae), we introduce a likelihood framework for fitting process‐based models of range dynamics to spatial abundance data. Location The fire‐prone Fynbos biome (Cape Floristic Region, South Africa). Methods Our process‐based models have a spatially explicit demographic submodel (describing dispersal, reproduction, mortality and local extinction) as well as an observation submodel (describing imperfect detection of individuals), and are constrained by species‐specific predictions of habitat distribution models and process‐based models for seed dispersal by wind. Free model parameters were varied to find parameter sets with the highest likelihood. After testing this approach with simulated data, we applied it to eight Proteaceae species that differ in breeding system (monoecy versus dioecy) and adult fire survival. We assess the importance of Allee effects and negative density dependence for range dynamics, by using the Akaike information criterion to select between alternative models fitted for the same species. Results The best model for all dioecious study species included Allee effects, whereas this was true for only one of four monoecious species. As expected, sprouters (in which adults survive fire) were estimated to have lower rates of reproduction and catastrophic population extinction than related non‐sprouters. Overcompensatory population dynamics seem important for three of four non‐sprouters. We also found good quantitative agreement between independent data and most estimates of reproduction, carrying capacity and extinction probability. Main conclusions This study shows that process‐based models can quantitatively describe how large‐scale abundance distributions arise from the movement and interaction of individuals. It stresses links between the life history, demography and range dynamics of Proteaceae: dioecious species seem more susceptible to Allee effects which reduce migration ability and increase local extinction risk, and sprouters seem to have high persistence of established populations, but their low reproduction limits habitat colonization and migration.     

The influence of sample size on two approaches to estimate Black Grouse Lyrurus tetrix population size using non-invasive sampling methods

Population size is an important parameter to monitor for species conservation and management. This is especially important for rare and endangered species, as declines can give information about anthropogenic impacts and the need for new conservation measures. To estimate population size, various methods of analysis can be used, for which sample size is an important factor. Sample size is particularly important to consider when applying non-invasive sampling strategies such as sampling faeces or feathers/hairs as a source of DNA, as a means to limit disturbance and stress for the species of concern. We investigated a Black Grouse Lyrurus tetrix population in the eastern part of the Alps, in East Tyrol (Austria), and estimated population size using two approaches: capture–recapture and rarefaction. With a set of 12 polymorphic microsatellite markers, we identified genotypes from faeces and feathers (backed up with 23 tissue samples) and checked for population substructure and gene flow among sampling sites. We estimated population size using four different models from the two approaches (molecular capture–recapture: TIRM, TIRMpart; rarefaction: hyperbolic function – Kohn, exponential function – Eggert). To evaluate the impact of sample size on the estimations, we used the full dataset of 500 samples (‘complete’ dataset) and half the dataset of 250 samples (‘half’ dataset). We also estimated the population size for each sex separately using complete and half datasets to check for sex-specific differences in population size. We found similar results in three of four models (capture–recapture: capwire TIRM, capwire TIRMpart; rarefaction: rarefaction Kohn). Using just half of the data increased the uncertainties in the estimation of population size in all models used and deviations were particularly large in females, which indicated a sex bias. Only the complete dataset of males had an observation rate of more than two observations/individual, and this observation rate meets the recommendation for using the capwire models. This indicates that, for species with different sex-specific detectability, larger sample sizes do not generally imply higher observation rates. We conclude that calculating the observation rates and population-size estimations for each sex separately can improve overall population-size estimation, especially in species with biased sex ratios and those that exhibit sex-specific behaviour.     

On the stock estimation for some fishery systems

In this work we address the stock estimation problem for two fishery models. We show that a tool from nonlinear control theory called “observer” can be helpful to deal with the resource stock estimation in the field of renewable resource management. It is often difficult or expensive to measure all the state variables characterising the evolution of a given population system, therefore the question arises whether from the observation of certain indicators of the considered system, the whole state of the population system can be recovered or at least estimated. The goal of this paper is to show how some techniques of control theory can be applied for the approximate estimation of the unmeasurable state variables using only the observed data together with the dynamical model describing the evolution of the system. More precisely we shall consider two fishery models and we shall show how to built for each model an auxiliary dynamical system (the observer) that uses the available data (the total of caught fish) and which produces a dynamical estimation of the unmeasurable stock state x(t). Moreover the convergence speed of towards x(t) can be chosen.     

PEtab—Interoperable specification of parameter estimation problems in systems biology

Reproducibility and reusability of the results of data-based modeling studies are essential. Yet, there has been—so far—no broadly supported format for the specification of parameter estimation problems in systems biology. Here, we introduce PEtab, a format which facilitates the specification of parameter estimation problems using Systems Biology Markup Language (SBML) models and a set of tab-separated value files describing the observation model and experimental data as well as parameters to be estimated. We already implemented PEtab support into eight well-established model simulation and parameter estimation toolboxes with hundreds of users in total. We provide a Python library for validation and modification of a PEtab problem and currently 20 example parameter estimation problems based on recent studies.     

Analysis and Simulation of Division- and Label-Structured Population Models

In most biological studies and processes, cell proliferation and population dynamics play an essential role. Due to this ubiquity, a multitude of mathematical models has been developed to describe these processes. While the simplest models only consider the size of the overall populations, others take division numbers and labeling of the cells into account. In this work, we present a modeling and computational framework for proliferating cell populations undergoing symmetric cell division, which incorporates both the discrete division number and continuous label dynamics. Thus, it allows for the consideration of division number-dependent parameters as well as the direct comparison of the model prediction with labeling experiments, e.g., performed with Carboxyfluorescein succinimidyl ester (CFSE), and can be shown to be a generalization of most existing models used to describe these data. We prove that under mild assumptions the resulting system of coupled partial differential equations (PDEs) can be decomposed into a system of ordinary differential equations (ODEs) and a set of decoupled PDEs, which drastically reduces the computational effort for simulating the model. Furthermore, the PDEs are solved analytically and the ODE system is truncated, which allows for the prediction of the label distribution of complex systems using a low-dimensional system of ODEs. In addition to modeling the label dynamics, we link the label-induced fluorescence to the measure fluorescence which includes autofluorescence. Furthermore, we provide an analytical approximation for the resulting numerically challenging convolution integral. This is illustrated by modeling and simulating a proliferating population with division number-dependent proliferation rate.     

Efficient parameter estimation for spatio-temporal models of pattern formation: case study of Drosophila melanogaster

MOTIVATION: Diffusable and non-diffusable gene products play a major role in body plan formation. A quantitative understanding of the spatio-temporal patterns formed in body plan formation, by using simulation models is an important addition to experimental observation. The inverse modelling approach consists of describing the body plan formation by a rule-based model, and fitting the model parameters to real observed data. In body plan formation, the data are usually obtained from fluorescent immunohistochemistry or in situ hybridizations. Inferring model parameters by comparing such data to those from simulation is a major computational bottleneck. An important aspect in this process is the choice of method used for parameter estimation. When no information on parameters is available, parameter estimation is mostly done by means of heuristic algorithms. RESULTS: We show that parameter estimation for pattern formation models can be efficiently performed using an evolution strategy (ES). As a case study we use a quantitative spatio-temporal model of the regulatory network for early development in Drosophila melanogaster. In order to estimate the parameters, the simulated results are compared to a time series of gene products involved in the network obtained with immunohistochemistry. We demonstrate that a (mu,lambda)-ES can be used to find good quality solutions in the parameter estimation. We also show that an ES with multiple populations is 5-140 times as fast as parallel simulated annealing for this case study, and that combining ES with a local search results in an efficient parameter estimation method.     

From Microscopic to Macroscopic Descriptions of Cell Migration on Growing Domains

Cell migration and growth are essential components of the development of multicellular organisms. The role of various cues in directing cell migration is widespread, in particular, the role of signals in the environment in the control of cell motility and directional guidance. In many cases, especially in developmental biology, growth of the domain also plays a large role in the distribution of cells and, in some cases, cell or signal distribution may actually drive domain growth. There is an almost ubiquitous use of partial differential equations (PDEs) for modelling the time evolution of cellular density and environmental cues. In the last 20 years, a lot of attention has been devoted to connecting macroscopic PDEs with more detailed microscopic models of cellular motility, including models of directional sensing and signal transduction pathways. However, domain growth is largely omitted in the literature. In this paper, individual-based models describing cell movement and domain growth are studied, and correspondence with a macroscopic-level PDE describing the evolution of cell density is demonstrated. The individual-based models are formulated in terms of random walkers on a lattice. Domain growth provides an extra mathematical challenge by making the lattice size variable over time. A reaction–diffusion master equation formalism is generalised to the case of growing lattices and used in the derivation of the macroscopic PDEs.     

On parameter estimation in population models III: time-inhomogeneous processes and observation error

Essential to applying a mathematical model to a real-world application is calibrating the model to data. Methods for calibrating population models often become computationally infeasible when the population size (more generally the size of the state space) becomes large, or other complexities such as time-dependent transition rates, or sampling error, are present. Continuing previous work in this series on the use of diffusion approximations for efficient calibration of continuous-time Markov chains, I present efficient techniques for time-inhomogeneous chains and accounting for observation error. Observation error (partial observability) is accounted for by joint estimation using a scaled unscented Kalman filter for state-space models. The methodology will be illustrated with respect to models of disease dynamics incorporating seasonal transmission rate and in the presence of observation error, including application to two influenza outbreaks and measles in London in the pre-vaccination era.     

Parameter estimation techniques for interaction and redistribution models: a predator-prey example

Summary The use of parameter estimation techniques for partial differential equations is illustrated using a predatorprey model. Whereas ecologists have often estimated parameters in models, they have not previously been able to do so for models that describe interactions in heterogeneous environments. The techniques we describe for partial differential equations will be generally useful for models of interacting species in spatially complex environments and for models that include the movement of organisms. We demonstrate our methods using field data from a ladybird beetle (Coccinella septempunctata) and aphid (Uroleucon nigrotuberculatum) interaction. Our parameter estimation algorithms can be employed to identify models that explain better than 80% of the observed variance in aphid and ladybird densities. Such parameter estimation techniques can bridge the gap between detail-rich experimental studies and abstract mathematical models. By relating the particular bestfit models identified from our experimental data to other information on Coccinella behavior, we conclude that a term describing local taxis of ladybirds towards prey (aphids in this case) is needed in the model.     

Estimating environmental effects on population dynamics: consequences of observation error

Within the paradigm of population dynamics a central task is to identify environmental factors affecting population change and to estimate the strength of these effects. We here investigate the impact of observation errors in measurements of population densities on estimates of environmental effects. Adding observation errors may change the autocorrelation of a population time series with potential consequences for estimates of effects of autocorrelated environmental covariates. Using Monte Carlo simulations, we compare the performance of maximum likelihood estimates from three stochastic versions of the Gompertz model (log–linear first order autoregressive model), assuming 1) process error only, 2) observation error only, and 3) both process and observation error (the linear state–space model on log‐scale). We also simulated population dynamics using the Ricker model, and evaluated the corresponding maximum likelihood estimates for process error models. When there is observation error in the data and the considered environmental variable is strongly autocorrelated, its estimated effect is likely to be biased when using process error models. The environmental effect is overestimated when the sign of the autocorrelations of the intrinsic dynamics and the environment are the same and underestimated when the signs differ. With non‐autocorrelated environmental covariates, process error models produce fairly exact point estimates as well as reliable confidence intervals for environmental effects. In all scenarios, observation error models produce unbiased estimates with reasonable precision, but confidence intervals derived from the likelihood profiles are far too optimistic if there is process error present. The safest approach is to use state–space models in presence of observation error. These are factors worthwhile to consider when interpreting earlier empirical results on population time series, and in future studies, we recommend choosing carefully the modelling approach with respect to intrinsic population dynamics and covariate autocorrelation.     

Stage duration distributions in matrix population models

Matrix population models are a standard tool for studying stage‐structured populations, but they are not flexible in describing stage duration distributions. This study describes a method for modeling various such distributions in matrix models. The method uses a mixture of two negative binomial distributions (parametrized using a maximum likelihood method) to approximate a target (true) distribution. To examine the performance of the method, populations consisting of two life stages (juvenile and adult) were considered. The juvenile duration distribution followed a gamma distribution, lognormal distribution, or zero‐truncated (over‐dispersed) Poisson distribution, each of which represents a target distribution to be approximated by a mixture distribution. The true population growth rate based on a target distribution was obtained using an individual‐based model, and the extent to which matrix models can approximate the target dynamics was examined. The results show that the method generally works well for the examined target distributions, but is prone to biased predictions under some conditions. In addition, the method works uniformly better than an existing method whose performance was also examined for comparison. Other details regarding parameter estimation and model development are also discussed.     

An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso

In this paper we present a numerical method for the bidomain model, which describes the electrical activity in the heart. The model consists of two partial differential equations (PDEs), which are coupled to systems of ordinary differential equations (ODEs) describing electrochemical reactions in the cardiac cells. Many applications require coupling these equations to a third PDE, describing the electrical fields in the torso surrounding the heart. The resulting system is challenging to solve numerically, because of its complexity and very strict resolution requirements in time and space. We propose a method based on operator splitting and a fully coupled discretization of the three PDEs. Numerical experiments show that for simple simulation cases and fine discretizations, the algorithm is second-order accurate in space and time.     

Estimation and inference of R0 of an infectious pathogen by a removal method

The basic reproductive ratio, R0, is a central quantity in the investigation and management of infectious pathogens. The standard model for describing stochastic epidemics is the continuous time epidemic birth-and-death process. The incidence data used to fit this model tend to be collected in discrete units (days, weeks, etc.), which makes model fitting, and estimation of R0 difficult. Discrete time epidemic models better match the time scale of data collection but make simplistic assumptions about the stochastic epidemic process. By investigating the nature of the assumptions of a discrete time epidemic model, we derive a bias corrected maximum likelihood estimate of R0 based on the chain binomial model. The resulting 'removal' estimators provide estimates of R0 and the initial susceptible population size from time series of infectious case counts. We illustrate the performance of the estimators on both simulated data and real epidemics. Lastly, we discuss methods to address data collected with observation error.     

Age estimation in a long‐lived seabird (Ardenna tenuirostris) using DNA methylation‐based biomarkers

Age structure is a fundamental aspect of animal population biology. Age is strongly related to individual physiological condition, reproductive potential and mortality rate. Currently, there are no robust molecular methods for age estimation in birds. Instead, individuals must be ringed as chicks to establish known‐age populations, which is a labour‐intensive and expensive process. The estimation of chronological age using DNA methylation (DNAm) is emerging as a robust approach in mammals including humans, mice and some non‐model species. Here, we quantified DNAm in whole blood samples from a total of 71 known‐age Short‐tailed shearwaters (Ardenna tenuirostris) using digital restriction enzyme analysis of methylation (DREAM). The DREAM method measures DNAm levels at thousands of CpG dinucleotides throughout the genome. We identified seven CpG sites with DNAm levels that correlated with age. A model based on these relationships estimated age with a mean difference of 2.8 years to known age, based on validation estimates from models created by repeated sampling of training and validation data subsets. Longitudinal observation of individuals re‐sampled over 1 or 2 years generally showed an increase in estimated age (6/7 cases). For the first time, we have shown that epigenetic changes with age can be detected in a wild bird. This approach should be of broad interest to researchers studying age biomarkers in non‐model species and will allow identification of markers that can be assessed using targeted techniques for accurate age estimation in large population studies.     

Estimating Water Quality Guidelines for Environmental Contaminants Using Multimodal Species Sensitivity Distributions: A Case Study with Atrazine

Species sensitivity distributions (SSDs) are used globally to generate water quality guidelines (WQGs). In Canada, a suite of models has been endorsed for describing SSDs. However, these models may not be suitable for substances with multiple modes of toxic action such as pesticides. Pesticides can produce multimodal SSDs where sensitive target organisms comprise one mode of the SSD and non-target organisms comprise the remaining mode(s). Guidelines from this type of SSD might be estimated using only the most sensitive taxa or using a multimodal distribution. The multimodal method presented here uses all data meeting data quality criteria and is thus in keeping with the concept that data comprising an SSD are a random sample from the population of interest rather than a subset thereof. The bimodal method can simultaneously emphasize the more sensitive portion of the dataset by allowing estimation of WQGs using a statistical subset of the data. In the case of the atrazine dataset example, this allowed estimating a WQG emphasizing more sensitive taxa whereas no parametric models fit only the more sensitive data and the small sample size (5) precluded the use of nonparametric methods.     

稻纵卷叶螟种群系统动态模拟

本文组建了稻纵卷叶螟种群系统模似模型,它由系统亚模型和总模型二个部分组成.系统亚模型包括发育、死亡和繁殖三个子模块.系统总模型是作者提出的新模型,它综合了前人所提出的种群模型的优点,以差分方程形式给出,以生理时间为单位,考虑了种群内个体间发育速率的差异,不仅能模拟种群数量动态,而且能模拟种群年龄结构,同时能预测发生期.模型有效性检验表明,模拟结果基本上能吻合实测结果.文中还对5个影响因子进行了灵敏性分析.     

Trait Substitution Sequence process and Canonical Equation for age-structured populations

We are interested in a stochastic model of trait and age-structured population undergoing mutation and selection. We start with a continuous time, discrete individual-centered population process. Taking the large population and rare mutations limits under a well-chosen time-scale separation condition, we obtain a jump process that generalizes the Trait Substitution Sequence process describing Adaptive Dynamics for populations without age structure. Under the additional assumption of small mutations, we derive an age-dependent ordinary differential equation that extends the Canonical Equation. These evolutionary approximations have never been introduced to our knowledge. They are based on ecological phenomena represented by PDEs that generalize the Gurtin–McCamy equation in Demography. Another particularity is that they involve an establishment probability, describing the probability of invasion of the resident population by the mutant one, that cannot always be computed explicitly. Examples illustrate how adding an age-structure enrich the modelling of structured population by including life history features such as senescence. In the cases considered, we establish the evolutionary approximations and study their long time behavior and the nature of their evolutionary singularities when computation is tractable. Numerical procedures and simulations are carried.      

Modelling mortality and dispersal: consequences of parameter generalisation on metapopulation dynamics

Modelling dispersal is a fundamental step in the design of population viability analyses. Here, we address the question of the generalisation of population viability analysis models across landscapes by comparing dispersal between two metapopulations of the bog fritillary butterfly ( Proclossiana eunomia ) living in similar highly fragmented landscapes (<1% of suitable habitat in 9 km²). Differences in dispersal patterns were investigated using the virtual migration (VM) model, which was parameterised with capture–mark–recapture data collected during several years in both landscapes. The VM model allows the estimation of 6 parameters describing dispersal and mortality as well as the simulation of dispersal in the landscapes. The model revealed large differences in the VM parameter estimates between the two landscapes and consequently, simulations indicated differential rates of emigration and dispersal mortality. Furthermore, results from crossed-simulations i.e. simulations performed in one of the landscape but using parameter estimates from the other landscape emphasize that dispersal parameters are very specific to each metapopulation and to their landscape. Hence, we urge conservation biologists to be cautious with such parameter generalisations, even for the same species in comparable landscapes.