Hierarchical Nonlinear Spatio-temporal Agent-Based Models for Collective Animal Movement

Modeling complex collective animal movement presents distinct challenges. In particular, modeling the interactions between animals and the nonlinear behaviors associated with these interactions, while accounting for uncertainty in data, model, and parameters, requires a flexible modeling framework. To address these challenges, we propose a general hierarchical framework for modeling collective movement behavior with multiple stages. Each of these stages can be thought of as processes that are flexible enough to model a variety of complex behaviors. For example, self-propelled particle (SPP) models (e.g., Vicsek et al. in Phys Rev Lett 75:1226–1229, 1995) represent collective behavior and are often applied in the physics and biology literature. To date, the study and application of these models has almost exclusively focused on simulation studies, with less attention given to rigorously quantifying the uncertainty. Here, we demonstrate our general framework with a hierarchical version of the SPP model applied to collective animal movement. This structure allows us to make inference on potential covariates (e.g., habitat) that describe the behavior of agents and rigorously quantify uncertainty. Further, this framework allows for the discrete time prediction of animal locations in the presence of missing observations. Due to the computational challenges associated with the proposed model, we develop an approximate Bayesian computation algorithm for estimation. We illustrate the hierarchical SPP methodology with a simulation study and by modeling the movement of guppies.Supplementary materials accompanying this paper appear online.     

Estimating Parameters in Delay Differential Equation Models

Delay differential equations (DDEs) are widely used in ecology, physiology and many other areas of applied science. Although the form of the DDE model is usually proposed based on scientific understanding of the dynamic system, parameters in the DDE model are often unknown. Thus it is of great interest to estimate DDE parameters from noisy data. Since the DDE model does not usually have an analytic solution, and the numeric solution requires knowing the history of the dynamic process, the traditional likelihood method cannot be directly applied. We propose a semiparametric method to estimate DDE parameters. The key feature of the semiparametric method is the use of a flexible nonparametric function to represent the dynamic process. The nonparametric function is estimated by maximizing the DDE-defined penalized likelihood function. Simulation studies show that the semiparametric method gives satisfactory estimates of DDE parameters. The semiparametric method is demonstrated by estimating a DDE model from Nicholson’s blowfly population data.     

Dynamic Models of Animal Movement with Spatial Point Process Interactions

When analyzing animal movement, it is important to account for interactions between individuals. However, statistical models for incorporating interaction behavior in movement models are limited. We propose an approach that models dependent movement by augmenting a dynamic marginal movement model with a spatial point process interaction function within a weighted distribution framework. The approach is flexible, as marginal movement behavior and interaction behavior can be modeled independently. Inference for model parameters is complicated by intractable normalizing constants. We develop a double Metropolis–Hastings algorithm to perform Bayesian inference. We illustrate our approach through the analysis of movement tracks of guppies (Poecilia reticulata).     

Imputation Approaches for Animal Movement Modeling

The analysis of telemetry data is common in animal ecological studies. While the collection of telemetry data for individual animals has improved dramatically, the methods to properly account for inherent uncertainties (e.g., measurement error, dependence, barriers to movement) have lagged behind. Still, many new statistical approaches have been developed to infer unknown quantities affecting animal movement or predict movement based on telemetry data. Hierarchical statistical models are useful to account for some of the aforementioned uncertainties, as well as provide population-level inference, but they often come with an increased computational burden. For certain types of statistical models, it is straightforward to provide inference if the latent true animal trajectory is known, but challenging otherwise. In these cases, approaches related to multiple imputation have been employed to account for the uncertainty associated with our knowledge of the latent trajectory. Despite the increasing use of imputation approaches for modeling animal movement, the general sensitivity and accuracy of these methods have not been explored in detail. We provide an introduction to animal movement modeling and describe how imputation approaches may be helpful for certain types of models. We also assess the performance of imputation approaches in two simulation studies. Our simulation studies suggests that inference for model parameters directly related to the location of an individual may be more accurate than inference for parameters associated with higher-order processes such as velocity or acceleration. Finally, we apply these methods to analyze a telemetry data set involving northern fur seals (Callorhinus ursinus) in the Bering Sea. Supplementary materials accompanying this paper appear online.     

Incorporating Telemetry Error into Hidden Markov Models of Animal Movement Using Multiple Imputation

When data streams are observed without error and at regular time intervals, discrete-time hidden Markov models (HMMs) have become immensely popular for the analysis of animal location and auxiliary biotelemetry data. However, measurement error and temporally irregular data are often pervasive in telemetry studies, particularly in marine systems. While relatively small amounts of missing data that are missing-completely-at-random are not typically problematic in HMMs, temporal irregularity can result in few (if any) observations aligning with the regular time steps required by HMMs. Fitting HMMs that explicitly account for uncertainty attributable to location measurement error, temporally irregular observations, or other forms of missing data typically requires computationally demanding techniques, such as Markov chain Monte Carlo (MCMC). Using simulation and a real-world bearded seal (Erignathus barbatus) example, I investigate a practical alternative to incorporating measurement error and temporally irregular observations into HMMs based on multiple imputation of the position process drawn from a single-state continuous-time movement model. This two-stage approach is relatively simple, performed with existing software using efficient maximum likelihood methods, and completely parallelizable. I generally found the approach to perform well across a broad range of simulated measurement error and irregular sampling rates, with latent states and locations reliably recovered in nearly all simulated scenarios. However, high measurement error coupled with low sampling rates often induced bias in both the estimated probability distributions of data streams derived from the imputed position process and the estimated effects of spatial covariates on state transition probabilities. Results from the two-stage analysis of the bearded seal data were similar to a more computationally intensive single-stage MCMC analysis, but the two-stage analysis required much less computation time and no custom model-fitting algorithms. I thus found the two-stage multiple-imputation approach to be promising in terms of its ease of implementation, computation time, and performance. Code for implementing the approach using the R package “momentuHMM” is provided.Supplementary materials accompanying this paper appear online.     

Agent-Based Inference for Animal Movement and Selection

Contemporary ecologists often find themselves with an overwhelming amount of data to analyze. For example, it is now possible to collect nearly continuous spatiotemporal data on animal locations via global positioning systems and other satellite telemetry technology. In addition, there is a wealth of readily available environmental data via geographic information systems and remote sensing. We present a modeling framework that utilizes these forms of data and builds on previous research pertaining to the quantitative analysis of animal movement. This approach provides additional insight into the environmental drivers of residence and movement as well as resource selection while accommodating path uncertainty. The methods are demonstrated in an application involving mule deer movement in the La Sal Range, Utah, USA. Supplemental materials for this article are available online.     

Multi-scale Modeling of Animal Movement and General Behavior Data Using Hidden Markov Models with Hierarchical Structures

Hidden Markov models (HMMs) are commonly used to model animal movement data and infer aspects of animal behavior. An HMM assumes that each data point from a time series of observations stems from one of N possible states. The states are loosely connected to behavioral modes that manifest themselves at the temporal resolution at which observations are made. Due to advances in tag technology and tracking with digital video recordings, data can be collected at increasingly fine temporal resolutions. Yet, inferences at time scales cruder than those at which data are collected and, which correspond to larger-scale behavioral processes, are not yet answered via HMMs. We include additional hierarchical structures to the basic HMM framework, incorporating multiple Markov chains at various time scales. The hierarchically structured HMMs allow for behavioral inferences at multiple time scales and can also serve as a means to avoid coarsening data. Our proposed framework is one of the first that models animal behavior simultaneously at multiple time scales, opening new possibilities in the area of animal movement and behavior modeling. We illustrate the application of hierarchically structured HMMs in two real-data examples: (i) vertical movements of harbor porpoises observed in the field, and (ii) garter snake movement data collected as part of an experimental design. Supplementary materials accompanying this paper appear online.     

Selecting the Number of States in Hidden Markov Models: Pragmatic Solutions Illustrated Using Animal Movement

We discuss the notorious problem of order selection in hidden Markov models, that is of selecting an adequate number of states, highlighting typical pitfalls and practical challenges arising when analyzing real data. Extensive simulations are used to demonstrate the reasons that render order selection particularly challenging in practice despite the conceptual simplicity of the task. In particular, we demonstrate why well-established formal procedures for model selection, such as those based on standard information criteria, tend to favor models with numbers of states that are undesirably large in situations where states shall be meaningful entities. We also offer a pragmatic step-by-step approach together with comprehensive advice for how practitioners can implement order selection. Our proposed strategy is illustrated with a real-data case study on muskox movement.Supplementary materials accompanying this paper appear online.     

Modeling Collective Animal Movement Through Interactions in Behavioral States

Animal movement often exhibits changing behavior because animals often alternate between exploring, resting, feeding, or other potential states. Changes in these behavioral states are often driven by environmental conditions or the behavior of nearby individuals. We propose a model for dependence among individuals’ behavioral states. We couple this state switching with complex discrete-time animal movement models to analyze a large variety of animal movement types. To demonstrate this method of capturing dependence, we study the movements of ants in a nest. The behavioral interaction structure is combined with a spatially varying stochastic differential equation model to allow for spatially and temporally heterogeneous collective movement of all ants within the nest. Our results reveal behavioral tendencies that are related to nearby individuals, particularly the queen, and to different locations in the nest.     

Computationally Efficient Statistical Differential Equation Modeling Using Homogenization

Statistical models using partial differential equations (PDEs) to describe dynamically evolving natural systems are appearing in the scientific literature with some regularity in recent years. Often such studies seek to characterize the dynamics of temporal or spatio-temporal phenomena such as invasive species, consumer-resource interactions, community evolution, and resource selection. Specifically, in the spatial setting, data are often available at varying spatial and temporal scales. Additionally, the necessary numerical integration of a PDE may be computationally infeasible over the spatial support of interest. We present an approach to impose computationally advantageous changes of support in statistical implementations of PDE models and demonstrate its utility through simulation using a form of PDE known as “ecological diffusion.” We also apply a statistical ecological diffusion model to a data set involving the spread of mountain pine beetle (Dendroctonus ponderosae) in Idaho, USA. This article has supplementary material online.     

Animal breeding systems and big game hunting: Models and application

We apply an age- and stage-structured model incorporating varying harem sizes, paternal care and infanticide to examine the effect of hunting on sustainability of populations. Compared to standard carnivore and herbivore models, these models produce different outcomes for sustainable offtake when either adults, or adult males are harvested. Larger harem size increases sustainable offtake whereas paternal care and infanticide lowers it. Where males are monogamous, populations are vulnerable to male offtake, regardless of paternal care. Surprisingly, an incidental take of 10% of other age-sex-classes has very little effect on these findings. Indiscriminate (subsistence) hunting of all age-sex classes has a dramatic effect on certain populations. Applying these behavior-sensitive models to tourist hunting in the Selous Game Reserve, Tanzania, we find that across the Reserve hunting quotas were generally set at sustainable rates except for leopard (Panthera pardus). In certain hunting blocks within the Reserve, however, quotas for eland (Taurotragus oryx), hartebeest (Alcelaphus buselaphus), lion (Panthera leo), reedbuck (Redunca arundinum), sable antelope (Hippotragus niger), warthog (Phacochoerus aethiopicus) and waterbuck (Kobus ellipsiprymnus) are set at unsustainably high rates. Moreover, particular blocks are consistently awarded high quotas. Behaviorally sensitive models refine predictions for population viability, specify data required to make predictions robust, and demonstrate the necessity of incorporating behavioral ecological knowledge in conservation and management.     

典型经验根系吸水函数的田间模拟检验及评价

根据田间实测的作物生理数据和土壤水动态观测资料,确定了冬小麦拔节－开花生长阶段两个根系吸水函数（Ｆｅｄｄｅｓ,ｖａｎ Ｇｅｎｕｃｈｔｅｎ）的特定解析式。在实验小区实测的气象数据和室内外试验得到的土壤水力特性参数基础上,对吸水函数进行田间模拟检验,并依据灵敏度分析结果比较两者间的差异、评价各自的特点。尽管两个根系吸水函数得到的模拟结果与土壤含水量实测值间的差异均较小,但Ｆｅｄｄｅｓ函数要比后者在土壤水分条件改变和根系吸水强度差异对作物根系吸水量的影响上反应更灵敏,故模拟结果应更趋合理、切合实际。     

Equation for high-rate gully erosion

An approach to gully erosion is presented in this paper. The approach is based on general equations derived from theoretical considerations. The equations apply to a situation of intense erosion rate, such as at peak discharge during the few critical rainstorms, able to generate or to widen gullies.Equations linking gully widening to gully deepening are derived. They do not depend on the way in which concentrated flow aggressiveness is estimated. The equation expressing gully width/depth relationship was successfully compared with data from the literature.When runoff aggressiveness was estimated through unit stream power and bottom flow shear stress, the width/discharge relationships found were similar to those expected on the basis of previous studies (e.g., Leopold and Maddock [U. S. Geol. Surv. Prof. Pap. 252 (1953) 57 pp.]) even if slope gradient explicitly appears in contrast with empirical evidence. Only threshold conditions for gullies indicate that flow shear stress (for laminar flow conditions) can explain the observed trends. This astonishing result most probably indicates that gully initiation needs more complex contexts to be explained than the one here used (based on a Montgomery and Dietrich [1994. Landscape dissection and drainage area–slope thresholds. In: M.J. Kirkby (ed.), Process Models and Theoretical Geomorphology. Wiley, 221–246] approach).A selection of the proposed equations have been arranged into a research model and an example of the outcome has been given for two situations typical of cropland in southern Tuscany (Italy). The results indicate that the spatial distribution of soil characteristics and of land use influences significantly gully generation and evolution. This further confirms that gully morphological thresholds cannot be explained by simple approaches.     

冬小麦灌溉随机模拟模型研究

为了考虑随机因素(气象因素等)对冬小麦灌溉的影响,用时间序列方法建立了蒸发蒸腾量的随机模型,并将其导入田间水量平衡方程,推导出冬小麦灌溉的随机模拟模型。将该模型用于商丘市李庄乡的冬小麦灌溉,其灌水量与实际情况较为符合,最大误差仅为4.7%,可用于制定冬小麦的灌溉计划。     

Models for strawberry inflorescence data

The flowers of strawberry plants grow on very variable branched structures called inflorescences, in which each branch gives rise to 0, 1, or 2 offspring branches. We extend previous modeling of the number of strawberry flowers at each individual level in the inflorescence structure conditional on the number of strawberry flowers at the previous level. We consider a range of logistic regression models, including models that incorporate inflorescence effects and random effects. The models can be used to summarize the overall structure of any particular variety and to indicate the main differences between varieties. For the data of the article, we show that models based on convolutions of correlated Bernoulli random variables outperform binomial regression models.     

Models for microbiological colony counts

Concentrations of microorganisms can be estimated from colony counts at different dilutions. However, complications can occur because of colony overlap or inhibition of colony growth. We develop a model of inhibition in which colonies fail to develop if spores are close to spores of other inhibitory species. The model has three parameters, but a limiting case of the model with only two parameters is shown to be more useful in practice. This latter model, which is a generalized linear model, is fitted to colony counts of the fungus Verticillium dahliae and contrasted with a model suggested in an earlier article.     

Equation of sediment transport for slope flows

The suggested model of transporting capacity for slope flows is based on the laws of physics and hydrodynamics. In this model, the probability constituent plays an important role for different types of sediment movement within the range of velocities close to critical values. Each type of movement is specified by an individual transportation coefficient independent of the sediment’s particle size. The model has been parameterized and verified using a large amount of the authors’ experimental results on the determination of transport capacity within a wide range of slope gradients (0–30%), as well as using the data obtained by Goncharov. The correlation coefficients are high (0.90–0.98), with the maximal relative errors fluctuating within 15–35%.