# Mathematical Modeling

## Science Center Objects

A formal framework that revolves around the construction of a mathematical expression using equations, statements or formulas as representation of a system, behavior, or process to  simulate the behavior of the system, a series of relationships, or process under study.

Royle, J. A., Chandler, R. B., Sollmann, R., and Gardner, B., 2014, Spatial capture-recapture: Amsterdam,  Elsevier.    xxix, 577  p. (Public domain.)

Spatial Capture-Recapture Models to Estimate Abundance and Density of Animal Populations

For decades, capture-recapture methods have been the cornerstone of ecological statistics as applied to population biology.  While capture-recapture has become the standard sampling and analytical framework for the study of population processes (Williams, Nichols & Conroy 2002) it has advanced independent of and remained unconnected to the spatial structure of the population or the landscape within which populations exist. Furthermore, capture-recapture does not invoke any spatially explicit biological processes and thus is distinctly non-spatial, accounting neither for the inherent spatial nature of the sampling nor of the spatial distribution of individual encounters.  Linking observed encounter histories of individuals to mechanisms of spatial population ecology will enable ecologists to study these processes using new technologies such as noninvasive genetics, remote cameras and bioacoustic sampling.

Hierarchical Model for population change and movement based on removal based on removal data of Veiled Chameleons (Public domain.)

Hierarchical Models for Estimation of Population Parameters

Much of wildlife research consists of the description of variation in data. Some of the variation results from spatial and temporal change in populations, while some results from biologically irrelevant sampling variation induced by the process of data collection. Distinguishing relevant from irrelevant variation is the first task of statistical analysis, but the job does not end there. Even if the true values of population parameters were known, without the contamination of sampling variation, the investigation of population processes would require an evaluation of pattern among parameters.

BBS Observer along Hughesville, MD BBS route (Credit: Heather Martley. Public domain.)

Implementing Cross Validation Approaches for Model Selection and Evaluating Goodness of Fit in Complex Hierarchical Models

Many critical wildlife surveys, such as the North American Breeding Bird Survey (BBS), are analyzed using complex hierarchical models. These models are generally multi-scale and contain random effects; the standard approaches to model selection and assessment of model fit are often inappropriate and no simple way exists to compare alternative models. However, a clear need exists for these assessments. Many alternative models new exist for analysis of BBS data, and simply presenting multiple results without clear guidance on which model is most appropriate will lead to confusion among users of BBS data and limit use of this important survey.

Log-linear hierarchical model of population change for BBS data (Public domain.)

Design and Analysis of Surveys for Estimation of Temporal and Spatial Change in Animal Populations

Population status information is required for management of migratory bird populations, and structured decision making and adaptive anagement place additional emphasis on the need for rigorous survey designs and robust estimation methods. The North American Breeding Bird Survey (BBS) and Christmas Bird Count (CBC) provide continent-scale information on breeding and wintering populations of >450 species of North American birds, and for many species these two surveys are our only data source for population status and trend information. Appropriate analyses of these important surveys require sophisticated methods to accommodate variation in survey efficiency over the large areas covered by the surveys and to control for factors that influence detection of birds. Factors such as observer quality and effort, if not appropriately controlled for in the analysis, can lead to biased estimates of population change.

Survival Probability (Public domain.)

Hierarchical Models of Animal Abundance and Occurrence

Research goals of this project are to develop models, statistical methods, sampling strategies and tools for inference about animal population status from survey data. Survey data are always subject to a number of observation processes that induce bias and error. In particular, inferences are based on spatial sampling – we can only ever sample a subset of locations where species occur --and imperfect detection – species or individuals might go undetected in the sample. Principles of hierarchical modeling can be applied directly to accommodate both features of ecological data. Prior to the development of hierarchical models at PWRC, studies of unmarked populations focused on simplistic descriptions of distribution patterns and temporal trends. Hierarchical models have advanced the field of population ecology by enabling the estimation of demographic and movement parameters that previously could only be obtained using costly field methods. Ecologists can now make inferences about population dynamics at broad spatial and temporal scales using models designed specifically for this task.

grams that were developed at PWRC and elsewhere for the analysis of animal populations for wildlife biologists/managers. (Public domain.)

Development of Computer Software for the Analysis of Animal Population Parameters

Biologists at USGS Patuxent, as well as cooperating agencies are constantly looking for new ways of answering questions about the status of animal populations and how animal populations change over time. To address these questions, data are collected on captures and or sightings of animals which can be used to estimate parameters which affect the population using legacy software. Over time, new questions and methods for addressing these questions arise which require new computer software.

(Public domain.)

Modeling, Estimation, and Adaptive Management of Florida Manatees

Florida manatees are threatened by watercraft-related mortality, the potential loss of warmwater habitat, red tide events, and other anthropogenic factors. The USFWS and the Florida Fish and Wildlife Conservation Commission have regulatory authorities under the Endangered Species Act (ESA), the Marine Mammal Protection Act (MMPA), and state statutes to recover manatees. To support management decision-making, these agencies need quantitative assessments of population status.

Endangered Karner blue butterfly, Lycaeides melissa samuelis (Public domain.)

Estimation of Density and Abundance of Biological Populations on National Parks and Wildlife Refuges through Distance Sampling

Assessing the status and trends of populations of biological organisms is an important management goal and a recurrent theme in USGS research. Often, the most basic question of “how many are there?” remains elusive, thus making management decisions more difficult. This study continues a long-term commitment of technical support for the use of distance sampling for wildlife population abundance estimation in our National Parks and Wildlife Refuges.

(Credit: Jeff Hatfield, USGS Patuxent Wildlife Research Center. Public domain.)

Vegetation Studies in National Parks, Wildlife Refuges, and Other Protected Areas

Forests and marshes provide critical habitat for numerous species of plants and animals. National Parks, Wildlife Refuges, and other protected areas are attempting to protect, manage, and in some cases, restore our forests and marshes, many of which provide critical habitat for declining, threatened, or endangered species, in addition to providing recreational opportunities across the U.S. These natural communities are being degraded by a variety of anthropogenic forces, including habitat destruction due to urbanization or conversion to agriculture, and the effects of invasive species, introduced diseases, and environmental contaminants. Maintaining healthy forests and marshes is important if we are to continue to provide valuable habitat for many of our native species.

(Credit: Ian Thomas, US Fish and Wildlife Service. Public domain.)

Survival and Reintroduction of the Laysan Teal

The Laysan Teal is an endangered, endemic, Hawaiian dabbling duck that has been pushed to the brink of extinction numerous times. The previous range includes the Main and Northwestern Hawaiian Islands, and its current range is less than 10 sq. km within the National Wildlife Refuges of Papahānaumokuākea Marine National Monument. This non-migratory waterfowl was eliminated from all the Hawaiian Islands except for Laysan by the 1860’s through anthropogenic effects (i.e., introduced rats, shipwrecked mariners, etc.). The Laysan Island population was threatened when guano miners inhabited the island, hunted the duck, and introduced rabbits, devastating the native habitat until they were removed in 1923. Presently, extreme events (e.g., tsunamis, hurricanes, drought, or flooding), disease (e.g., Avian Botulism), sea-level rise, accidental predator or competitor introductions, are ongoing threats to this duck’s survival.