Incorporating Activity Time in Harmonic Home Range Analysis

Previously used methods to analyze radio tracking data have not generally taken advantage of all the information associated with location fixes. Most early analyses were preoccupied with size and shape of home range. Analysis strategies with options determined by study objectives, type of data and species-specific variables are described. Descriptions can include cartographical, graphic and tabular depictions of distributions for spatial, habitat, and activity type utilization, direction and orientation as well as interaction potential measures. A sample set of data from a red fox radio tracking study was used to demonstrate methods of analysis. Analysis was performed interactively, which permitted the researcher to interrogate the data base with few inherent assumptions. Computer algorithms for measuring area and depicting borders of home ranges were compared for consistency, sensitivity and richness for biological implications. Two methods employing cell frequency counts were found to be useful for investigating the internal anatomy of home ranges. Emphasis is placed on describing approaches to answering questions on how animals use their home areas rather than the computer program package developed. A review of home range analysis and literature is also included.

Many analyses of animal movements assume that an animal's position at time t + 1 is independent of its position at time t, but no statistical procedure exists to test this assumption with bivariate data. Using empirically derived critical values for the ratio of mean squared distance between successive observations to mean squared distance from the center of activity, we demonstrate a bivariate test of the independence assumption first proposed by Schoener. For cases in which the null hypothesis of independence is rejected, we present a procedure for determining the time interval at which autocorrelation becomes negligible. To illustrate implementation of the test, locational data obtained from a radio-tagged adult female cotton rat (Sigmodon hispidus) were used. The test can be used to design an efficient sampling schedule for movement studies, and it is also useful in revealing behavioral phenomena such as home range shifting and any tendency of animals to follow prescribed routes in their daily activities. Further, the test may provide a means of examining how an animal's use of space is affected by its internal clock.

Abstract We examine the use of conventional sampling data to estimate the area extent of an animal's home range. Popular methods of estimation based on polygons and capture radii are discussed. A simple example shows they are not directly comparable. Furthermore, when home ranges lack circular symmetry, methods based on capture radii suffer large biases. On the other hand, methods based on polygons have sample size biases. A new method, based on the determinant of the covariance matrix of the capture points, is introduced. We show that this method, under specified assumptions, is free from both forms of bias. Tables that identify the biases in the polygon methods are provided. An important use of the tables, that of making comparisons between values obtained by the various methods of estimation, is illustrated. The statistical stability of each method is given by means of a table or formula for its relative variance as a function of sample size. Finally, the methods are evaluated. For purposes of estimating home range area, the determinant method is recommended, the old convex polygon method is defended, and the use of methods based on capture radii is discouraged.

In home range studies of wild animals and birds, statistical analysis of radio telemetry data poses special problems due to lack of independence of successive observations along the sample path. Assuming that such data are generated by a continuous, stationary, Gaussian process possessing the Markov property, then a multivariate Ornstein-Uhlenbeck diffusion process is necessarily the source and is proposed here to be a workable model. Its characterization is given in terms of typical descriptive properties of home range such as center of activity and confidence regions. Invariance of the model with respect to choice of an observational coordinate system is established, while data for twin deer are used to illustrate the manner in which the model may be used for study of territorial interaction. An approximate maximum likelihood procedure is proposed with results being reported for deer, coyote, and bird tracking data.

A new method of calculating centers and areas of animal activity is presented based on the harmonic mean of an areal distribution. The center of activity is located in the area of greatest activity; in fact, more than one “center” may exist. The activity area isopleth is related directly to the frequency of occurrence of an individual within its home range. The calculation of home range allows for the heterogeneity of any habitat and is illustrated with data collected near Corvallis, Oregon, on the brush rabbit (Sylvilagus bachmani.)

A new technique for estimating home range size from a sequence of independent observations of animal positions is introduced. This method describes home range in a probabilistic sense, makes no assumption about the underlying distribution, and therefore is nonparametric. The procedure is based upon an existing nonparametric technique for density estimation which uses the Fourier transform (Tarter and Kronmal 1970). A comparison is made with two commonly used methods; the minimum convex polygon method (Southwood 1966) and bivariate normal method (Jennrich and Turner 1969), as well as another recent nonparametric method developed by Ford and Krumme (1979). All these methods have certain problems, making none of them suitable for every situation. The trade—offs among these methods and some fundamental limitations of home range estimation are discussed. Evidence is given which indicates that in some situations the proposed method is preferable to earlier methods.

The minimum-convex-polygon method for estimating home-range area, in which the outermost points are connected in a particular way, is extremely sensitive to sample size. Existing methods for estimating home-range area that correct for sample size fail to encompass all the important kinds of biological variation in the home-range utilization. (The home-range utilization describes the relative degree to which different units of space are frequented by an animal.) Although previous methods have assumed specific unimodal distributions, such as the bivariate normal, home-range utilizations may resemble funnels or pies as well as hills. A regression method is introduced that uses data from well-sampled individuals whose true home ranges are assumed approximately known to predict home-range areas for less well-sampled individuals. Appendix 5 summarizes this method. Sizes of home ranges estimated by the regression method are half or less than sizes estimated by previous methods in which utilization distributions are assumed to be all of a particular statistical type.

Abstract Seven major types of sampling for observational studies of social behavior have been found in the literature. These methods differ considerably in their suitability for providing unbiased data of various kinds. Below is a summary of the major recommended uses of each technique: In this paper, I have tried to point out the major strengths and weaknesses of each sampling method. Some methods are intrinsically biased with respect to many variables, others to fewer. In choosing a sampling method the main question is whether the procedure results in a biased sample of the variables under study. A method can produce a biased sample directly, as a result of intrinsic bias with respect to a study variable, or secondarily due to some degree of dependence (correlation) between the study variable and a directly-biased variable. In order to choose a sampling technique, the observer needs to consider carefully the characteristics of behavior and social interactions that are relevant to the study population and the research questions at hand. In most studies one will not have adequate empirical knowledge of the dependencies between relevant variables. Under the circumstances, the observer should avoid intrinsic biases to whatever extent possible, in particular those that direcly affect the variables under study. Finally, it will often be possible to use more than one sampling method in a study. Such samples can be taken successively or, under favorable conditions, even concurrently. For example, we have found it possible to take Instantaneous Samples of the identities and distances of nearest neighbors of a focal individual at five or ten minute intervals during Focal-Animal (behavior) Samples on that individual. Often during Focal-Animal Sampling one can also record All Occurrences of Some Behaviors, for the whole social group, for categories of conspicuous behavior, such as predation, intergroup contact, drinking, and so on. The extent to which concurrent multiple sampling is feasible will depend very much on the behavior categories and rate of occurrence, the observational conditions, etc. Where feasible, such multiple sampling can greatly aid in the efficient use of research time.

When an investigator records an observation by nature according to a certain stochastic model, the recorded observation will not have the original distribution unless every observation is given an equal chance of being recorded. A number of papers have appeared during the last ten years implicitly using the concepts of weighted and size-biased sampling distributions. In this paper, we examine some general models leading to weighted distributions with weight functions not necessarily bounded by unity. The examples include: probability sampling in sample surveys, additive damage models, visibility bias dependent on the nature of data collection and two-stage sampling. Several important distributions and their size-biased forms are recorded. A few theorems are given on the inequalities between the mean values of two weighted distributions. The results are applied to the analysis of data relating to human populations and wildlife management. For human populations, the following is raised and discussed: Let us ascertain from each male student in a class the number of brothers, including himself, and sisters he has and denote by k the number of students and by B and S the total numbers of brothers and sisters. What would be the approximate values of B/(B+S), the ratio of brothers to the total number of children, and (B+S)/k, the average number of children per family? It is shown that B/(B+S) will be an overestimate of the proportion of boys among the children per family in the general population which is about half, and similarly (B+S)/k is biased upwards as an estimate of the average number of children per family in the general population. Some suggestions are offered for the estimation of these population parameters. Lastly, for the purpose of estimating wildlife population density, certain results are formulated within the framework of quadrat sampling involving visibility bias.

The term utilization distribution is used to refer to the relative frequency distribution for the points of location of an animal over a period of time. Nine univariate and bivariate probabilistic home- range models were compared to determine their relative suitabilities as an approximation of a utiliza- tion distribution. The differences between univariate and bivariate models and methods were clarified. Since the utilization distribution is a bivariate distribution itself, a bivariate probability distribution gen- erally will be more appropriate. J. WILDL. MANAGE. 39(1):118-123 One important question in any modeling effort is: What is the purpose of the model in terms of the biological information de- sired? The answer to this question must be considered prior to developing and testing a model. In the home-range models I shall compare, the biological information of in- terest is: (1) size and shape of home range, and (2) the effect of various factors on size and shape of home range. A second important question is: What properties of the real system are to be modeled? The answer requires reflecting on how much detail of the system is to be lumped in the interests of emphasizing cer- tain dominant features. For home-range models this means deciding what aspects of our knowledge concerning the behavior of the animal and the characteristics of the habitat are to be included in the model and what aspects are to be ignored.

A robust estimation procedure is proposed to identify and reduce the influence of extreme locations for the bivariate normal home-range method. Tests are proposed for validating the underlying probability distribution from observed animal locations. Location data from a black bear (Ursus americanus) are used to demonstrate the effect of outliers on size and orientation of home-range estimates and to illustrate the goodness-of-fit tests. J. WILDL. MANAGE. 49(2):513-519 Burt (1943:351) defined the home range as area traversed by the individual in its normal activities of food gathering, mating, and caring for young. He believed that occasional sallies and exploratory moves outside the area should not be included as part of the home range. However, a lack of standard conventions for identifying such extreme locations has resulted in potentially arbitrary home-range estimates (Schoener 1981). Hayne (1949) recognized that biological understanding of an animal's home range required information about the intensity of use within the area. Furthermore, he believed that knowledge of the use pattern was important to define the limit of the home range. This pattern of use has subsequently been termed the utilization distribution or UD (Jennrich and Turner 1969, Van Winkle 1975, Anderson 1982). When the observed UD matches a simple probability distribution, we can readily obtain estimates of home-range size, shape, and orientation. These parameters provide a basis for important behavioral and ecological interpretations. Three commonly used methods to estimate home range are the minimum convex polygon, circular bivariate normal, and general bivariate normal methods. These approaches and their underlying probability distributions are discussed by Metzgar (1973); their biological assumptions, sample size biases, and sensitivity to extreme locations have been criticized by several authors (Jennrich and Turner 1969, Dixon and Chapman 1980, MacDonald et al. 1980, Schoener 1981, Anderson 1982). Typically, home-range methods are applied without evaluating the fit of the assumed probability distribution to the observed data. The method selected seems to be based on tradition rather than underlying properties of the data. I this paper we describe a robust estimator t minimize the problems of outliers for the general bivariate normal method and propose procedures for testing the goodness-of-fit of the assumed probability distributions. Location data from a black bear are used to illustrate the good ess-of-fit test and to demonstrate the effect of outliers on size and orientation for bivariate normal and minimum convex polygon home ranges. We wish to thank R. K. Steinhorst for valuable assistance during the development of the statistical methods. Data for our example were provided by J. Unsworth and the Idaho Dep. of Fish and Game. L. J. Nelson, D. F. Stauffer, and D. H. Johnson provided comments on the manuscript. Computer time for this project was provided by the Computing Cent., Univ. of Idaho. This is Contrib. No. 270 from the For., Wildl. and Range Exp. Stn., Univ. of Idaho. This content downloaded from 157.55.39.132 on Thu, 15 Sep 2016 04:56:52 UTC All use subject to http://about.jstor.org/terms 514 HOME RANGE * Samuel and Garton J. Wildl. Manage. 49(2):1985 UNIFORM DISTRIBUTION Metzgar (1973) described the frequency distribution of locations for an animal with equal probability of occurrence per unit of area throughout its home range. This bivariate uniform distribution assumes that the animal has no area of highest activity (center of activity), although an arithmetic center exists. A uniform UD may be appropriate for animals that perceive the environment in a fine-grained fashion (Schoener 1981), whose home ranges are uniform, and that lack a center of activity. The minimum convex polygon (MCP) appears to be an appropriate method to represent the homerange boundary of a uniform UD. The MCP method defines a distinct boundary as Metzgar (1973) suggested for the uniform UD, and Stickel (1954) found that home ranges of uniform use were accurately represented by similar polygon methods. However, because the MCP method does not consider the distribution of use within the home range (Macdonald et al. 1980, Voigt and Tinline 1980), the calculation of potential interaction measures (Macdonald et al. 1980, Voigt and Tinline 1980) by proportional home-range overlap (Owings et al. 1977, Nelson and Mech 1981, Seegmiller and Ohmart 1981) assumes an underlying uniform use pattern and may produce dramatically different values from that of a bivariate normal (Macdonald et al. 1980).

Home range and activity patterns were studied on kangaroo rats, Dipodomys ingens , equipped with blinking-light transmitters. Patterns of space use are presented as three-dimensional utilization distributions (UD) with intensity of use as the third axis. Home ranges, calculated from these UDs ranged from 60 to 350 m2. Study animals were active on average less than 20 min/night, but continued active on nights of rain, snow, and bright moonlight. Males and females did not differ significantly in the size of their home ranges or patterns of activity. Measures of overlap between individual home ranges suggest strict intrasexual avoidance but moderate levels of intersexual contact.

SUMMARY (1) A distinction is made between the domain and utilization distribution of an animal's space-use pattern, and the need for a reappraisal of model-based approaches to the description of utilization distributions is indicated. In particular, the symmetry of the widely used bivariate normal model is considered to provide an inadequate characterization of individual space-use intensity in many ecological situations. (2) A field worker is often aware of the position of biological attraction points, termed nuclei, of which there may be several for a particular home range. There is evidence that such nuclei have important influences over the space utilization intensity of a given individual. The home range model proposed here assumes a circular normal distribution of activity about each nucleus. The strength and range of the attractive fields about each nucleus are parameters which may be estimated. (3) Independent fixes of an individual's locations are used to obtain model parameter estimates using the Maximum Likelihood method, enabling a description of the utilization distribution, and calculation of probability domains. (4) The model is applied to field data collected from the grey squirrel (Sciurus carolinensis, Gmelin) and a hypothetical data set. The resulting domain and utilization distribution properties are discussed in relation to the bivariate normal model, and various advantages of the present model are demonstrated.

Shape of the home range and the distribution of activity within the home range were analyzed for 15 female and seven male Peromyscus leucopus. Ten of the 22 home ranges were circular. In all of the home ranges, the frequency with which animals were recorded in any unit of area in the home range decreased normally away from the center of activity. During the last 20 years, investigators of home range behavior in small mam- mals have been increasingly concerned with the way activity is distributed within home ranges. The statistical analysis of home range activity, an ap-

(1) Animals generally use space disproportionately within the boundaries of their home range. Areas receiving concentrated use by resident animals can be termed core areas. Identifying these core areas is an important part of understanding the ecological factors that determine use. (2) Comparison of the observed space-use pattern with that expected from a uniform pattern of use is our basis for defining core areas. The difference in ordered cumulative distribution functions can be tested using a one-sided goodness-of-fit procedure. Core areas are delineated by enscribing those areas within the home range where use exceeds that expected from a uniform distribution. (3) This method of estimating core areas is illustrated with a hypothetical data set and applied to radio-telemetry locations of a black bear (Ursus americanus, Pallas). The bear used two distinct core areas, which enclosed 34% of the total home range and included 76% of the animal locations. (4) Patterns of core area use are demonstrated for male western tanagers (Piranga ludoviciana, Wilson). Although 90% home range boundaries overlap, core areas for four adjacent males are shown to closely resemble exclusive-use territories. (5) Interspecific use of core areas of western tanagers and chipping sparrows (Spizella passerina, Bechstein) are compared. Core areas for chipping sparrows tended to be a larger proportion of the home range and intensity of core area use was greater for western tanagers. (6) The size and location of core areas depend on the method of determination of home range size. Alternative home range methods may have a substantial influence on the estimation of core areas due to differences in the estimated home range boundary and underlying use distribution.