Elsevier

Ecological Modelling

Volume 177, Issues 1–2, 15 September 2004, Pages 209-232
Ecological Modelling

A mechanistic home range model for optimal use of spatially distributed resources

https://doi.org/10.1016/j.ecolmodel.2004.01.015Get rights and content

Abstract

Home ranges of animals are associated with the spatial distribution of limiting resources on a landscape, yet no mechanistic models representing this relationship exist. We present models of how animals might choose patches for their home ranges in ways that are optimal with respect to spatially distributed resources. The models assume that animals choose patches for their home ranges based on resource benefits discounted for travel costs. Animals might select patches to maximize resources within their home ranges over random use of the landscape (resource maximization), or to satisfy a minimum resource threshold needed for survival or reproduction (area minimization). We evaluated how landscape configuration structures home ranges of animals by performing individual-based, spatially explicit computer simulations using each model on simulated landscapes that differed only in the spatial continuity of resources among patches (from over-dispersed to clumped). The most important factor determining quality, efficiency, resource content, and spatial distribution of home ranges was the extent to which resources were clumped on a landscape. Characteristics of resource-maximizing home ranges were determined only by the distribution of resources, and differed from those of area-minimizing home ranges depending upon the magnitude of the resource threshold required. An increase in resource threshold increased area and total resource content for area-minimizing home ranges, but did not change their quality or efficiency. Because animals can consume or protect resources within their home ranges, they can depress the value of resources available to other animals and hence how those animals will choose their home ranges. Depression of resource values on a landscape by animals should result in different configuration and spatial distribution of home ranges on a landscape than would be predicted in its absence. We modified the two home range models to depress the value of resources available to other animals within patches selected for each home range. We generated home ranges with the new models on the simulated landscapes and evaluated how home ranges with resource depression varied with landscape configuration and with the number of home ranges on the landscape. We compared characteristics and spatial distribution of home ranges with resource depression to those of home ranges that do not. For the number of simulations we performed, resource depression resulted in home ranges that differed little in configuration and landscape interactions from those without, except that they were distributed more evenly on the landscapes and overlapped each other less. As the number of home ranges on a landscape increased, resource distributions declined in quality and heterogeneity, and home ranges became larger, less efficient, and of lower quality. Our results suggest that, in addition to landscape configuration, the extent to which animals depress resources included in their home ranges should determine the evenness of spatial dispersion, overlap, and home range structure, especially where animals select home ranges to satisfy a specific resource threshold and the density of animals is high. Because resource depression sets a limit on the number of home ranges a landscape with fixed resource levels can support, our models provide a means of assessing carrying capacity of a landscape.

Introduction

The home ranges and territories of animals, and the spatial distribution of animals within a population, are commonly thought to reflect the distribution of one or several limiting resources on a landscape (Ebersole, 1980, Hixon, 1980, Schoener, 1981, Powers and McKee, 1994, Powell et al., 1997, Powell, 2000). A territory is that part of an animal’s home range where the animal excludes conspecifics to protect resources (Ostfeld, 1990, Wolff, 1993, Powell et al., 1997, Powell, 2000) or offspring (Wolff, 1997). The relationship between resources and territories has been investigated extensively (Brown, 1969, Carpenter and McMillen, 1976, Ebersole, 1980, Gill and Wolff, 1975, Hixon, 1980, Kodric-Brown and Brown, 1978, Powers and McKee, 1994, Stenger, 1958, Schoener, 1983), primarily using economic analyses of fitness trade-offs between benefits gained from resources and costs of defending them.

In contrast, the factors structuring home ranges of animals have received little attention, partly because definitions for home ranges (and the costs and benefits that might define their structure) are imprecise. Burt (1943, p. 351) described a home range as:

…that area traversed by an individual in its normal activities of food gathering, mating, and caring for the young. Occasional sallies outside the area, perhaps exploratory in nature, should not be considered part of the home range.

Burt’s definition is conceptually complete but difficult to evaluate analytically because terms are vague and difficult to quantify (Powell et al., 1997, Powell, 2000). Although many studies of home ranges exist, little has been done to evaluate, quantify, or improve upon Burt’s definition. Much research and debate has focused on statistical approaches to estimating home ranges from location data (Worton, 1987, Loehle, 1990, White and Garrott, 1990, Gautestad and Mysterud, 1993, Gautestad and Mysterud, 1994, Gautestad and Mysterud, 1995, Bascompte and Vila, 1997, Powell, 2000), but such approaches are descriptive and have limited theoretical or predictive value because they are not mechanistic (Moorcroft et al., 1999). The sole mechanistic home range model to date (Lewis and Murray, 1993, Moorcroft et al., 1999) used correlated random walk structured by scent marking to estimate home ranges of carnivores. No general, mechanistic model exists relating home ranges to the resources that structure or facilitate the “normal activities” described by Burt (1943). Although the importance of food as a limiting resource is cited in many home range studies (Harestad and Bunnell, 1979, Lindzey and Meslow, 1977, Lindstedt et al., 1986, Litvaitis et al., 1986, Jones, 1990, Holzman et al., 1992, Joshi et al., 1995), particularly for females (Young and Ruff, 1982, Ims, 1987, Powell et al., 1997), little is known about how a home range is structured with respect to these or any resources. Researchers, therefore, have had to assume that an animal’s life requisites are satisfied by the resources available within its observed home range.

We hypothesize that home ranges, like territories, are structured primarily by the fitness-driven need for efficient accumulation of resources required for survival and reproduction (Powell, 2000). Because home ranges based solely on accumulating as many resources as possible would be limitless in size, it is clear that animals with defined, finite home ranges accumulate spatially distributed resources under limiting constraints. If home ranges, like territories, are a function of the availability and distribution of limiting resources, limited by the costs of resource acquisition, then the home range choices of animals seeking to maximize reproductive fitness can be modeled as an optimization function. We hypothesize that an animal maximizes resource accrual per unit area of its home range through the optimal selection of resource-bearing patches, analogous to optimal foraging for food items in a diet (Stephens and Krebs, 1986, Krebs and Kacelnik, 1991). This differs from the common view of the home range as the sum of an animal’s movements (Worton, 1987, Loehle, 1990, Gautestad and Mysterud, 1993, Lewis and Murray, 1993, Gautestad and Mysterud, 1995, Bascompte and Vila, 1997) with a focus instead on the spatially distributed resources that structure an animal’s movements (i.e., its cognitive map; Peters, 1978).

We developed spatially explicit, individual-based models for optimally selecting patches for a home range from a landscape comprising patches that contain limiting resources. Each model assumes that animals select patches of the highest quality available for their home ranges. The models differ in the point at which this patch selection would stop, i.e., when a home range contains sufficient resources. For understanding different ways animals might determine sufficiency, we envision a spectrum of behaviors. At one extreme are animals for which survival and reproduction increase monotonically with the efficient accumulation of spatially distributed resources. Such an animal would seek to balance the benefits of accumulating as many resources as possible against the costs of including the patches that contain them, causing it to seek the most efficient accumulation of resources in a home range that a resource distribution can offer. At the other extreme are animals for which survival and reproduction asymptote with the efficient accumulation of spatially distributed resources. At some point, a biological threshold is reached beyond which adding new, resource-rich patches to a home range has no benefit; such animals should seek to accumulate efficiently only the resources necessary to survive or reproduce in their home range. Based on these extremes, we modeled two alternative strategies for determining when patch selection should end in home range construction: (1) maximizing resources within a home range over random use of patches, or (2) accumulating resources sufficient to satisfy a pre-set minimum threshold. The first strategy maximizes the difference between selective and random use of a resource distribution and therefore is optimal with respect to the resources themselves. The second strategy minimizes the area needed to satisfy a resource threshold sufficient for an animal’s survival and reproduction and therefore is optimal with respect to this biological threshold. We evaluated how home ranges of selective animals pursuing these strategies might differ by performing computer simulations for each model on resource distributions of known characteristics.

In addition to strategies of patch selection by animals, spatial structure of home ranges and their distribution on a landscape should also be determined by the distribution of resource-containing patches. To evaluate this relationship, we applied our models to five simulated landscapes differing in how patches of varying resource value were distributed. From 100 simulations for each model on each landscape, we developed hypotheses about how the spatial distribution of resources should determine the structure and distribution of home ranges.

The distribution of animals on a landscape may not be solely a function of landscape structure. Other factors such as social interactions (e.g., territoriality, hierarchical antagonism) and depletion of resources by individuals using patches (e.g., consumption of foods, causing prey to be vigilant, occupation of den sites) could also strongly affect how animals are distributed in space. We hypothesized that these factors can depress the perceived value of resources contained in a patch and, thereby, affect patch selection by an animal establishing a home range. In this case, we suggest the value of resources in a patch depends on (1) the inherent quantity and quality of those resources, (2) average costs incurred in traveling to that patch, (3) the number of animals using that patch, and (4) the extent to which those animals depress the value of resources to other animals. From this it follows that use of a landscape by an animal modifies the distribution of resources available to other animals; as the number of animals using a landscape increases, the resource distribution available to successive animals changes and, therefore, the home ranges constructed on those resources and the distribution of home ranges on the landscape should also change. Consequently, on identical landscapes one would expect important differences between an equal number of home ranges with resource depression and home ranges without. Further, one would also expect important changes in characteristics of both resource distributions and home ranges as the number of animals establishing home ranges on a landscape changes.

We were interested in how resource depression in patches selected for home ranges might affect home ranges of animals pursuing both the resource-maximizing and area-minimizing home range strategies we have hypothesized. To evaluate how resource depression can affect home ranges developed under each strategy and their distribution on a landscape, we modified the models so that resources in patches selected for simulated home ranges were devalued. We then used these modified models to simulate home ranges on each of five simulated landscapes differing in their distributions of resources. By comparing these home ranges to those developed using models without resource depression, we develop hypotheses about how social interactions and resource depletion, in addition to the spatial distribution of resources, affect the structure and spatial distribution of home ranges of animals inhabiting a landscape.

Our purpose was to learn how optimal use of spatially distributed resources might underlie the collection of movements and behaviors that ultimately define an animal’s home range (Powell, 2000), and how effects of animals on their resource base might influence the distribution of home ranges on a landscape. Accordingly, our models emulate the selection of patches by an individual animal over a time period, and do not depict the animal’s day-to-day time budget, movements, or foraging. Our models can make predictions about home ranges that differ from traditional depictions based on movements, particularly on disjunct resource distributions where selected patches may not form a contiguous area. We emphasize that our models are not intended to portray the movements of animals within their home ranges but rather the reasons for those movements, i.e., the resource-bearing patches between or within which the animals move. To that end, we defined a home range as the patches an animal selects to use (Powell, 2000). In the following sections we describe the concepts underlying our models and the resulting design of our simulations. We then present the results of our simulations, evaluate these results, and from them develop testable predictions of home range behaviors.

Section snippets

Patches

Patches can be defined in two ways. Traditional ecological understanding is that a patch is a discrete area having some internal characteristic that distinguishes it from its surroundings (Wiens, 1995). Such patches vary in size and shape but are defined by discrete, internally homogeneous resource values. True spatial distributions of resources are often continuous, however, and defining patches through traditional patch classification can be arbitrary, if not impossible (Mitchell and Powell,

Model MR, maximizing resource density

Animals that use patches randomly on a landscape will, on the average, accrue resources within a home range at a rate proportional to the mean availability of resources per patch for that landscape. A reasonable hypothesis is that a selective animal should choose high quality patches for its home range, exceeding mean availability of resources by as much as possible, until adding more patches begins to reduce this difference. Therefore, a home range that maximizes the density (V′/area) of

Landscapes

We simulated five landscapes on a 109×131 matrix of cells, or patches. V for each patch was a value between 0 and 1 inclusive, and the means and standard deviations of V for the five landscapes were approximately equal (Table 1), although maintaining the same variance for highly clumped values of V was not possible. We constructed the landscapes to differ in spatial continuity (i.e., covariability between neighboring patches), which we quantified using Moran’s I (Cliff and Ord, 1981). Moran’s I

Results and discussion

We discarded 117 of the 3,000 simulated home ranges because they exceeded the size or iteration limits; the number of home ranges discarded did not vary consistently across simulated landscapes or home range models (Mitchell, 1997). For all home range parameters we discuss, the magnitude of differences between resource-maximizing (MR and MRD) and area-minimizing (MA and MAD) home ranges depended on the arbitrarily-set minimum resource threshold. Changing the threshold alters how home ranges

Acknowledgements

M.L. Gumpertz, and M. Fortin, provided valuable statistical advice. J.G. Kie, J. Fryxell, L.L. Rogers, and 4 anonymous reviewers provided valuable comments on early versions of the manuscript.

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