Relating Populations and Risk to Habitats using Resource Selection Functions. Lyman McDonald January 11-12, 2003

1. Relating Populations and Risk to Habitats using Resource Selection Functions. Lyman McDonald January 11-12, 2003

2. Reference Boyce, Mark S. and Lyman L. McDonald. 1999. Relating Populations to Habitats using Resource Selection Functions. Trends in Ecology and Evolution 14: 268-272.

3. Summary: Resource Selection Functions • Habitat use can be characterized by resource selection functions (RSFs) that are proportional to the probability of an area being used. • Consider two procedures that have been developed to relate RSFs to population density under certain assumptions. • Interface RSF models with geographical information systems (GIS). • map the probability of use, and ultimately populations, across landscapes.

4. Summary: Resource Selection Functions • RSFs offer a quantitative characterization of resource use. • RSFs can accommodate virtually any type of resource being selected including both categorical and scalar variables. • RSF models easily accommodate spatial structureand can be interfaced with GIS to facilitate rapid analysis and use of remote sensing data. • Highlight assumptions for estimating population size. • Suggest some ways that violations of assumptions might be overcome.

5. RSFs are proportional to the probability of use of a resource unit • If we know the distribution of RSF values among habitats in an area of known population size, we can estimate the relative density of animals by habitat type, under the assumption that all units are equally available. • If similar use patterns can be assumed to occur in another area, we can predict the population by applying densities by habitats. • Example: extrapolate the number of wolves in the northeastern United States. • In an area occupied by wolves, a RSF was developed to estimate the relative probability that a unit would be used by a wolf pack. • When the model was applied to an unoccupied area, units that yielded predictions above 50% were defined to be potential wolf habitat.

6. Estimating population size using RSFs • Assume that our resource selection function, w(x), can be characterized by an exponential or logistic regression model: • e.g., w(x) = exp(β0 + β1x1 + β2x2 + . . . + βkxk) • where the xidenote i = 1, ..., k independent habitat variables, and the βis are selection coefficients. • Estimate the coefficients in the model, based on samples or census of • used and unused units, • used and available units. • Unused and available units.

7. Assume populations size = 100 animals w(x) = Expected Habitat Type Area=A(x) rel. prob. Density Selection I 1.0 ha 0.1 II 1.0 ha 0.2 III 1.0 ha 0.3 IV 1.0 ha 0.4 TOTAL 1.0 Assume that all habitat types are equally accessible. What are the relative values of each Habitat Type?

8. Assume population size = 100 animals • Value = Relative Expected Expected • Type A(x) w(x) A(x)*w(x) value Number Density • I 100 ha 0.1 10 0.25 25 0.25 ha • II 50 ha 0.2 10 0.25 25 0.5 ha • III 50 ha 0.3 15 0.375 37.5 0.75 ha • IV 12.5ha 0.4 5 0.125 12.5 1 ha • Total 1.0 40 100 • Questions. • What is the value of losing 10 ha of Type I habitat to development? • What is the value of losing 10 ha of Type II habitat to development? • Would you trade 2.0 ha of Type IV habitat for 10 ha of Type I?

9. V = 5 V = 15 V = 10 Volume = 10 The volume under the surface is proportional to the relative value of the habitat.

10. Assume population size = 100 animals • Value = Relative Expected Expected • Type A(x) w(x) A(x)*w(x) value Number Density • I 100 ha 0.1 10 0.25 25 0.25 ha • II 50 ha 0.2 10 0.25 25 0.5 ha • III 50 ha 0.3 15 0.375 37.5 0.75 ha • IV 12.5ha 0.4 5 0.125 12.5 1 ha • Total 1.0 40 100 • Questions. • What is the value of losing 10 ha of Type I habitat to development? • 0.1*10 = 1.0 • What is the value of losing 10 ha of Type II habitat to development? • 0.2*10 = 2.0 • Would you trade 2.0 ha of Type IV habitat for 10 ha of Type I? • Value of 1.0 ha of Type IV = 0.4*2.0 = 0.8 • Value of 10.0 ha of Type I = 0.1*10.0 = 1.0

11. Forage Site Selection by a Pair of Northern Spotted Owls • Determine a random sample of sites used by a radio tagged pair of owls. • Determine a sample of “available” foraging sites with a GIS. • Fit a “foraging site selection function” based on variables like: • habitat type (old growth forest, clearcut, etc.) • distance to nest • density of roads • etc.

12. Results: Models • Top models (hypothetical):

14. Candidate Units to Clearcut 1 2 3

16. “Risk” Indexes • Conclusion: Unit #1 has the least “risk” of the three units.

17. Estimating value of habitat or population size using RSFs • For the ith habitat type with area A(xi) and habitat vector xi the relative value is: • U(xi) = w(xi)A(xi)/w(xj)A(xj) • The relative volume under the RSF surface for each type. • If every habitat unit has a unique value of the variables xi then the sum is over the number of units in the study area. • The number of animals expected in the ith habitat type is: • Ni= N(U(xi)), and • density of animals, D(xi), in the ith habitat type is obtained by dividing the expected number by the area: • D(xi) = N*U(xi)/A(xi).

18. Resource selection probability functions • A RSF permits us to calculate w, which is proportional to the probability of use for a resource unit. • A resource selection probability function (RSPF) is scaled so that we can calculate w*, the probability of use. • If the used units are individual territories (or NON-OVERLAPPING home ranges), we can sum the probabilities of use over an area to estimate total population size.

19. Estimating resource selection probability functions (RSPFs) • The easiest approach for estimating an RSPF, w*(x), is to use logistic regression on a census of used and unused units, where the selection function is modeled by a logistic function of k independent variables, xi, hypothesized to influence resource selection and βi are coefficients to be determined when fitting the model to data: • w*(x) = exp(β0 + β1x1 + β2x2 + . . . + βkxk)/[1 + exp(β0 +β1x1 +β2x2 + . . . +βkxk)].

20. Northern spotted owl nest site selection in the Pacific Northwest of USA. • Sample unit = 2-km2 circle surrounding an owl nest site, approximately the core territory for a pair of owls. • Potential predictor variables included measures of landscape pattern such as patchiness, isolation, contagion and fractal dimension from GIS. • Measure details of habitat within 2-km2 circles at 50 owl sites (from 445 known pairs, Pu= 50/445 = 0.1124 ) and 50 random locations out of 18079 possible 2-km2 plots (Pa = 50/15148 = 0.00663) (Meyer et al. 1998) • A RSPF can be estimated given these estimated sampling fractions and data on used and available units.

21. Model for the RSPF: Northern spotted owl nest site selection. • w* = exp[-ln(Pu/ Pa) + β0 + β1(OldGrowth) + β2(ElevRange)] • OldGrowth is the area within the 2-km2 circle in old-growth forest, and • ElevRange is the range in elevation at the site. • Shortcut: Estimate the coefficients, β0, β1, and β2 from a logistic regression between used sites (1) and available sites (0) under the assumption that a “small” proportion of resource units are used. • Pu/ Pa = (0.1124/.00663) = 16.95, -ln(Pu/ Pa) = -2.83 • β0 = -2.545 • β1 = 5.507 • β2 = 0.0046

22. Model for the RSPF: Northern spotted owl nest site selection . • Observed occupancy rates from long term study sites in western Oregon average 0.85. • Multiply this proportion by w*, to obtain the probability that a 2-km2 area is an occupied owl site. • (0.85)w* = (0.85) exp[-5.375+5.507(OldGrowth)+0.0046 (ElevRange)] • Sum over all units in a prediction of the future state of the forest to estimate the total predicted population size, etc. • Boyce, M.S., J.S. Meyer, and L.L. Irwin. 1994. Habitat-based PVA for the northern spotted owl. Pp. 63-85 in Statistics in Ecology and Environmental Monitoring, Otago Conference Series No. 2, University of Otago Press, Dunedin. • Meyer, J.S., L.L. Irwin and M.S. Boyce. 1998. Influence of habitat abundance and fragmentation on northern spotted owls in western Oregon. Wildlife Monographs No. 139, The Wildlife Society.

23. Assumptions of RSF-based population estimation or value of habitat. • Know the limiting factors that influence the distribution and abundance of the study organism. • Data are available on key predictor variables. • Organisms have free and ready access to available resource units. • When resource units are sampled, we presume that these are sampled randomly and independently. • RSFs and resources do not change during the study. • Can be relaxed if sufficient data are available to estimate RSFs repeatedly. • Effects of spatial variation, are the same in areas where predictions are being made.

24. Assumptions of RSF-based population estimation or value of habitat. • Available habitat is well defined. • Varying the size of the study area or excluding certain areas from the domain of the study can result in different models. • Often we obtain robust models that are relatively insensitive to variation in availabilities. • Models are robust with respect to habitats which are rarely used. • The model will change little if rarely used habitats are included or not included in the study. • In the spotted owl models, the amount of old growth forest was a good predictor even in quite different habitats

25. Changing availability of units. • We can model the relation between model coefficients and availabilities explicitly over a range of studies, thereby taking availabilities into account. • Develop a model for βs in the RSF as a function of availabilities of units. • This approach is possible for categorical variables. • Difficult or impossible for multiple continuous and discreet variables. • Understanding how RSF coefficients might change as resource availabilities change on the landscape is fundamental to the reliability of population projections.

26. Assumption of equilibrium population dynamics • If populations are changing rapidly, we cannot expect RSF models to hold constant. • Model RSFs at varying population densities to see how the coefficients vary in a density-dependent fashion. How do the coefficients change as the population is increasing? Decreasing? • Model mean population densities assuming that population fluctuations are attributable to stochastic fluctuations around some long-term average.

27. Other discussion. • The value of habitats is not necessarily based upon the total time of use. • Habitats used for sleeping might not be in short supply nor all that crucial for survival. • Access to water might be crucial but only a few minutes each day might be spent drinking. • Model the reproduction or survival of individuals as a function of habitat variables. • An understanding of the ecology of the species must be used in the interpretation of RSF models. • RSFs are statistical descriptions of the distribution and use of landscapes. • They do not necessarily help us to understand why organisms are where they are.

28. Other Discussion • RSFs take a broad or top-down perspective characterizing general patterns on the landscape. • RSFs offer a framework from which one can explore the ecological processes that shape distribution and abundance. • Interactions with other species, influences of a variety of physical and biotic factors, and the structure of the population can all be built into a RSF model explicitly. • RSFs can be constructed at alternative scales. • RSFs and GIS technology makes spatial modeling much easier.

29. Some Applications that have been made in conservation and ecological management • Model populations of spotted owls (Strix occidentalis caurina) in the Pacific Northwest of USA. • Anticipate future timber wolf (Canis lupus) populations in the Northern Great Lakes states and New England. • Base a population viability analysis of California gnatcatchers (Polioptila c. californica), and • Anticipate the distribution and abundance of grizzly bears (Ursus arctos horribilis) in the Selway-Bitterroot wilderness area of Rocky Mountains.

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