Forging a Bayesian link between habitat selection and avoidance behavior in a grassland grouse

Focal species

The Lesser Prairie-Chicken is a medium-sized grouse endemic to, broadly, the shortgrass prairie ecosystem of the south-central United States, where it is found only in southwestern Colorado, western Kansas, northwestern Oklahoma, the Texas panhandle, and eastern New Mexico. As with almost all open-country grouse of temperate environments, the Lesser Prairie-Chicken forms leks at which a cluster of males (in this species, usually 5–12 individuals) display vigorously and female visit to assess males with an intent to secure sperm to fertilize her eggs, which she will lay and raise without male help. Outside of male lekking (mid-March to mid-May) and female nesting (late April to early July), birds congregate is small flocks to forage, at times on grain remains in farmed fields but typically, as through the rest of the life cycle, restricting themselves to native prairie. Accordingly, this species has three distinct aspects of habitat selection: general occurrence, lek placement, and nest placement.

Data

Lesser Prairie-Chicken were tracked at two study sites, one in Roosevelt County in east-central New Mexico, U.S.A., the other in Beaver, Harper, and Ellis Counties in northwestern Oklahoma, U.S.A. Birds tagged with VHF transmitters were tracked from April 1999–March 2006 in New Mexico and from March 1999–July 2013 in Oklahoma^11,12,13,14. Study periods differed chiefly because of funding, which for New Mexico was insufficient after 2006. The study sites differ markedly in land tenure history. Parcel size in New Mexico averages 1300 ha versus 180 ha in Oklahoma¹¹. The difference largely stems from settlement patterns over the past two centuries. New Mexico was part of the Spanish land grant system, which tended to yield huge parcels. In our study area, parcels approach 8 km² (> 1900 acres). By contrast, during the “land rush” era of the late 19th Century most of northwestern Oklahoma was parceled into 65-ha (160-acre) plots as part of the United States’ Homestead Act. Smaller parcels translate to a higher density of roads, fences, buildings, and powerlines¹¹.

Radiotracking typically yielded a triplet of coordinate readings, from which we had to triangulate a grouse’s location. We estimated latitude and longitude using a maximum likelihood estimator (MLE), although it some cases the MLE algorithm failed to converge. If it failed, we instead used the Andrew and Huber methods¹⁵. R code for the estimation procedure can be found at https://github.com/henry-dang/triangulation/blob/master/lenth_triang.R.

From these data we used kernel density methods (R package ks¹⁶) to estimate annual home ranges (235 in New Mexico, 263 in Oklahoma). Tracking data included lek (12 in New Mexico, 23 in Oklahoma) and nest (122 in New Mexico, 128 Oklahoma) locations. For home range centroids, the outer contour of home ranges, leks, and nests, we estimated distance to seven anthropogenic features: roads (highways, primary, and secondary roads only; small farm roads or one-lane gravel roads were excluded), powerlines (overhead only, with buried or trunk lines excluded; https://hifld-geoplatform.opendata.arcgis.com/datasets/electric-power-transmission-lines), oil wells, gas wells (for both types of wells, http://www.occeweb.com and http://www.emnrd.state.nm.us/ocd), outbuildings (barns, grain silos, poultry houses, and similar large structures; chiefly the TIGER database), and fences (Bureau of Land Management) in both states, plus private houses in New Mexico and railroad tracks in Oklahoma. We placed 2000 random points on each study area to estimate distances to each of these same anthropogenic features, which provided an estimate of feature density on the landscape.

Analyses

The initial step was to estimate the probability of a grouse occurring a certain distance from a feature. We treated New Mexico and Oklahoma data separately, giving us a replicate assessment because these populations have been isolated from each other for > 100 years¹⁷ and, as noted above, land tenure history differs strikingly between the states¹¹. We estimated probabilities of grouse occurrence, π_i, via a Bayesian model with binomial likelihood and flat prior (i.e., no assumption of the central tendency of occurrence probability at a given distance from a feature):

y_i ~ binomial (π_i, n) with y_i the cumulative count of grouse at distance i (i.e., the data) and n the total number of home ranges, leks, or nests. Distances, i, were binned to the smallest extent possible, from 10 to 100 m, to allow the Markov chain Monte Carlo (MCMC) algorithm to converge in a reasonable number of iterations (e.g., < 100,000). (In Bayesian statistics, MCMC is used to build a posterior probability distribution from the product of the prior and the likelihood without having to integrate the often intractable denominator in the Bayes’ theorem formulation. The algorithm “allows one to characterize a distribution without knowing all of the distribution’s mathematical properties by randomly sampling values out of the distribution”¹⁸.) We used the same process for random points to estimate probability of feature occurrence at a given distance, the idea being to establish a null expectation of density of class in each study area. We plotted resultant bird and random curves to get an idea of whether grouse avoided the feature (Fig. 3A). If grouse neither avoided nor preferred a feature, then its curve and the random curve would trace similar arcs (within credible intervals; see below). If instead grouse avoided a feature, then its curves would intersect the random one at some distance from the origin (Fig. 3A); the converse holds for preference.

To assess if and to what extent grouse avoid a feature, we extended the Bayesian model to calculate a difference in probabilities between random points and grouse occurrence (Fig. 3B¹⁹). (At root, our method resembles a resource selection ratio²⁰, but it was derived independently within a Bayesian framework.) We used the posterior probability distributions of these differences to search for the point at which grouse and random curves converged, the threshold distance (T). Our algorithm was.

1. (1)

   calculate differences per bin, π_Ri–π_Gi, with associated posteriors;

2. (2)

   use the extent (q) to which a posterior distribution overlapped 0 to estimate the probability (p) that the threshold occurred in bin k, with p_k = 1 − 2|q_k− ½|;
3. (3)

   locate the apparent crossover region given calculated differences; usually the crossover was unambiguous (Fig. 3C), but if credible intervals overlapped 0 before the bulk of the distribution crossed (Fig. 3D), we conservatively used the region with the shorter distance so as not to overestimate distance;

4. (4)

   estimate T as a weighted average of probabilities p_k ± 500 m of apparent crossover, with p_k as weights and for the kth bin (small boxes in Fig. 3C,D).

This algorithm yielded an estimate of T as a posterior probability distribution, from which we calculated threshold distance as the median and uncertainty as the highest density credible intervals, the broadest probability density that includes the mode. (A Bayesian credible interval is calculated directly from the posterior distribution and states that, say, 95% of parameter estimates lie between upper and lower bounds.) If the lower interval overlapped or closely approached 0, we could reasonably conclude the feature was not avoided. We supplemented threshold analyses with a Bayesian estimate, with log-normal likelihood and flat prior, of prediction intervals for grouse occurrence relative to a feature. From these estimates we calculated a coefficient of variation to evaluate whether areas with higher feature density had greater uncertainty for distance estimates. Lastly, we compared feature density at each study site by means of a Bayesian analog to a two-sample t-test²¹, with t likelihood and flat prior.

Our final analysis assessed the extent to which avoidance of one feature might suggest preference for another or an instance of “co-avoidance”; i.e., it assesses colocation in a manner related to a cross-K function²². Available data were distance from a grouse (home range centroid only; sample size was otherwise too low) or random point to a given feature. We estimated distance, d, between one feature and another feature by means of the cosine rule:

where A is the distance from a grouse to the first feature, B is the distance from a grouse to the second feature, and θ is the angle between them (Fig. 4). We built a Bayesian model to estimate d for grouse (d_G) and for random points (d_R), with log-normal likelihood and θ as a flat prior. We used estimated distances to estimate a dispersion ratio, d_G/d_R, for which values less than one (i.e., below the lower credible interval of the estimate) implied a clumped distribution, values greater than one (i.e., beyond the upper credible interval) implied a dispersed distribution, and values between (i.e., within the credible intervals) implied a random distribution. The logic behind this ratio is that if avoidance of one feature was stronger than avoidance of another, then the ratio would increase: if B is unchanged, a decrease in A will increase d (Fig. 4), implying that strong aversion of B “pushes” organisms nearer to A as an effect of their preferences. The reverse situation implies that an organism is equally averse to the features or prefers one feature to another even if both are avoided (i.e., they are “lesser evils”).

Each of our Bayesian models was coded in JAGS and run via package rjags²³ in R, with associated R code to support hierarchical flow (i.e., to manage data between some stages of the analyses).

Source: Ecology - nature.com