Allen, Max L.

Norton, Andrew S.

Roberts, Nathan M.

Van Deelen, Timothy R.

Illinois Digital Environment for Access to Learning and Scholarship Repository

Maximilian L. AllenNathan M. RobertsAndrew S. NortonTimothy R. Van DeelenUsing Bayesian State-Space Models with Age-At-Harvest Data to Estimate Black Bear Abundance and Demographic Parameters INTRODUCTIONPopulation estimates are essential for making decisions about the management and conservation of many species,but are difficult or expensive to obtain across large geographical scales (Skalski et al. 2005). This is particularly trueof mammalian carnivores, which are cryptic and difficult to count directly. Consequently, carnivore managers oftenbase their population estimates on extrapolations from small data sets and adjust harvest quotas based on subjectiveopinion from the public and experts (Hristienko et al. 2007). The importance and challenges of estimating wildlifepopulations has led to many different estimation methods (Skalski et al. 2005). Using age-at-harvest data may bemost practical across large scales (Skalski et al. 2005, Norton 2015).Several statistical models have been developed that use age-at-harvest data to accurately estimate populations.Models based on frequentist statistics include statistical population reconstruction (SPR) that integrate hunter effort(e.g., Skalski et al. 2011), and state-space models that integrate food availability (e.g., Fieberg et al. 2010). Modelsbased on Bayesian statistics include data augmentation models that are similar to SPR and integrate tag recovery(e.g., Conn et al. 2008), and state-space models that integrated known-fate survival data (e.g., Norton 2015). To datethere has not been a model developed that creates accurate estimates without integrating auxiliary data, whichmakes it necessary for large field projects to collect demographic data.Black bears (Ursus americanus) are a K-selected and widely distributed solitary carnivore (Garshelis et al.2016). In Wisconsin, black bears are a popular game animal whose population and harvest have increased over thelast few decades. The Wisconsin Department of Natural Resources (WDNR) manages bear populations over a 35day season in 4 distinct zones with unique quotas and hunting regulations. Most of the bear population resides in thenorthern half of the Wisconsin (zones A, B, and D). Since 1985, the WDNR has estimated bear populations using adeterministic accounting model (MacFarland 2009). However, an independent capture-recapture estimate generatedfrom tetracycline marking found that the current model underestimated the population size by nearly 2/3(MacFarland 2009). Independent population estimates have allowed the WDNR to more accurately assess the blackbear population in the state (MacFarland 2009), but these are expensive and often conducted years apart.METHODSRESULTSSUMMARY• Several rigorous statistical approaches that use age-at-harvest and auxiliary data to accurately estimate populations haverecently been developed, but these are often dependent on a) accurate prior knowledge about demographic parameters ofthe population, b) auxiliary data, and c) initial population size. We developed a two-stage state-space Bayesian model forpopulations with age-at-harvest data, but little demographic data and no auxiliary data available.• The posterior estimates for abundance and demographic parameters from our state-space models using age-at-harvestdata were reasonable for our statewide estimate and for each management zone, and our statewide model estimates werealso similar to an independent estimates from tetracycline marking and the trend of catch-per-unit-effort for the state.• Our model was also robust to bias in the prior distributions for all parameters, except for reporting rate, and was robust toeven extreme bias in initial population size.• The Bayesian state-space modelling approach allows the modeler to transparently provide biologically supportedinformation and constraints on parameters as priors, but the models use these as a starting point and the posterior valuesare not dependent on the prior values provided. Drawbacks of Bayesian models are that they can be more complex anddifficult to comprehend and more computationally intensive to implement than simpler models.• The Bayesian state-space models using age-at-harvest data appear effective for making zone-specific abundanceestimates for the management of harvested species, and could result in better decision-making about populations andharvest quotas, and lead to more effective monitoring and management.• Our state-space model created a precise estimate of the black bear population trend in Wisconsin based on age-at-harvestdata and potentially improves on previous models by using little demographic data, no auxiliary data, and not beingsensitive to initial population size.The posterior estimates for abundance and demographic parameters were reasonable foreach management zone. The abundance estimates indicated a decreasing trend in zone A,and a generally stable trend in B, C and D. The posterior means and distributions wereimproved from the prior distributions for litter sizes, initial population size, harvestseason survival, and non-harvest season survival, with a notable increase in precision forthe survival values.Our state-space model consisted of two process models whose likelihoods were jointly modeled(Buckland et al. 2004, Norton 2015). The population process model was based on the unobserved(latent) population state process (that progresses from the initial state density to sub-state transitionaldensities [hunting season, non-hunting season, recruitment]), and the observation state process wasbased on observed harvest data (Newman et al. 2009, Norton 2015). We used Markov Chain MonteCarlo (MCMC) methods to approximate posterior distributions and based our inference on posteriorsummaries of the MCMC samples.The goal of our model was to estimate abundance (N) of the black bear population immediatelyprior to the hunting season. We used black bear harvest data including total annual observed harvest(O), and the number of harvested animals with known age and sex (C) in our observation stateprocess.In our statewide model, we used 8 years (2009-2016) of harvest data, and based the initialpopulation size on the WDNR estimate from 2009. In our models for each management zone, we used6 years of harvest data (2011-2016), and based our initial population sizes on independent capture-recapture estimates generated from tetracycline marking from 2011 (NA = 6,009, NB = 4,468, NC =4,841, and ND = 6,425). We assumed that the harvest was proportional to the population among ageclasses after visually assessing the harvest proportion by age class over 30 years and finding similarproportions despite increases in harvest over time.We used reasonably informative prior distributions for the demographic parameters of the model (seetable to right). Our population process model was constructed as a two-sex, ten-stage (1.5-year-olds,2.5-year-olds, …, 9.5-year-olds, >10.5-year-olds; excluding 0.5-year-old cubs that cannot beharvested legally) population projection matrix (Caswell 2001). Age-specific fecundity values (asnumber of 1.5-year-olds entering the model, per female) were calculated as:!"#$ = &'$ × )*$ × +,-'. × +,-'-and multiplied by the number of females in each age class of the previous year to determine thenumber of 1.5-year-olds entering the population and by SPs to determine the proportion by sex. Thenumber of 0.5-year-olds were back-calculated in the population model as:/0.2,4,567 = /7.2,4,5 / +,-'-We fit our models in Program R (R Core Team 2017) using JAGS (Plummer 2003) and the Rpackage rjags. We ran 220,000 iterations with 3 chains, a burn-in of 20,000, and a thinning rate of 4.We visually assessed the convergence and mixing of the chains, and then used Gelman-Rubinstatistics on the annual abundance estimates to determine convergence.LITERATURE CITEDAn early draft of a Directed Acylic Graph, highlighting the prior models (population projection) in blue and the harvest observation model in red.Our population projection model, using the prior distributions for each variable.Right: The boundaries of the four black bear management zones in Wisconsin.Below Annual harvest of black bears in each management zone of Wisconsin from 1986-2017.Because demographic information from Wisconsin was sparse, we set up a quasi-study area with similar habitats across North America (above left).We then reviewed all peer-reviewed literature from this area to create the prior distributions for our model (above right).We ran models in 3 steps: A) Our statewide state-space model (as proof of concept).B) To understand how the prior values affected the accuracy of our model, we compared the results of our statewide state-space model to models run with 10% positive and negative bias in 9 individual parameters to determine Percent Relative Change (PRC).C) Our zone-specific state-space models for each of the management zones. For full results, including estimates of demographic parameters see Allen et al. (2018).The statewide population estimates (from the SSM) indicated a decreasing trend in the blackbear population from 2009 to 2017 and reasonable demographic parameters. The annualvariation and 95% credible intervals were similar, but increased slightly in the final two yearsof estimation. The population abundance estimate for 2011 was visually similar to theindependent tetracycline estimate for 2011. The population trend estimate had a significant andstrong correlation with CPUE (df = 8, R2 = 0.93, p <0.0001).Based on the PRC values, our population model estimates were mostsensitive to potential bias in the reporting rates (10% underestimate = PRCof 7.33). The model was robust to potential bias in all other parameters,which had PRCs of < 2.00. Since most population models are sensitive to theinitial population estimate, we also assessed the sensitivity to extreme bias inthis parameter. A 50% underestimate had a PRC = -7.60%, and a 50%overestimate had a PRC = 11.88%.Allen, M. L., A. S. Norton, G. Stauffer, N. M. Roberts, Y. Luo, Q. Li, and T. R. Van Deelen. 2018. A Bayesian state-space model using age- at-harvest data for estimating the population of black bears (Ursusamericanus) in Wisconsin. Scientific Reports 8:12440.Buckland, S. T., K. B. Newman, L. Thomas, and N. B. Koesters. 2004. State-space models for the dynamics of wild animal populations. Ecological Modelling 171:157–175.Caswell, H. 2001. Matrix population models: construction, analysis, and interpretation. 2nd edition. Sinauer Associates.Conn, P. B., D. R. Diefenbach, J. L. Laake, M. A. Ternent, and G. C. White. 2008. Bayesian analysis of wildlife age-at-harvest data. Biometrics 64:1170–1177.Fieberg, J. R., K. W. Shertzer, P. B. Conn, K. V. Noyce, and D. L. Garshelis. 2010. Integrated population modeling of black bears in Minnesota: implications for monitoring and management. PLoS ONE 5.Garshelis, D. L., B. K. Scheick, D. L. Doan-Crider, J. J. Beecham, and M. E. Obbard. 2016. Ursus americanus. The IUCN Red List of Threatened Species 8235:IUCN 2008: T41687A114251609.Hristienko, H., J. E. McDonald, and J. E. M. Jr. 2007. Going into the 21st century: a perspective on trends and controversies in the management of the American black bear. Ursus 18:72–88.MacFarland, D. M. 2009. Population estimation, habitat associations and range expansion of black bears in the upper midwest. University of Wisconsin.Newman, K. B., C. Fernández, L. Thomas, and S. T. Buckland. 2009. Monte Carlo inference for state-space models of wild animal populations. Biometrics 65:572–583.Norton, A. S. 2015. Integration of harvest and time-to-event data used to estimate demographic parameters for white-tailed deer. University of Wisconsin, Madison.Plummer, M. 2003. JAGS: A Program for Analysis of Bayesian Graphical Models Using Gibbs Sampling. Proceedings of the 3rd International Workshop on Distributed Statistical Computing  3:1–10.Skalski, J. R., J. J. Millspaugh, M. V. Clawson, J. L. Belant, D. R. Etter, B. J. Frawley, and P. D. Friedrich. 2011. Abundance trends of American martens in Michigan based on statistical population reconstruction.Journal of Wildlife Management 75:1767–1773.Skalski, J. R., K. E. Ryding, and J. J. Millspaugh. 2005. Wildlife demography: analysis of sex, age, and count data. Elsevier Academic Press.

Using Bayesian state-space models with Age-At-Harvest data to estimate black bear abundance and demographic parameters.

https://www.ideals.illinois.edu/bitstream/handle/2142/101872/AllenIBA_Ljubljana2018Poster.pdf?sequence=2&isAllowed=y

Using Bayesian state-space models with Age-At-Harvest data to estimate black bear abundance and demographic parameters.

Abstract

Similar works

Full text

Available Versions

Illinois Digital Environment for Access to Learning and Scholarship Repository