The package is in branch Vital-as-data-core and analysis can be found in vital-rate-as-data
White tailed deer (Odocoileus virginianus) is one of the most important game species in north America. However, over abundance of this species also cause problems, including over grazing one landscape, conflict with human and spread diseases like Crown Waste Disease. In suburban area, rapid growth of deer population increased the probability of human-deer conflict. In Chicago suburban area, increasing deer density promote concern of deer-vehicle collision (Etter et al. 2000, Jones and Witham 1995 ). In 1992, 2 human were killed and 145 injured due to deer-vehicle collision (Etter et al. 2002, Etter 2001). Over growth of deer population also raise public health concern of zoonoses in this area (Miller 2001, Etter 2001).
Researchers and Officials from Illinois Department of Natural Resources started to try intensive culling as population control method in Cook and DuPage Counties, Illinois (Complex 1) from 1992 to current. Reconstruction of population dynamic is needed to evaluate this method and its outcome.
Various method for reconstruction populations using multiple data sources were proposed recently (Iijima et al. 2013, Wheldon et al. 2013). Bayesian method is one of the most promising approach. Stochastic matrix projection model is one of the most famous projection models for population with age structure (Leslie 1945). Bayesian framework of reconstruction allow us to fuse multiple data sources with different quality. Here we proposed and tested a Bayesian framework reconstruction modified from Wheldon et al. 2013 in order to understand the population’s dynamic under culling.
The study area encompassed 334,934 ha in the western suburbs of the greater Chicago metropolitan area, in Cook and DuPage Counties, Illinois. Land cover was dominated by urban/developed land (57.5%) and associated urban grassland (14%). Other cover types included forested/woodland (14.2%), cropland (4.9%), rural grassland (4.1%), wetland (3.3%), open waters (1.8%), and barren/exposed land (0.2%; Illinois Department of Natural Resources 1996).
Population of study occupied complex 1 included Argonne National Laboratory (ANL), Waterfall Glen (WFG) and Woodridge (WDG) forest preserves located in southeast DuPage County, and Black Partridge forest preserve (directly adjacent to Woodridge) located in southwest Cook County, Illinois. Habitat area of this complex remained constant through culling years.
Prior to 1992, no deer were legally harvested in Complex 1. Culling in Complex 1 started in WFG in 1992. In 1993 and 1994, we culled deer from WFG and WDG, and beginning in 1995 we culled deer from ANL and all other forest preserve in Complex 1. Deers were culled during October–April by sharpshooting, and capture and euthanasia as described by (DeNicola and Williams 2008 Etter 2001). Antlerless deer were prioritized, but deer were culled on a first opportunity basis.
Culled deers were aged into 8 age classes for female, 3 for male and recorded from 1992 to 2006. Initial estimation and their uncertainty which also serves as prior mean of vital rates and used to determine hyperparameters came from previous study in the same area (Etter 2001). Initial estimation of harvest rate was the ratio between previous reconstruction study in the same population and total culling count, while we assume harvest rate of fawns was half of harvest rate of adult since the protocol of culling was strongly biased toward non-fawns.
Parameters of interest are time and age specific fecundity and survival,
as well as harvest rate and population size of female deers in study
area. We later on use for culling counts at age a and time
t,
for survival,
for fecundity,
harvest rate,
for latent living population size after culling,
for the post harvest aerial count estimation,
for the
aerial count detection,
). Underline of
certain parameters means the best estimation and data we have currently
that will be used in the model.
We assume a time inhomogeneous stochastic proportional harvest. Further,
harvest rate of fawns (age 0.5) is assumed to be different from yearling
and adults (age >0.5). We used a diagonal harvest matrix
to model the harvest in the projection to seperate
harvest and other mortality. Growth of living individuals are projected
using standard stochastic Leslie matrix model (Leslie 1945). Leslie
matrix contains
and
was noted by
.
Since harvest happened after reproduction, we left multiply the harvest
matrix. Vital rates’ distribution as described in the prior part of the
Bayesian framework.
Formally the projection model of harvest count vector
is given by:
We also have aerial count data as estimation of post-harvest population with imperfect detection. Model for aerial count is given below.
In which I is identity matrix. Note that
solves the living
individual after culling that undergone reproduction at time t+1. Also
note that baseline year should be trait differently since there is no
L0 needed. A graphical illustration of the dynamics is given in
Fig below.
We generally followed Bayesian reconstruction framework of (Wheldon et al. 2013), except we had a new set of parameters regarding the harvest rate and aerial count. Further, likelihood used Poisson distribution rather than log normal since we have 0 harvests. Reconstruction is equivalent of estimating vital rates s, f, H and population population counts X. We used the same 4 level setting to count for uncertainty of initial estimation.
Relationship between data and parameters were shown in Fig below. Note that alpha and beta are hyperparameters which encode the prior knowledge we have for the uncertainty of certain vital rate or culling count. Initial estimations of survival (logit transformed), fecundity (log transformed), sex ratio at birth (SRB, logit transformed), harvest rate (logit transformed), as well as aerial count probability (logit transformed) were used as corresponding transformed prior mean of the normal distribution whose sigma was invGamma distributed with parameter alpha and beta. Culling counts as well as aerial counts served in the likelihood part of the model, prediction of the projection model served as expected value of the Poisson distribution to evaluate the likelihood.
Determination of hyperparameters alpha and beta for vital rates except for harvest rate were based on previous study’s error estimation, use the same method in (Wheldon et al. 2013), but more conservative. Harvest rate’s hyperparameter were set to be enough conservative that has .95 quantile greater than 2. Detail hyperparameter setting is shown in
| parameters | Survival | Fecundity | SRB | Harvest | Aerial detection |
|---|---|---|---|---|---|
| alpha | 1 | 1 | 1 | 1 | 1 |
| beta | .05 | .01 | .05 | .05 | .05 |
We draw samples from posterior distribution using Markov chain Monte
Carlo (MCMC) method and diagnose using R package coda
. The algorithm generally followed Wheldon et al. 2013. We
updated variance (sigmas) from conjugated full conditional
distributions as proposed density in Metropolis-Hastings algorithm.
Other vital rates including culling rate were updated using
Metropolis-Hastings steps and a symmetric normal proposal. Algorithm was
tuned by hand to achieve a reasonable acceptance rate. Follow Wheldon et
al., “iteration” was defined as one complete sweep through all age- and
time-specific parameters and variance parameters.
After we reconstruct the culling dynamics and harvest
rate
, we solve the living individual after culling
using eqn:
The model is implemented in R 3.6.0 modified from package
popReconstruct (Weldon et al. 2013) Source code is available on
GitHub under MIT
license.
In this frame work, we use DIC as model selection criterion (Gelman 2002). Lower DIC means higher support of the model by data.
Total 15 years of culling count was used in this study (1992-2006, 1992
as baseline). In 15 years total 3827 individuals were culled. Fecundity
and survival estimation was homogeneous for all age
| Culling counts | mean | Standard Error |
|---|---|---|
| Absolute Difference | 3.38 | 1.65 |
| Posterior Standard Deviation | 3.25 | 0.151 |
| Aerial counts | ||
| Absolute Difference | 2.25 | 0.33 |
| Posterior Standard Deviation | 14.20 | 0.240 |
Four models were tested to chose the best supported:
- Four Harvest full model: Survival and Fecundity rates are time sex and age dependent, with 4 harvest rates: female fawn, femal adult male fawn and male adult which are time dependent.
- Three Harvest full model: Survival and Fecundity rates are time sex and age dependent, with 3 harvest rates: fawn, femal adult and male adult which are time dependent.
- Solely Density Dependent model: Survival and Fecundity rates are sex age and density but not time dependent, with 4 harvest rates, which are time dependent.
- Three age classes: All similar to 1 but fecundity and survival has only 3 age classes for both female and male.
| Model | effective n parameters | DIC | Best Model |
|---|---|---|---|
| 1 | 187.11 | 649.38 | |
| 2 | 196.3 | 631.48 | * |
| 3 | 108.81 | 835.74 | |
| 4 | 148.36 | 732.35 |
After reconstruct the culling dynamic, we solved the total living individuals’ posterior distribution as described by eqn.1. Results were shown in Fig below.
Density dependency is one of the key features controlling population growth (ueno 2010). To address whether there exist density dependency we did linear regression between vital rates and reconstruct living population size before reproduction (i.e. living individual counts after culling of the previous year). Results were summarized in Table below.
| Age | 0.5 | 1.5 | 2.5 | 3.5 | 4.5 | 5.5 | 6.5 | 7.5 |
|---|---|---|---|---|---|---|---|---|
| Fecundity | * | * | * | * | * | * | * | |
| p-value | 0.00203 | 2.79e-5 | 2.27e-5 | 2.82e-4 | 7.71e-4 | 1.84e-3 | 0.0147 | 0.0343 |
| adj R^2 | 0.525 | 0.762 | 0.770 | 0.654 | 0.593 | 0.532 | 0.353 | 0.265 |
| beta | -6.32e-05 | -1.01e-03 | -6.37e-04 | -4.96e-04 | -4.09e-04 | -3.28e-04 | -2.70e-04 | -2.49e-04 |
| SE beta | 1.61e-05 | 1.55e-04 | 9.540e-05 | 9.81e-05 | 9.16e-05 | 8.26e-05 | 9.49e-05 | 1.04e-04 |
| Survival | * | * | ||||||
| p-value | 0.365 | 0.150 | 0.886 | 0.130 | 0.514 | 0.00705 | 2.63e-4 | 0.0546 |
| adj R^2 | 0 | 0.0949 | 0 | 0.112 | 0 | 0.423 | 0.657629475 | 0.213787859 |
| beta | -8.10e-05 | 2.64e-04 | -9.31e-06 | -1.69e-04 | 5.60e-05 | -4.42e-04 | -2.12e-04 | -9.33e-05 |
| SE beta | 8.61e-05 | 1.72e-04 | 6.39e-05 | 1.04e-04 | 8.32e-05 | 1.36e-04 | 4.16e-05 | 4.38e-05 |
| *: p:0.01 |
For males:
| Age | Fawn | Yearling | Adult |
|---|---|---|---|
| Survival | * | ||
| p-value | 0.000441 | 0.402 | 0.0805 |
| adj R^2 | 0.628 | 0 | 0.169 |
| beta | -6.53e-04 | 7.731e-05 | 1.55e-04 |
| SE beta | 1.36e-04 | 8.89e-05 | 8.17e-05 |
| *:p:0.01 |
By conducting intensive culling in Chicago’s suburban area, we successfully controlled the overabundant deer population in this area. Density dependency may reduce our ability to control population when size is small. Our reconstruction shows that survival has no density dependency probability since the source of mortality stays similar before and after intensive culling. Fecundity shows a weak negative density dependency for older individuals. This suggest in lower population density, further culling of older individual is needed to keep the same amount of control outcomes.
We assumed the population is closed to female which is generally appropriate since deer is male dispersion. But if we are willing to include sex ratio in this reconstruction we may need to consider migration rates from the population for males. Sex ratio at birth is also not considered here for a female reconstruction, however, it is interesting to evaluate the sex ratio change under intensive culling.
The statistical model we modified can also be generalized to reconstruct proportional harvesting dynamics using uncertain estimation of vital rates and age-at-harvest data. It can also include relationships between vital rates with environmental variables.
All the supplementary figures and result summaries, as well as the source code of this report are available here.