Population statistics for each year, summarized from all simulations under scenario 2

Model input and output for one example simulation






% confidence intervals

View or edit adult direct mortality counts


View or edit first-year direct mortality counts


View or edit adult WNS survival probability multiplicative reductions


View or edit first-year WNS survival probability multiplicative reductions


View or edit WNS reproduction probability multiplicative reductions




% confidence intervals

View or edit adult direct mortality counts


View or edit first-year direct mortality counts


View or edit adult WNS survival probability multiplicative reductions


View or edit first-year WNS survival probability multiplicative reductions


View or edit WNS reproduction probability multiplicative reductions





Population statistics for each year, summarized from all simulations under scenario 1

Model input and output for one example simulation





Population statistics for each year, summarized from scenario 1 and 2 simulation differences

Tab definitions

This information tab contains definitions of all tabs, a brief description of the BatTool, definitions of model inputs and model outputs, a description of the demographic matrix model, instructions for saving and loading model inputs and outputs, and a bibliography with selected references. The glossary contains definitions for abbreviations used.

The model inputs tab contains all inputs to the BatTool demographic model for population projection as well as a few options for plotting and calculating summary statistics. See model inputs definitions below.

After the Run Scenario 1 button has been clicked, the scenario 1 results tab will show a plot, a table of summary statistics of the simulations from the model, and one example simulation (and for scenario 2 results similarly). The scenario comparison results tab shows a plot and table of summary statistics for the difference between simulations for scenarios 1 and 2.

BatTool description

The BatTool R package provides functions implementing a demographic matrix model with associated Shiny graphical user interface. The user can project the abundance of a bat population of several species using its estimated rate of population growth, including stochasticity and the effects of stressors in the form of direct mortality and reductions to vital rates. Stressors can model the impacts of white-nose syndrome, wind energy mortality, extreme weather events, flying hazards, etc. The model runs an ensemble of simulations to include all sources of uncertainty, projecting abundance into the future and summarizing the resulting trajectories and vital rate parameters. The Shiny application allows users to compare multiple scenarios with different model inputs without any knowledge of R. In addition to the following sections, see Wiens and others (2022) for the most recent publication detailing the BatTool R package and demographic model, Erickson and others (2014) for the original BatTool publication, and Thogmartin and others (2013) for further details on the demographic model implemented in the package.

Definitions of model inputs

Number of years
The number of years into the future for which population abundance is projected
Number of simulations
The number of replicates for which population abundance is projected into the future
Species (# of pups per annual litter)
The species of the population being modeled; the number of pups per annual litter is used when mapping lambda to 12 vital rates
Run Scenario 1/2 buttons
Clicking this button runs the demographic model using the model inputs specified for scenario 1 or 2, producing simulations projecting population abundance into the future, calculating statistics and displaying plot and tables
Starting population lower and upper bounds
The starting abundance of each simulation is an iid draw from a uniform distribution with these bounds
Carrying capacity
If population abundance reaches this limit then the population growth rate is set to 1
Starting adult proportion
The proportion of the total starting population that are adults, the remaining being first-years
Y-axis limit
Sets the upper y-axis limit in the plot
Extirpation threshold
The number of bats below which the population is considered extinct, for calculating the probility of extirpation in results summary table
Environmental stochasticity
Defines the upper bound E of the uniform distribution U(-E,E) from which iid perturbations to vital rates are drawn for each year and simulation. Units of probability of survival, probability of reproduction, and probability of breeding success
Demographic stochasticity checkbox
Indicates whether births and deaths in simulations are deterministic based on vital rates (if unchecked/FALSE) or randomly drawn from binomial distributions parameterized by vital rates (if checked/TRUE)
Checkbox option to add specified % confidence intervals
If checked, adds confidence intervals around the median population abundance line in the plot
Starting λ lower and upper bounds
The starting population growth rate λ of each simulation is an iid draw from a uniform distribution with these bounds
Vital rate range for scenario 1/2
This table shows the minimum and maximum values of the combinations of vital rates in which λ is within the starting lower and upper bounds
Direct mortality, adult and first-year
Positive (negative) values in this table indicate bats will be subtracted from (added to) the population in the specified season and year for each simulation. Can be applied to adults or first-years. Units of bats
Year of WNS arrival
Defines the year in which WNS impacts begin to be applied in the model, through reductions to vital rates
WNS survival probability multiplicative reductions, adult and first-year
For each year, defines the lower and upper bounds of a uniform distribution from which iid draws are taken for each simulation. The adult or juvenile winter survival (AWS or JWS) is reduced by this multiplicative factor (in the interval [0,1], inclusive), with 1 corresponding to no reduction from the winter survival rate defined by the starting λ value
WNS reproduction probability multiplicative reductions, adult and first-year
For each year, defines the multiplicative reduction to adult or juvenile breeding success (AB or JB). This is a point estimate instead of a uniform distribution of the impact to breeding success. There is a safeguard in the code so that the reduced breeding success rates does not become less than 0 or greater than one

Definitions of model outputs

Median, mean, lower CI, and upper CI N(t)
Statistics summarizing the population abundance over time of all simulations for a given scenario
Median, mean, lower, upper λ̃(t)
Statistics summarizing the population growth rate over time of all simulations for a given scenario
Pr(extirpation)
The proportion of simulations with projected abundance less than the extirpation threshold in a given year
Pr(survival)
The proportion of simulations with projected abundance greater than or equal to the extirpation threshold in a given year
Pr(growth)
The proportion of simulations with projected abundance greater than the starting population in a given year
Median time to extirpation
The time to extirpation of a simulation is the number of years into the projection when the abundance falls below the extirpation threshold. If the abundance does not fall below the extirpation threshold within the simulation period, the time to extirpation is set to Inf in R. The median time to extirpation is the median of the time to extirpation values for all simulation for a given scenario
D1,2(t)
A metric quantifying the percent difference in median abundance between two scenarios 1 and 2 at time t
1,2(t)
The difference in median derived growth rate between scenarios 1 and 2 at time t

Units of model inputs and outputs

Inputs and outputs in units of bats (total, males and females)
Starting population lower and upper bounds, carrying capacity, extirpation threshold, y-axis limit, direct mortality, results table abundance projections N(t)
Inputs and outputs in units of population growth rate
Starting λ lower and upper bounds, results table growth rate projections λ̃(t), scenario median growth rate difference ℓ1,2(t)
Inputs and outputs in units of vital rates (probability of survival, reproduction, and breeding success)
Vital rate tables, WNS impacts tables, and environmental stochasticity
Output in units of percent population difference
D1,2(t), the scenario median percent abundance difference

Demographic model description

The population model works at each time step by multiplying the adult and first-year abundances by the projection matrix.

Using a seasonal, two-stage model for adults and first-year individuals can effectively capture the demography of several cave bat species who hibernate in the winter, roost in the summer, and migrate in spring and fall. The model tracks females, and offspring production is assumed to conform to a 1:1 sex ratio. This means that the input starting abundance (in units of total bats, males plus females) is halved before projection, and then the population trajectory results are doubled in the output. Spatial structure is not explicitly accommodated so the modeled population is assumed to be closed; however it could represent a population at any spatial scale given certain assumptions.

Let xt = (FJ, FA)tT be the population abundance vector at time t, where FJ is the the abundance of first-year individuals (juveniles) and FA is the abundance of adults. The demographic model projects the population abundance at time t to time t+1 by multiplication with the projection matrix A through the equation xt+1 = A xt.

The projection matrix A is defined by the life cycle of the organism; see below and Thogmartin and others (2013) for the bat species life cycle diagram for this model.



We construct a seasonal model, denoting survival rates with φ, reproductive propensity rates with p, and birthing success rates with b. Subscripts on these rates denote whether they correspond to first-years/juveniles (J), adults (A), pups (P), or the combination of non-breeding adults and first-year individuals (N). Superscripts W, S, and F denote the vital rate corresponding to winter, summer, and fall, respectively. Thus, the projection matrix equation can be written as follows



All vital rates are contained in the range [0,1] (inclusive), except fecundities (bJ and bA) when the annual pups per litter of a species exceeds one. For example, Myotis sodalis (MYSO), M. lucifugus (MYLU), and M. septentrionalis (MYSE) have one pup, whereas Perimyotis subflavus (PESU) can have two. This parameter is the only varying factor when modeling different cave-hibernating bat species.

To project a population using the demographic model, one must specify the starting abundance (FJ, FA)Tt=0 and the twelve vital rates making up the projection matrix A. Estimating stage specific vital rates can be a difficult task, especially for elusive species such as bats. An alternative to direct estimation is to create a mapping between the growth rate of a population, λ, and the set of vital rates in the model. To create a look up table encoding this mapping, as was done in Thogmartin and others (2013), one can sample the set of vital rates from appropriate distributions, fill the projection matrix A, and calculate its leading eigenvalue, which corresponds to the growth rate λ. Because the mapping is not one-to-one, different combinations of vital rates can result in the same value of λ. Thus, to inform the vital rates in the model given a range of λ values, one can draw a row from the look up table such that λ resides in the desired range.

Environmental stressors on the population can be incorporated into the model using two methods. The first method acts by reducing the vital rates in the model. This method has been used to model the effects of WNS, decreasing adult winter survival rates as well as adult and first-year fecundity rates, reflecting observed and hypothesized effects of the disease, respectively. Other rate-based estimates of stressors could be incorporated into the model in this manner.

Direct mortality is the second method of applying a population stressor, accomplished by subtracting individuals directly from the population abundance in a given season and year. This option allows for seasonal mortality events such as threats during spring or fall migration or habitat loss during the summer or winter. This approach could quantify impacts of wind energy development, flying hazards, extreme weather, and other sources of mortality.

The population model is deterministic with this set of input parameters, but both environmental and demographic stochasticity can be incorporated. Environmental stochasticity is reflected in the model by annually perturbing the vital rates defining the demographic model. The annual vital rates are drawn from a uniform distribution of user-input width centered at the values prescribed in the λ look up table. There is a safeguard applied here that does not allow the vital rates to exceed their maximum theoretical values.

After these vital rates are drawn, the projection matrix A is defined for each year. Demographic stochasticity can then optionally be included in the simulation. Demographic stochasticity reflects the discrete nature of birth and death processes by using statistical distributions to model survival and reproduction instead of treating them as deterministic. In this case, we use a binomial distribution to simulate each birth and death in the model, which assumes multiple births by the same individual are independent events.

Finally, a carrying capacity may be included in the model to ameliorate unrealistic exponential growth. If the total population of adult and first-year individuals exceeds the carrying capacity, the projection matrix A becomes the identity matrix for that time step, before effects of stressors are added.

Thus, the model inputs include the starting abundance and population growth rate, carrying capacity, the option of demographic stochasticity and level of environmental stochasticity, and annual/seasonal stressor schedules consisting of vital rate decreases and direct mortality estimates. Hereafter the set of inputs to the demographic model and its outputs will be referred to as a scenario.

For each scenario, uncertainty is captured in the model by producing an ensemble of simulations. A range of abundance values and population growth rate values can be supplied, and each simulation begins by drawing a starting abundance and growth rate uniformly from the set ranges. Vital rates are specified using the λ look-up table, and the stochastic projection matrix is applied to the population vector in conjunction with the specified impacts of stressors.

The set of simulations for a scenario take into account the variation included in the starting population, growth rate, mapping to vital rates, stressors, and environmental and demographic stochasticity.

For a given scenario, each simulation projecting abundance into the future is an annual time series with total population size N(t) = (FJ)t + (FA)t. For each scenario, we report several metrics including the median population abundance over all simulations and confidence intervals specified by the user, the median and confidence intervals for the derived population growth rate λ̃(t) = N(t)/N(t-1), as well as the probability of survival (percent of simulations with greater than or equal to the extirpation threshold), probability of growth (percent of simulations with greater than the starting abundance at time step 0), and probability of extirpation (percent of simulations with less than the extirpation threshold). The median time to extirpation is also reported, and the extirpation threshold can be set by the user.

Also reported is the percent difference in median abundance at the end of the simulation period between scenarios i and j as

where medk{Nk(i)(t)} is the median population abundance over simulations k = 1, ..., K at time t for scenario i. We also report the difference in median derived growth rate between scenarios at the end of the simulation period

Note that the input λ value defining the vital rates is not necessarily the derived growth rate of a given simulation. The addition of stressors and stochasticity in the model can cause the derived growth rate λ̃ to deviate from the input growth rate λ. We distinguish between the input growth rate λ and the derived growth rate λ̃; the derived population growth rate is reported in the model output for comparison.

Saving and loading model inputs and outputs

In the results tab for each scenario there is a button which allows the user to save all inputs and outputs as CSVs in a zipped directory. The CSVs whose filenames begin with “input” can be uploaded in model inputs tab. The plot displayed can also be saved as a PNG within the app.

Files include

input_atake.csv, input_jtake.csv
Input adult and first-year direct mortality counts, 4 seasons x years, point estimates
input_wns_surv_prob_a.csv, input_wns_surv_prob_j.csv
Input adult and first-year WNS survival probability multiplicative reductions, years x lower and upper bounds
input_wns_birth_prob.csv
Input adult and first-year WNS breeding success multiplicative reductions, years x adult and juvenile reductions, point estimates
output_sim_results.csv
Simulation abundance results: simulations x years, with values of abundance. Also included is a column with the geometric mean (over time) of the annual derived growth rate for each simulation
output_pop_stats.csv
Statistics summarizing simulation results: years x abundance N(t) statistics, growth rate &lambdã(t) statistics, probability of extirpation, survival, growth, and median time to extirpation (the results table in this Shiny application)
output_vital_rates.csv
The λ and vital rates used for each simulation: simulations x λ and corresponding vital rates (before WNS impacts and direct mortality are applied)
output_wn_birth_a.csv, output_wn_birth_j.csv, output_wns_sur_a.csv, output_wns_sur_j.csv
The winter survival and breeding success probability reductions used for each simulation: simulations x years, values are the probability reductions used in each simulation

Bibliography


  1. Wiens, A. M., Schorg, A., Szymanski, J., and Thogmartin, W. E. (2022) BatTool: projecting bat populations facing multiple stressors using a demographic model. In press.
  2. Erickson, R. A., Thogmartin, W. E. and Szymanski, J. A. (2014) BatTool: an R package with GUI for assessing the effect of White-nose syndrome and other take events on Myotis spp. of bats. Source Code for Biology and Medicine, 9, 1–10.
  3. Thogmartin, W. E., Sanders-Reed, C. A., Szymanski, J. A., McKann, P. C., Pruitt, L., King, R. A., Runge, M. C. and Russell, R. E. (2013) White-nose syndrome is likely to extirpate the endangered Indiana bat over large parts of its range. Biological Conservation, 160, 162–172.
  4. Thogmartin, W. E., King, R. A., McKann, P. C., Szymanski, J. A. and Pruitt, L. (2012) Population-level impact of white-nose syndrome on the endangered Indiana bat. Journal of Mammalogy, 93, 1086–1098.
  5. Cheng and others (2021) The scope and severity of white-nose syndrome on hibernating bats in North America. Conservation Biology. https://conbio.onlinelibrary.wiley.com/doi/abs/10.1111/cobi.13739.



Glossary

CI
Confidence interval
iid
Independent and identically distributed
WNS
White-nose syndrome
YOA
Year of WNS arrival at a given location
JWS
Juvenile winter survival
AWS
Adult winter survival
NSS
Non-reproducing juvenile and adult summer and fall survival
JSS
Reproducing juvenile summer survival
ASS
Reproducing adult summer survival
PFS
Pup fall survival
JFS
Juvenile fall survival
AFS
Adult fall survival
JP
Proportion of juveniles reproducing
AP
Proportion of adults reproducing
JB
Births per juvenile
AB
Births per adult
CSV
Comma separated value file format
PNG
A type of image file format (Portable Network Graphics)