Exam Instructions

For this exam, you will render this Rmarkdown document into a PDF as you have done with your labs. You can write your answers below each question and show your work using the code chunk. Note that small coding errors will cause your render to fail and Rmarkdown is not as easy to debug as a typical R script that can be run one line at a time. We recommend (a) rendering your exam before you start, and (b) rendering after each answer, and/or (c) using a separate script to debug your code and then transferring the working code into the Rmarkdown code chunk.

Read each question carefully and make sure that you answer completely. SHOW ALL YOUR WORK to get partial credit for incorrect answers. Use graphs and illustrations any time, but it will be clear if you need to use them. USE APPROPRIATE AXIS LABELS on your graphs. If you aren’t sure about a question, give it your best shot and explain what you did and why.

Short Answers: Write your answers to the questions below without using a code chunk (15 points total)

1. What does “risk of quasi-extinction” mean? Why don’t we just always assume that extinction = 1 animal? (3 points)

Risk of quasi-extinction means that a population has reached size that is too small to recover also known as QET. This is a substitue for absolute extinction. QET is the population size where depensation or the Allee effect begins to occur. We do not assume that extinction is 1 animal because populations can start experiencing depensation at higher population numbers. This is when the population is too small to recover and this usually occurs at a population size much larger than 1 animal.

2. What is a Lincoln-Peterson mark recapture analysis and what does it estimate? How? What are 3 critical assumptions of the analysis? (3 points)

##A Lincoln-Peterson mark recapture analysis is a mark-recapture method that is used to estimate population size. This is done by capturing and marking organisms during a sampling collection. Then there is a second sampling collection where more animals are collected and marked. Estimates of population size are determined by the the proportion of marked animals in the surveys compared to the proportion of marked animals in the population. The equation N/M = C/R -> N = M*C/R is used to determing this where N=population estimate, M=marked on 1st survey, R=recaptured individuals, and C=total sample for the second survey. The three critical suumptions for this analysis are there are no births or deaths in the population, no immigrating or emigrating individuals to the population, and that marking has no affect on mortality.

3. What is the difference (mathematically and conceptually) between a continuous and a discrete rate of population increase? (3 points)

A continuous population increase is represented by r or continuous=r. This assumes that population growth is continuous. Contionous population growth is modeled with the equation dN/dt = B – D where dN/dt is the population growth rate and B-D is births - deaths. Continous population increase is the net reproduction per individual per time, which is the slope of the population trend. Discrete rate of population increase is the population change at finite time periods usually increases or decreases per year. Discrete population=lambda. The relationship between the two growth rates is expressed as r = ln(lambda)

4. How is a p-value different from an R2, used, for example, in a regression analysis to calculate population trend? (3 points)

##The p-value is the probability that there is not a relationship between the x and y axis values or is the slope statistically different from 0. R2 is how well the regression line fits the data. The closer the data is to the regression line then the closer the R2 value is to 1. R2 is not dependent on slope but is a measure of how far the data points are from the regression line.

5. How is the intrinsic rate of increase in the logistic equation different from the r we calculated from a population trend? (3 points)

The intrinsic rate of increase is the maximum per capita growth rate of a population at a low density. It stays constant and does not change with with time or with population size. The intrinsic rate of increase is used for logistic growth or density dependence.

R used for a population trend is a continous rate of increase for exponential growth. r is used to determine the growth rate at a specific point and time and r can change over time.

Analysis: use the code chunks below each question to show your work. Use comments within the code chunk to describe your work as needed. Err on the side of showing ALL work that led to your answers.

6. If a population is declining at 12% per year and the current population size is 300 animals, how many animals will be in the population after one year? How about 7 years from now? Show your work. (5 points)

1 year=264

7 years=122.6

300*(.88)^1 ##after 1 year
## [1] 264
300*(.88)^7 ## after 7 years
## [1] 122.6027

7. An elk population is censused for 6 years and the population time series is 140, 170, 190, 228, 215, 251. Use the code chunk below to: (10 points total)

a) What is lambda, and how do you calculate it from census data like these? Make the calculation for each of the years sequentially in this time series. (5 points)

elk=c(140, 170, 190, 228, 215, 251)
lambda=elk[2:6]/elk[1:5]
lambda
## [1] 1.2142857 1.1176471 1.2000000 0.9429825 1.1674419

b) What is the average value of lambda, and what does that tell you about this population? (5 points)

It tells us there is an average population increase of 12.84% a year.

mean(lambda)
## [1] 1.128471

8. Dr. Levi researches spider monkeys in the Amazon, which are very sensitive to overharvest due to their low maximum intrinsic growth rate of r = 0.07. (15 points total)

a) If a previously overhunted spider monkey population from low density starts with 100 individuals at t = 0, how many individuals will there be at times t = 1, 2, and 3 years assuming geometric population growth? (5 points)

100*exp(.07)*1
## [1] 107.2508
107.2508*exp(.07)
## [1] 115.0274
115.0274*exp(.07)
## [1] 123.3678

b) What size would the spider monkey population be at t = 20 years if this growth continues? (5 points)

100*exp(.07)^20
## [1] 405.52

c) How would you determine how long it will take the spider monkey population to triple from its initial size? Write an equation for this, and then make the calculation, given N0 = 100 and r = 0.07. (Do this calculation in the code chunk) (5 points)

##16 years

#100*exp(.07)^t

#100*exp(.07)^16

9. The data in the code chunk below show index catches of juvenile Lost River suckers (a type of fish) from Klamath Lake - the number of fish caught per hour by one 100 foot seine net. (25 points total)

a) Look at this plot of log(N) versus time. What information could you obtain from the slope of a linear regression line plotted through the data? (10 points)

#knitr::include_graphics("Exam1_Q9a.png")

##The population is declining and also the population does not fit the linear regression line that is plotted well. While the population looks to be declining there is a lot of variability from year to year and the data doesn’t fit the line well.

b) Describe what the shape of the plot of the population trend would look like if you plotted N versus time instead of log(N)? (5 points)

The shape of the plot would be variable because of the jumps from year to year.

#plot(xlab=population,y=time)

c) Look at the regression output below. How confident are we (statistically) in this population trend? Is it increasing or decreasing? Provide three pieces of evidence to support your findings. (5 points)

We are confident statistically in the population trend because the p-value is significant at P= 0.02491. The population is decreasing with a slope of -0.26813.

reg_log=lm(log(SuckerCatch\(catch)~SuckerCatch\)Year) summary(reg_log) #Call: # lm(formula = log(SuckerCatch\(catch) ~ SuckerCatch\)Year) # # Residuals: # Min 1Q Median 3Q Max # -1.3275 -0.8013 -0.0525 0.7962 1.3376 # # Coefficients: # Estimate Std. Error t value Pr(>|t|)
# (Intercept) 538.66373 199.56816 2.699 0.0244 # SuckerCatch$Year -0.26813 0.09978 -2.687 0.0249 # — # Signif. codes: 0 ‘’ 0.001 ‘’ 0.01 ‘’ 0.05 ‘.’ 0.1 ‘ ’ 1 # # Residual standard error: 1.047 on 9 degrees of freedom # Multiple R-squared: 0.4452, Adjusted R-squared: 0.3835 # F-statistic: 7.221 on 1 and 9 DF, p-value: 0.02491

coef(reg_log)[2] # SuckerCatch$Year # -0.2681318

summary(reg_log)$coefficients[2,4] # 0.02491362

confint(reg_log) # 2.5 % 97.5 % # (Intercept) 87.2091862 990.11826873 # SuckerCatch$Year -0.4938588 -0.04240484



#### d) What are lambda and the percent change per year? Show your work. (2 points)

## the percent change is -26.81% a year.  

```r
##coef(reg_log)[2] 
# SuckerCatch$Year 
#       -0.2681318

##We should give the population special conservation status because we are statisically confident the population is decreasing. We need to know what QET is for this species to help determine if protection is needed and support a conservation listing.

10. The logistic model is a common first approximation for population dynamics that has two parameters, r and K. (10 points)

a) Under the logistic model, what population abundances are associated with the highest overall growth rate (i.e., what is the value of N when is dN/dt greatest)? What population abundance is associated with the lowest overall growth rate?

##The population abundances with the highest growth rates are ones that are at half the carrying capacity. The population abundance associated with the lowest overall growth rate are populations that have exceeded carrying capacity.

b) Would the logistic model be a good approximation for this bird population, based on the figure provided below? Why or why not?

##No because we don’t know what the carrying capacity is.

#knitr::include_graphics("Exam1_Fig1.png")

c) What do you predict the breeding success would be if density increased even higher? Why?

##I think the breeding success would start to level off. As more individuals are competeing to breed then breeding success will decrease.

d) What are some explanations for why breeding success decreases with colony density? What is the term for this general phenomenon?

##There is more competition for mates. More competition means that less individuals will have success in being able to breed per capita. The term for this is density dependent. The density of the population affects the breeding success.

11. Use the monthly capture-recapture data for eulachon fish below to estimate the population size using the Schnabel estimator. “mark” refers (10 points)

The dataset includes the number of individuals caught each day (Catch), the number of those that are recaptures (Recaptures), and number of individuals newly marked (Newmarks). Hint: Does the Schnabel estimate use the NEW marked each day or some other metric? Make sure your results make sense given the data as a sanity check.

As a reminder:

n = sum(Mt*Ct) / sum(Rt)

Where: n = an estimate of the true number (N) of individuals in the population at the time of initial marking. Mt = the number of individuals in the population that have been marked at the time the t sample was taken. Ct = the number of individuals captured at the sampling that occurred at time t. Rt = the number of recaptured, marked individuals taken at time t.

ct<-eulachon\(Catch Rt<-eulachon\)Recaptures nmk<-eulachon$Newmarks mt<-c(0,cumsum(nmk)[1:4]) mt [1] 0 29 49 67 79 N_hat<-sum(mt+ct)/sum(Rt) N_hat #[1] 8.780488

#eulachon <- data.frame(Month=1:5,Catch=c(29,34,28,20,25),Recaptures=c(0,14,10,8,9),Newmarks=c(29,20,18,12,16))
#ct<-eulachon$Catch
#Rt<-eulachon$Recaptures
#nmk<-eulachon$Newmarks
#mt<-c(0,cumsum(nmk)[1:4])
#mt
#N_hat<-sum(mt+ct)/sum(Rt)
#N_hat
# N_hat
#[1] 8.780488

12. The Florida Nutmeg Tree, Torreya taxifolia, is endemic to a few ravines along the eastern side of the Appalochicola River in northern Florida and Southern Georgia. Timeseries for two populations. The generation time is 30 years. According to the IUCN criteria, how would you list each population? (10 points)

Risk and IUCN status criteria. The International Union for the Conservation of Nature (IUCN) uses the following general criteria for “listing” species: http://www.iucnredlist.org/technical-documents/categories-and-criteria/2001-categories-criteria

“>=50% chance of extinction within 10 years or three generations, whichever is the longer (up to a maximum of 100 years) = list as ‘Critically Endangered’”

“>=20% chance of extinction within 20 years or five generations, whichever is the longer (up to a maximum of 100 years)= list as ‘Endangered’”

“>= 10% chance of extinction in 100 years = list as ‘Vulnerable’”

“<10% chance of extinction in 100 years = no listing”

Species or populations are always listed at the highest level for which they meet the criteria.

a) Given this output from a quasi-extinction analysis, how would you list this population of Nutmeg tree? Note: in the plots, the solid red line = 10%, dashed line = 20%, and dotted line = 50% of populations.

##I would list this population at endangered because it has a greater that 20% chance of falling below QET within 20 years. By less than 50 years almost 100% of the population would be below QET.

#knitr::include_graphics("Exam1_Q12a_TS.png")
#knitr::include_graphics("Exam1_Q12a_QET.png")

b) Here is output from a quasi-extinction analysis of another population of Florida Nutmeg Tree, how would you list this population? Note: in the plots, the solid red line = 10%, dashed line = 20%, and dotted line = 50% of populations.

##I would give this population a vulnerable listing because there is between 10 and 20% of the population below QET at 100 years.

#knitr::include_graphics("Exam1_Q12b_TS.png")
#knitr::include_graphics("Exam1_Q12b_QET.png")