In this lab, you will continue to learn about the application of matrix population models to natural resource management. You will use demographic rate estimates to build a matrix model and to explore how harvest of seeds and adult influences the population dynamics of a long-lived tree. Your goal will be to set harvest limits that allow for sustainable use of the trees by multiple users.

Objectives

Use functions from the popbio package to learn about the ecology of a commercially important tree species
Conduct a sensitivity analysis and simulations to make management recommendations to allow sustainable harvest of seeds and adults
Practice visualizing data using ggplot2
R functions used in this exercise:

Set up

Log in to R Studio Cloud
Create a new project called LastnameFirstname-Lab7
Create a new R Markdown file from the Homework-7 template:

3a) Click File -> New File -> R Markdown

3b) Click From Template and scroll down to WILD3810-Homework7

3c) In the Name box, type LastnameFirstname-Homework7

3d) Click Ok
Ensure that you can Knit the file

4a) Change “YOUR NAME HERE” in the header section to your name

4b) Click Knit to make sure you can knit the .Rmd file
Complete the assignment as directed in the .Rmd file
Follow the instructions for submission to upload your completed assignment to Canvas

Note: This homework assignment is another written assignment. As for the first writing assignment, you can receive 4 bonus points by incorporating feedback from the science writing center into your final version

In this lab, we’ll examine the dynamics of a commercially-important but imaginary species of palm tree, Thrinax imaginarium. T. imaginarium is a tropical palm found throughout Central America and parts of the Caribbean. The large, edible seeds are an important source of food for many local communities throughout its range and are collected for both consumption and sale. Over-harvest of seeds in some areas has led to concern about the sustainability of this practice and some local governments are now considering limits on the number of seeds that can be collected each year.

Adult trees were also an important source of wood for building homes and as fuel for cooking fires, though in recent years demand for wood has grown as construction of large buildings and resorts has increased. Increased harvest of adult trees has added to concerns about population viability and has led to calls for limits on the number of adult trees that can be cut. Of course, competition between timber operations and seed collectors has led to conflict, with both groups blaming the recent declines in T. imaginarium abundance on the other.

Thrinax radiata, a real palm that is closely related to the made up species Thrinax imaginarium. Photo courtesy of Drew Avery via Wikicommons

As a leading expert on tree demography and silviculture, you have been asked by a government agency to come up with sustainable harvest limits for both seeds and adult trees of T. imaginarium. You will use available demographic data to build a matrix model for T. imaginarium, explore the effects of different harvest regimes, and write a report with recommendations that allow both local communities and timber operations to sustainably benefit from this resource.

Demography of Thrinax imaginarium

To write your report, you will rely mainly on published estimates of T. imaginarium demography. Several studies have published demographic rates for the species and at least one study reported the age distribution of a single T. imaginarium population. This information is summarized in the figure below.

Demography of Thrinax imaginarium life stages. Values inside the circle/squares are the observed number of individuals in each stage class from a single demographic study of the species. Values next to the arrows are the estimated transition probabilities for each stage

In the chunk named demography in the homework template, translate these demographic rates into a matrix object. Call that object A.

Ecology of Thrinax imaginarium

During the last lab, we calculated several properties of a demographic matrix by hand, including the asymptotic growth rate and stable stage distribution. Fortunately, there is an R package called popbio that will do these calculations (and many more) for us. For example, you can calculate $\lambda$ and the stable stage distribution of the matrix you just created using:

library(popbio)
lambda <- lambda(A)
ssd <- stable.stage(A)

The popbio package has a number of other useful functions, for example estimating the reproductive value of each stage and the generation time of a population:

rv <- reproductive.value(A)
gt <- generation.time(A)

Look at the output from the functions above. What can you say about the ecology and life history of this species from this information?

Sensitivity and elasticity

In lecture, we learned about the concepts of sensitivity and elasticity. Remember that sensitivity is the change in $\lambda$ caused by a small change in a vital rate and can be estimated analytically using:

\[\Large s_{i,j} = \frac{\delta \lambda}{\delta a_{i,j}} = \frac{v_iw_j}{\sum_{k=1}v_kw_k}\]

We also learned how to do this by hand but luckily that’s not necessary. We can instead use:

sens <- popbio::sensitivity(A)

As we saw in lecture, sensitivity is not always the most useful metric for understanding which vital rates are most important for influencing $\lambda$. Instead, most natural resource management studies focus on elasticity, which is the change in $\lambda$ caused by a small proportional change in a vital rate. As you probably guessed, popbio also does this calculation:

elas <- popbio::elasticity(A)

Based on the estimated elasticities, which vital rates do you expect to have the biggest influence on population growth of T. imaginarium?

Simulating the effects of harvest

To examine the effects of seed and adult harvest, you will simulate 50 years of population dynamics of a T. imaginarium population with different levels of seed collection and adult tree harvest. In the homework template, there is a chunk named sim_params where you can adjust three parameters: the number of seeds collected each year (s), the number of adult trees harvested each intervals (h), and the length of the adult harvest interval (harvest.interval; 1 = annual harvest, 2 = every other year, 3 = every third each, etc.). Initial values for each parameter are provided but you will need to change these by trial and error to find a combination that achieves the state management objectives

Next, you will need to complete the code in the chunk named simulation. The first few lines are completed for you, these set the number of years in the simulation, create matrices to store the annual abundances and harvest, and a data frame to keep track of how many seeds/adults are harvested in each year. You will need to do the following before running the code:

Create a vector object called n0 that contains the initial abundance for each stage
Assign the first column of the N matrix to be the n0 vector
In side the for loop, fill in the code necessary to project abundance each year. To incorporate harvest, note that:

\[\large N_{t} = A \times N_{t-1} - H_{t-1}\]

Change eval = FALSE to eval = TRUE in the chunk options
Fill the ggplot2 code in the adult.fig and seed.fig chunks to visualize the number of adult trees and seeds in each year
Change eval = FALSE to eval = TRUE in the adult.fig and seed.fig chunk options

Based on your understanding of matrix algebra, think about what’s happening in the equation above. What type of object is H? What are its dimensions? What are its values?

Based on the equation, are individuals harvested before or after reproduction?

Notice that after running the simulation there are several lines of code that calculate the average annual growth rate of the population for the final 6 years of the simulation (to avoid any transient dynamics early in the simulation) and the total profit made through each harvest. The timber operation nets $10 per tree while seeds net $0.05 per seed. Once you have completed all of the above steps, you should be able to knit the document. When you do so, you should see the figure with the number of adult trees and seeds in each year. The title of each plot also provides information on the total revenue of each harvest.

Decision context

For this assignment, you will need to choose the harvest limit (h), harvest interval (harvest.interval), and seed collection limit (s) to accomplish three competing goals:

ensure long-term persistence of the T. imaginarium population;
maximize profit from the timber harvest (but not at the expense of goals 1 & 3)
maximize profit from the seed collection (but not at the expense of goals 1 & 2)

To make this recommendation, you can change h, harvest.interval, and s by trial and error until you find levels that in your judgement meet the stated goals. Use these results to provide a recommendation for which management actions should be taken. Note that there is no specific mandate for relative profits from adult harvest vs. seed collection so you will need to decide what a fair balance looks like. This decision needs to be stated explicitly, though as long as you provide a reasonable rationale there is no wrong answer.

Use the code chunks in the homework 7 template to complete all code necessary to make your recommendation. Use the same outline that you used for homework 4 to structure your recommendation. In particular, be sure your recommendation includes:

a description of the management challenge
the specific objectives you are addressing
a brief description of the ecology of T. imaginarium based on your analysis of the matrix model
a specific management recommendation, including a rationale for the relative profits from seed vs. adult harvest
an interpretation of how the ecological properties of the T. imaginarium population (e.g., generation time, elasticities, reproductive values) relate to the ability to sustainably harvest adults and seeds
figures visualizing the predicted trajectories for adults and seeds under the recommended harvest limits
potential limitations and/or risks of your recommendation (including reasons it might not work as predicted)

As you put together your report, here are a few tips:

each time you change h, harvest.interval, or s note the values of each parameter and the resulting $\lambda$ and profits. Keeping notes of each change will help you figure out which combination leads to the best outcomes
The code automatically creates a number of objects that store information useful for the report (including lambda, elas, gt, timber.profit, timber.profit, h, harvest.interval, and s). Remember that you can insert these objects into the text of your report using, for example, `r timber.profit`. When you knit the document, that code will display the value of timber.profit, saving you a lot of typing. Even better, if you change the values of h, harvest.interval, or s, the report will reflect those change when you knit so you don’t have to retype those values.

Lab 7: Applications of matrix population models - Sustainable tree harvest