lab9_source-sink.RmdIn this lab, you will continue to learn about the dynamics of multiple interacting populations. Specifically, you will explore the dynamics of a source-sink metapopulation and how these dynamics influence management decisions.
Objectives
Model the dynamics of a source-sink metapopulation using matrix models
Manipulate the demographic and movement rates of sources and sinks to measure the impact on metapopulation growth
<<<<<<< HEAD To complete this week’s homework assignment, you will follow along with the instructions below, entering code and answering questions in the .Rmd file for homework 9. From within Rstudio cloud: ======= To complete this week’s homework assignment, you will follow along with the instructions below, entering code and answering questions in the .Rmd file for homework 8. From within Rstudio cloud: >>>>>>> 5574edd7fe912a3269261d268fc8f26b0196581a
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As we learned in lecture, we can approach modeling source-sink dynamics using the principles we used to model stage-structured population dynamics. In the source-sink model, the diagonal elements of the matrix represent the self-recruitment rate (\(R_i\)) of each population \(-\) that is, the degree to which each population can sustain itself through internal births and recruitment. In the simplest terms, we can estimate \(R_i\) as:
\[\large R_i = 1 + b - d\] In words, the self-recruitment rate is the birth rate minus the death rate plus one. Note that this is the same definition we used for \(\lambda\) in the sections on unstructured population dynamics.
Using the information above, answer question 2 on the homework assignment
In the matrix version of the source-sink model, the off-diagonals represent the movement of individuals from one population (represented by the column number) to another population (represented by the row number). This is equivalent to the growth/survival rates in the stage-structured population models. The total emigration rate from population \(i\) (\(E_{i}\)) can be calculated as:
\[\large E_{i} = \sum_{j \neq i} D_{ij}\]
where \(D_{ij}\) is the dispersal rate from \(i\) to \(j\).
The tables below provides demographic and movement rates for 3 populations.
| Population 1 | Population 2 | Population 3 | |
|---|---|---|---|
| Birth rate | 0.20 | 0.28 | 0.12 |
| Death rate | 0.22 | 0.25 | 0.20 |
| 1 | 2 | 3 | |
|---|---|---|---|
| 1 |
|
0.03 | 0 |
| 2 | 0 |
|
0 |
| 3 | 0.05 | 0.09 |
|
In your assignment file, there is a chunk named matrix. Inside of that chunk:
Use the table to calculate all of the relevant \(R_i\) values. Be sure you create an object corresponding to each value and give it an easily-understood name.
Using the rates estimated in step 1, create a matrix describing the dynamics of this metapopulation. Name this matrix B and make sure that there is one row and one column for each population.
Use the popbio::lambda() function to calculate the long-term growth rate of this source-sink metapopulation. Store this value in an object named lambda1
Use the popbio::stable.stage() function to calculate the long-term “stable stage distribution” of this source-sink metapopulation. Store this value in an object named ssd1
Answer questions 3a-3d on the homework assignment
The impact that habitat loss or loss of connectivity has on the metapopulation depends on whether those changes effect sources or sinks. For example, what happens connectivity of the source population is lost due to degradation of the matrix habitat?
Assume that the metapopulation experiences some type of habitat degradation that reduces connectivity among the patches. Specifically, no individuals can disperse from patch 2 to patch 1 and the dispersal rate from patch 2 to patch 3 is reduced to 1%. In the R chunk of your homework assignment named source:
Create a new matrix describing the dynamics of the metapopulation. Call the matrix B2
Use the popbio::lambda() function to calculate the new long-term growth rate of the source-sink metapopulation. Store this value in an object named lambda2
Use the popbio::stable.stage() function to calculate the new long-term “stable stage distribution” of this source-sink metapopulation. Store this value in an object named ssd2
Answer questions 4a-4c on the homework assignment
What if instead of changes to connectivity, the source population experiences declines in its self-recruitment rate?
Using the matrix B as a starting point, what happens if the birth rate of population 2 declines to 0.24. In the R chunk of your homework assignment named source2:
Create a new matrix describing the dynamics of the metapopulation. Call the matrix B3
Use the popbio::lambda() function to calculate the new long-term growth rate of the source-sink metapopulation. Store this value in an object named lambda3
Answer questions 5a-5b on the homework assignment
What happens if vital rates within the sink populations decreases?
Assume that population 3 experiences some type of habitat degradation that reduces the birth rate by 25%. In the R chunk of your homework assignment named `sink:
Create a new matrix describing the dynamics of the metapopulation. Call the matrix B4
Use the popbio::lambda() function to calculate the new long-term growth rate of the source-sink metapopulation. Store this value in an object named lambda4
Use the popbio::stable.stage() function to calculate the new long-term “stable stage distribution” of this source-sink metapopulation. Store this value in an object named ssd4
Answer questions 6a-6c on the homework assignment
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