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BIOL 416 – CONSERVATION BIOLOGY, SPRING 2007
DISCUSSION PERIOD:
POPULATION VIABILITY ANALYSIS, Tuesday 20 March
Learning objectives:
(1) Understand how matrix population models work, and how they are used as a fundamental and
important tool to guide conservation management.
(2) Learn practical skill of how to take a given population projection matrix and conduct a
population viability analysis by simulating the consequences of management interventions that
influence different demographic processes.
Main sources of information:
Lecture material on population viability analysis.
Relevant student knowledge from lectures:
• Population viability analysis.
Outline of discussion period:
Introduction
We are going to conduct a population viability analysis (PVA) for the Giant Burrowing Duck (GBD), a
large and rare duck that nests in burrows that it digs in open woodland habitat. Conservation biologists
have studied GBD in great detail, and are gravely concerned about one island population in particular,
where only 45 individuals remain: 20 juveniles, 15 non-reproductive adults, and 10 reproductive adults.
For a larger population on the mainland, demographic data have been collected over a period of 4
years, resulting in 3 population transition matrices. We will assume that demographic processes
operate in a similar way on the island as on the mainland. The transition matrices were identical across
the three year-to-year transitions, except for the number of offspring produced by reproductive adults,
which was observed to be 1, 2 or 3 in the three different years.
Throughout this document, several questions appear in bold. You will hand in answers to these
questions at the next discussion period.
PVA in a spreadsheet
An Excel spreadsheet, “BIOL416_PVA.xls”, has been prepared to allow us to conduct this PVA. On
the first worksheet, “baseline”, the three transition matrices are shown on the left.
Connect three boxes below to create a life-cycle diagram, showing the non-zero transitions
between life stages.
Juveniles
Nonreproductive
adults
Reproductive
adults
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In Transition Matrix 1, what is the mortality rate of reproductive adults? Explain how you
calculate this.
The basics of matrix population models
To the right of the transition matrices, the spreadsheet is set up to simulate 20 years of population data
(from left to right) calculated in the following way:
• First, we start at time zero with the present population sizes shown in red bold: 20 juveniles, 15
non-reproductive adults, and 10 reproductive adults.
• In each subsequent year we select one of the three transition matrices at random to determine
that year’s population changes, and using matrix algebra we multiply the transition matrix A by
the vector of abundances Nt, and round to the nearest integer to give the next year’s vector of
abundances Nt+1. In our example, to calculate N1 using transition matrix 1, we do the
following:
A
0
0.2
0
•
0
0.2
0.5
1
0.2
0.5
x
N0
=
N1
x
20
15
10
=
10
9
12.5
Since we can’t have 12.5 reproductive adults, we round to the nearest integer: 13. So, using
transition matrix 1, we would expect to find 10 juveniles, 9 non-reproductive adults, and 13
reproductive adults at time 1. Make sure you understand how these calculations work.
Given N1 with 10 juveniles, 9 non-reproductive adults, and 13 reproductive adults, show
calculations based on A that would yield the number of juveniles in the next time step.
Simulating population dynamics
We will use logical/mathematical formulas and random numbers in Excel to simulate population
dynamics through time. Let’s examine the formulas in cells I4, I5 and I6 of the Excel spreadsheet to
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see how these calculations are implemented. Here’s the formula in cell I4, which calculates the number
of juveniles at time 1:
=ROUND(IF(I1<0.33,MMULT($C4:$E4,H4:H6),IF(I1<0.66,MMULT($C11:$E11,H4:H6),MMULT(
$C18:$E18,H4:H6))),0)
This formula works as follows:
• The “ROUND” command with a “0” just before the last parenthesis indicates that whatever
value is calculated we want it rounded to zero decimal places – i.e., to an integer value.
• Recall that we want to use a randomly chosen transition matrix at each time step in the model.
The random numbers in row 1, columns H-AB vary from zero to one. If the value of the
random number in this column “I1” is less than 0.33 (which should be true 1/3 of the time) then
we will use transition matrix 1. That is, to calculate the number of juveniles we multiply the
first row of transition matrix 1 “C4:E4” by the vector of abundances from time 0 “H4:H6”. The
“MMULT” function means this is a Matrix MULTiplication, and the “$” symbols mean that if
we paste this formula in other cells, which normally then shifts the cells referred to in the
formula, these cell references will remain the same.
• If I1 is not less than 0.33, then if I1<0.66 (i.e., if 0.33<I1<0.66, which will be true 1/3 of the
time) we will use transition matrix 2; if neither of these is true (i.e., if I1>0.66) we will use
transition matrix 3.
• Effectively, this formula selects one of the three transition matrices at random with equal
probability, and calculates new population abundances by multiplying the randomly chose
matrix by the initial vector of abundances, rounding the result to the nearest integer.
If we now copy cells I4:I6, and paste them in each of the next 19 columns to the right, any cell
reference not preceded by a “$” symbol will also shift to the right. That is, each set of three cells will
take the preceding year’s abundance vector and multiply it by a randomly chosen transition matrix to
calculate that year’s abundance vector. So, highlight cells I4:I6 and press ctrl-C to copy the cell
contents. Now highlight cells J4:AB6 and press ctrl-V to paste them. Look at the formulas in these
cells to confirm that cell references have shifted to refer to the previous column in each case.
You have just simulated 20 years of population change for the Giant Burrowing Duck!
Conducting replicate simulations and analyzing the results
We would now like to store the results of 20 separate simulations to look at the range of possible
outcomes. The block of cells below, H15:AB34, is where we will put these results. Since reproductive
adults can be most easily surveyed, and are the only reproductive individuals in the population, we will
focus on the number of reproductive adults – the row highlighted in yellow.
Highlight the row of cells in yellow and press ctrl-C to copy their contents. Move the cursor to cell
H15 and from the Edit menu, chose Paste Special, click Values on the dialogue box that opens, and
click Okay. Since the cells we have copied contain formulas, we need to do this Paste Special
operation so we get only the numbers pasted.
Since pasting will have caused new random numbers to be generated, the yellow cells will now have
changed, though they are still copied to the clipboard. So, we can just highlight the next cell down in
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our output table (H16), and past the values again. Repeat this operation 20 times to fill up the data
table with simulation results.
The data table now contains 20 simulations, each of 20 years of data on adult population numbers for
the Giant Burrowing Duck, and the data are plotted in the figure below.
Based on the available demographic data, do we expect the population to grow or shrink in the
future?
The data should be identical for the first three years in all simulations. Why is that the case?
An adult population size of less than 5 is considered to be virtually guaranteed to go extinct.
What proportion of the 20 simulations showed adult populations <5? What was the average,
maximum, and minimum adult population size after 20 years?
USING PVA TO GUIDE MANAGEMENT
Three management options have been identified that may increase population growth for the GBD,
each targeted at a different life stage.
(1) Protecting nest sites with small fences to prevent predatory snakes from destroying eggs could
double fecundity of reproductive adults from 1, 2 or 3 offspring to 2, 4 or 6.
(2) Supplementing shrub cover to reduce avian predation could increase reproductive adult survival by
35%, with the reproductive adult – non-reproductive adult transition increasing from 0.2 to 0.27 and the
reproductive adult – reproductive adult transition from 0.5 to 0.675.
(3) Supplementing food availability for juveniles could increase their survival by 75%, from 0.2 to
0.375.
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Three Excel worksheets have been set-up for you to implement these hypothetical management
scenarios: (1) “high fecund”, (2) “high rep adult surv”, (3) “high juv surv”.
You now know how to implement a PVA in this spreadsheet environment. So, as with the baseline
data, incorporate these changes into the transition matrices and conduct PVAs for the three different
scenarios.
For each scenario, is the population expected to grow, shrink, or remain stable?
What is the probability of extinction (adult population < 5 in each case)?
Which scenario would you chose if only one could be implemented? Why?
If altering shrub cover were much more expensive than the other two options, would that change
your preferred management strategy?