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BIOL 416 – CONSERVATION BIOLOGY, AUTUMN 2010

DISCUSSION PERIOD:

POPULATION VIABILITY ANALYSIS, Tuesday 28 September

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. Answer these questions as you go along. There will be a short quiz at the end of the tutorial 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.


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:
  • 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 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 (For more recent versions of Excel, in the Home menu, click the arrow under Paste, and chose Paste Values). 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 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 87.5%, from 0.2 to 0.375.

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?

One proposal that has been put forward is the introduction of individuals raised in captivity onto an island where the Giant Burrowing Duck is not currently present. The starting population would include 20 juveniles, 2 non-reproductive adults, and 0 reproductive adults. Based on model projections over the next 5-10 years, does this change in the starting population abundance vector alter which of the three management strategies you recommend to implement during this reintroduction? Explain your answer.

For the “baseline” demographic scenario, the deterministic population growth rates for the three matrices are 1 = 0.84129, 2 = 0.93083, and 3 = 1.0. With “stochastic population growth” – that is, with one of these three matrices selected at random each year – what is the long-term expected population growth rate? Show your calculation.

Extra questions (if you have time)

Use the spreadsheet “BIOL416_PVA.xls” to answer the following questions.

Note: you will need to make modifications to the spreadsheet.

A matrix population model has been developed for the one remaining population of the critically endangered plant, the bog bearberry. This is a perennial plant, and researchers recognize four life stages: seedlings, small plants, medium-sized plants, and large plants. Only large plants produced seeds. The following three population transition matrices were estimated using four years of data:

(i) / seedling / small / medium / large
seedling / 0 / 0 / 0 / 60
small / 0.1 / 0.2 / 0.1 / 0
medium / 0 / 0.1 / 0.5 / 0.1
large / 0 / 0 / 0.2 / 0.55
(ii) / seedling / small / medium / large
seedling / 0 / 0 / 0 / 60
small / 0.1 / 0.2 / 0.1 / 0
medium / 0 / 0.15 / 0.5 / 0.1
large / 0 / 0 / 0.2 / 0.6
(iii) / seedling / small / medium / large
seedling / 0 / 0 / 0 / 100
small / 0.1 / 0.2 / 0.1 / 0
medium / 0 / 0.1 / 0.5 / 0.1
large / 0 / 0 / 0.2 / 0.4

The current number of individuals in each stage is as follows:

seedling / 500
small / 30
medium / 20
large / 10

(1) Using these transition matrices in a population viability analysis, do you expect the total population to grow, shrink, or remain constant in size? Show and explain in detail the modifications you have made to the spreadsheet to answer this question, including any Excel formulas. Show calculations used to determine your answer.

(2) If research funding had been ended early in the study, when only the first of the three matrices had been estimated, would you have predicted the total population to grow, shrink, or remain constant in size? Show calculations used to determine your answer.