Age & Stage Structure: Elephant Model

Transcription

1 POPULATION MODELS Age & Stage Structure: Elephant Model Terri Donovan recorded: January, 2010 Today we're going to be building an age-structured model for the elephant population. And this will be the model that Tony had described with the African elephants. You ll see how easy it is to set up this kind of model [Age and Stage Structure ] on a spreadsheet. Let s begin by entering the word Year in cell A10. And then let s begin our series by entering the number 0 and then 1. And then selecting cells A11 and A12 and we ll simulate this population for 80 years. For each of those years we need to keep track of the number of animals in each age class. And so let s enter the word Age in cell A9. And we ll start by entering a 0 in cell B9. And then the number 1 in C9. And we ll grab those two cells and pull those out. And we ll model the number of animals in each age class up to age 59. Because, remember at age 60 all animals [elephants] drop dead according to our modeling world, and we are no longer considering those in our population. Just to keep this a little bit clean, let s go ahead and select the upper left box in the spreadsheet and center our values. Now each of these ages belongs to an age group. So let s enter the word Group in cell A8. And we called those animals that were 0 years-old, newborns. And we called the animals that had lived to be 1 years-old, calves. Now remember that Tony indicated that animals start breeding at the year of 12. And so, that means that 2 year-olds through 11 year-olds are considered to be non-breeding adults. So let's select cells D8 through M8, merge and center them. Give them some kind of heading color and label these age classes Non-breeding Adults. And let s do something similarly for the age classes 12 all the way through age 59. These represent our breeding adults. Let s merge and center those and give them some kind of a heading. And we ll label those Adults. Now let s take a just a minute to remind ourselves what these numbers actually mean. This may seem a little rudimentary, but it s really important to go through this. In a model that has discreet time steps, like what we are doing here, we re modeling the number of animals in year 0, 1, 2, 3 - but we re counting these animals on a particular day. And on that particular day that we re doing all of our censusing, animals graduate to the next age class. So that means that we do our census, and we re counting the number of newborns, and that means that those 1

2 animals were literally just born. The number of calves or the number of 1 year-olds that just had their first birthday. The number of 2 year-olds is the number of animals that just celebrated their 2 nd birthday, and so on. So this is what is called a post-breeding census. We re counting animals after the breeding has occurred. And so we re counting animals that are both brand new animals into the population, newborns, as well as all the other animals in the population. Okay? So what we need to do first is start in the year 0, and we re going to enter our initial age distribution. So we re going to select the cells in row 11, going out through the 59 th age class. And we re just going to enter what we think would be a starting initial age distribution. These are model inputs, and so let s go ahead and shade those green. As Tony said this is something we re not certain about. And so we re going to forge ahead and enter some numbers here. We ll start off with 40 newborns. 20 calves. 20, 2 year-olds. 20, 3 year-olds. 20, 4 year-olds. 20, 5 year-olds. And in all of the other age classes we ll have just 4 animals each. Remember this is going to be a female only model. So these represent 40 females newborns and 20 female calves and so on. Let s go ahead and fill this number 4 all the way across by selecting cell H11 and then dragging the corner of that box all the way out to the age class 59. Now over here in cell BJ, let s go ahead and use this column to calculate our totals. So let s enter the word Total in cell BJ10. And then we ll enter a SUM function. And we ll just sum all of the animals that were in year 0, our population size for our starting year of our model. And there are 356 animals present. Our goal here will be to enter equations in this row, row 12, that will project how many 0 yearolds, how many 1 year-olds, 2 year-olds, 3 year-olds and so on, are in each of these different age classes in the year one. And before we can do that, we need to enter in our parameters up in this left-hand section or our page. PAUSE the video: What were some of those parameters? Well, Tony indicated that we have a calving interval. So let s write the word Calving Interval in cell A1. Tony indicated that the calving interval was somewhere between 3.1 and 3.3 years. So let s go ahead and enter 3.1 as a starting place. But we might want to keep a note to our self that this is a parameter that has very careful research associated with it. And so we want to, when we do our sensitivity analysis, keep in mind what these bounds are around this particular number. So let s enter that as a note to our self by clicking on cell A1. And then right-clicking, 2

3 and then go to the Insert Comment function. And we ll just keep a little note to our self here, this is somewhere between 3.1 to 3.3. And that comment will reappear when you hover your mouse over that particular cell. This means a female elephant has a calf about every three years. So from that we can calculate fecundity. Well write the word Fecundity in cell A2. And we ll actually calculate the fecundity as just 1, divided by, the calving interval. This calving interval is a model input, so we ll shade that green. But let s keep the fecundity value unshaded just to remind ourselves that this is a value that is linked to the calving interval. So it will automatically change as we change the calving interval Since we re modeling only females it s important for us to track the female-to-male ratio. And so let s enter female-to-male ratio in cell A3. And as Tony indicated, we re going to assume that when a female has an offspring, it s a chance whether it will be a male or female, an equal chance of either sex. If there was a sex skew for every 100 newborns, and 70 of those hundred were female, then we would enter.7, to represent that 7 out of 10, or 70 out of a hundred of those offspring would be female. So this is a parameter we can also change and do a sensitivity analysis on. Let s set that at.5. Next we had a survival rate. And this is a survival rate to calf, so from a newborn to a calf. And Tony indicated that this was somewhere between.8 and.9. And let s split that difference and start off with a value of.85. This is a model input. And this is another type of parameter in which there is some information known. So let s remind ourselves by adding a little comment. And we ll put,.8 to.9 as a comment. And this essentially bounds the values that we would explore in a sensitivity analysis later on. Finally the survival rate of all other age classes. Tony indicated that we don t know what the value is, but we know it s very very high. And so we re going to take a stab at that by entering some high value here, and we can change that a little bit later. Now we have the information we need to actually begin modeling our population. Let s start with the easiest ones first. Let s begin by calculating and entering an equation in cell C12, that will calculate how many calves would be present in year 1. What would that be? PAUSE the video: Enter an equation to calculate how many calves would be present in year one. 3

4 Well, it s simply the number of newborns present in year 0 that also survived to become calves. And so since B4 is a survival rate and a parameter of the model, let s go ahead and anchor that, so that we re always referencing that particular cell. This just means we re counting our animals in year 0. Forty brand new newborns were counted. We ll count again, on the same day in year 1, that means that we count on the birthdays. 85% of these 40 newborns survived to become calves. We re left with 34 animals that are exactly 1 years old, when we re counting them on their birthday. Now let s count how many 2 year olds there would be. What would that formula be? PAUSE the video: Insert the formula for two year olds Right, it would just be the number of 1 year-olds that were counted in year 0 that survived to become 2 year-olds in that period. And the reference in cell B5 needs to be anchored as an absolute reference. And when we re done we press the Enter button. 90% of these 20, 1 yearolds survived and are counted on their birthday. And there are 18 of those individuals. Now this particular formula is the same for all the remaining age classes. It s just the number of animals in the preceding age class multiplied by the survival rate. So since that formula applies all the way across we can simply grab the right-hand Fill Handle and pull that all the way over to the 59 th age class. You might not want to look at the decimal points here. But you do need to keep in mind that we need to calculate those decimal points. We need to keep those as - because we re modeling a population rather than individuals, it s important that we consider the decimal points as well. And so we ll go ahead and format these as Numbers. And we ll hide the decimal points, so we re only viewing the integer side of the equation. Now we can drag our total population equation down. And we re left with 314 animals. And again, let s make that a Number, and show no decimal points. Now at this point we re able to calculate lambda, or the growth rate of the population. And we ll calculate lambda as simply ratio of two populations sizes. It s the population size in year t + 1, divided by the population size in year t. So let s go ahead and enter that equation in cell BK12. Now in this case we absolutely do want to see the decimal points, multiple decimal points if possible. And this is telling us what s the relationship between a population size in one year with a population size in a preceding year. If the population size is exactly the same from one time step to next, this ratio lambda would be 1. If the ratio between this time step and the previous 4

5 time step is greater than 1, then you know the population is increasing. And if it is less than 1, as it s showing here, you know the population has decreased, because this number is less than the number it divides by. So this population is 88% of what it was in the preceding time step. What kind of growth loss is that? Well, we could just subtract the two. 1 minus 88%, tells us that it s a 11% decline in the population size from one to the next. So, we re not going to actually calculate that through our model. But I just wanted to point out that is how you interpret the value there. Now let s scroll back over. What we have forgotten to do, is to count in the newborns to this equation. Let s go ahead and do that now. Think about this carefully. We re counting on a day, when we know that these newborns are brand new newborns. So in order for them to have been alive, and counted in year 0, we know that their mothers had to have been alive to give them birth. Let s go ahead and follow the convention that Tony used, which was to count and use the number of breeding adults in this time step. And we re going to do that by just summing all of the breeding adults in this actual time step, because we know that those females lived to actually give birth to their babies. That s the sum of the breeding adults in that time step. And each one of those animals has a fecundity rate given in cell B2. And that needs to be anchored. And furthermore, we re only going to count the females that are born. And so, we ll multiply that through by cell B3, and anchor that as well. There we have our number of newborns. And again, if you would like to not display those decimal points, just choose the Number option, and then choose the Decrease Decimal button. Let s scroll all the way back to the right side of our spreadsheet and take a look at how our population has changed. We started out with 356 animals near year 0. And we ended up with 343 animals near year 1. The ratio of these two number is Which means this number is 96% of the previous number. That represents a growth decline of 3.7 percent. Let s go back, it s worth me repeating, one more time about how these adults are calculated. As I said, when you are dealing with a model that has discreet time steps, it s really important to think through when the counting occurs. And the logic behind how many animals can be produced. If we had just said let s let the number of newborns be a function of how many adults are in the preceding time step. We would have gotten a very different number. 5

6 In fact, let s just do this up here in a blank spot on our spreadsheet. Let s count the number of animals in the preceding time step. We ll multiply those numbers, that sum, by the fecundity rate. And then multiply by the male to female ratio. In that case, we would have counted 31 animals. 31 brand new babies, as opposed to 28. But if we were to do that, that would assume that all 30, all of these animals in the time step, in green, would have survived to actually deliver their baby on this counting day in year 1. And we know, that not all those animals actually made it to produce that newborn. We know that only 90% of them did. So if we go back up here and adjust this number, and multiply it by the survival rate, we should end up by the same number that we had done before. So that is just two different ways of thinking through this problem, this very critical problem of when does the counting occur, when do the births occur, and making sure that everything is order with the biology of the species that you re modeling. The only thing you need to do now, is out model is basically working, and so we just need to double-check our formula, and make sure that our inputs are anchored where needed. And once you are convinced that you have this model and the equations are correct. Select the equations that are provided in row 12 and drag those down to row 91. And we might as well grab the lambdas and totals, as well, and that will fill out our model. One thing that s often done is to create an output file. And we re going to go ahead and list our outputs in this area here of our spreadsheet. One thing we can calculate or record as an output is the final lambda. So lets go ahead and write down the final lambda. And what is often done is to record the very last lambda of the projection. And Tony will speak to this issue right after you are done with this particular video. So lets calculate and enter the lambda that is recorded for the final year of our model. So we go all the way down to time step 80, and all the way over to where we had calculated the lambda. And that is our final model lambda. Lambda as we indicated earlier, we need to see those decimal points, so let s go ahead and show all of those. So you have a clear idea, this is a model output, so let s go ahead and shade that as a blue color. And let s also record the Final Population Size. And the final population size will be the number of animals after our 80 years of projection. And so we will use our mouse and scroll down and record the number of animals. This also is a model output, so let s shade that blue, as well. Now that your model is working, we re going to go ahead and graph how lambda changes as a function of time. To do that, let s start by selecting cell A10, and go ahead and grab cells A10 6

7 all the way through year 80 in your projection. That s going to be one series that we ll display. Press the Ctrl-Key down, because the second series that we want to display is the lambda value that is recorded in column BK. We ll start by selecting cell BK10, dragging down through the last year of the projection. I can release my Ctrl-Key and go up to the Ribbon and choose Insert, Scatter graph. And we ll choose the Scatter graph with Smooth Lines and Markers. And here we have a very funny looking graph that displays our lambda value through time. Now remember when you have a graph that is selected, Excel will add a new section up to your Ribbon called Chart Tools. And there are three tabs that are affiliated with the Chart Tools option. What we need to do is to put labels on these axes. Let s choose the Layout tab and then go over to the axis title. For the horizontal axis we ll choose Title Below Axis. And here we re just simply reflecting and displaying the year of the simulation. To clean this up a bit more. We began our simulation in the year 0 and we ran it out for 80 years. So there is no need for us to display the years 80 through a 100. Click on the axis itself, and then right-click, go to Format Axis. Under the Axis Options, let s choose the Minimum to be 0, for year 0. And lets choose Fixed, and make the Maximum that we display be 80. And then choose Close. Our Chart is still selected, so now let s go back to up to Axis Titles, once again. And this time, let s label the Vertical Axis. This represents lambda. Lambda, as we said, is a ratio of two population sizes. What we would like to do is to make sure that we reflect ranges of populations sizes that encompassed the number 1. Because 1 indicates a stable population size. Let s choose this axis, right-click and go to Format Axis. And let s let this axis have a Minimum of.94. And we ll let the Maximum, go to - let s just put a number in here - we know the population is increasing so we want to reflect numbers at least out to 1.1. Tony had indicated previously that the population was growing by about 5%, per year. So then, let s press Close. And now our graph is updated. Since only a single value, a single series is displayed here, and it s clearly labeled what this is, we no longer need this legend. And so, we can choose it and press the delete key to get rid of that value. We may also choose to have a more informative title. This could be the Lambda Over Time for Elephants. 7

8 Now we re ready to explore our model a bit more. We re going to let Tony come back on, and he is going to talk to you a little bit about properties about stage-based models and age-based models. < 00:27:00 END > 8