Today you are going to learn to estimate growth rate from age-structured or stage-structured population models. Demographic monitoring is considered the state of the art in population monitoring, because demographic rates are mechanistically related to population growth, and monitoring demographic rates provides a great deal of information about the reasons for population change. Using demographic rates in a matrix population model greatly enhances the usefulness of demographic monitoring. For example, over the next two activities, you will learn to use matrix population models to:
We could in principle do much of this using life tables instead, but using the matrix approach allows us to get estimates of growth rate with shorter studies than cohort life tables, and with less burdensome assumptions to meet than period life tables.
We are going to work with a stage-structured population model for ravens in the Mojave Desert. This will be a matrix version of the life table model we did last time, except for a couple of new wrinkles.
First, real raven populations aren't made up just of breeders and youngsters that aren't ready to breed yet. Non-breeding adults are also present in the population, and non-breeders can become breeders when they obtain a mate and find a suitable territory.
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Second, ravens can choose to breed in a variety of different environments within the desert. Towns in the desert give ravens access to artificial nest substrates (like the one on the left), as well as food subsidies (like trash and road-killed carrion), as well as water sources. We'll call ravens nesting in towns the "urban" population (granted, the town of Mojave, CA, is stretching the definition of "urban"). |
Ravens nesting away from towns in undeveloped desert will typically nest in Joshua trees, which are the only natural nest substrates in most places, or on cliffs if they are available. Away from towns ravens do not have access to human food subsidies, and have to find naturally occurring foods. We'll call the ravens living in undeveloped areas the "desert" population.
Between urban and desert habitats are undeveloped areas that are near enough to nest in natural substrates, but be close enough to developments for ravens to have access to at least some human food and water resources. The transition between two cover types is an "ecotone", so we will call these the "ecotonal" population.
To construct a matrix model, we either need to be able to age the ravens, or at least be able to assign them to a developmental stage. The stages that we use should be a) unambiguous, and b) internally homogeneous.
Ambiguous means unclear, or difficult to tell apart, so unambiguous stages are ones that we can tell apart for sure. Ages have an unambiguous definition - a one year old is between its first and second birthday - but once a raven is an adult, it isn't possible to tell its age anymore (a four year old and a ten year old look alike). Since we don't know how old the ravens are, we can't use an age-structured model, so no Leslie matrix for us.
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It is possible, however, to tell hatch year birds from juveniles, and juveniles from adults, so we can use a stage-based model - we will use a Lefkovitch matrix to capture the stage structure in the population. Hatch year ravens have pink mouth linings and gray irises. |
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Juveniles have brown feathers in the wings and tail, instead of glossy black like adults have. The mouth lining of juveniles is also pink, often through the second year of life. Juvenile eye color is in transition from the gray color of hatch year birds to the brown color of adults over the first two years of life as well. Between the plumage color and iris color it's possible to tell juveniles from adults |
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Adult ravens have glossy black feathers and brown irises. Once they become adults ravens can no longer be reliably aged. |
The stages we will recognize for this model will thus be the ones we can tell apart: hatch year, juvenile, and adult, with adults split between non-breeding and breeding. A diagram of this life history looks like the life history diagram below.
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From this diagram, we can identify all the demographic rates needed for our Lekovitch matrix, and where they should go.
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Note that some of these entries appear low because they are dividing a survival probabilities between two
different paths. For example, once ravens are in the non-breeding adult population their survival probability is
0.8, but this probability is split between the ravens that survive and remain non-breeders (0.7) and those that
survive and becoming breeders (0.1).
If you download this file, you'll see that the first worksheet, Urban, has this matrix in it.
Switch to the Ecotonal sheet and you'll see that ecotonal ravens have a lower hatch year survival and lower fecundity than urban birds. Ecotonal adult survival probabilities are the same as in an urban population, but breeding adults have a 10% chance of becoming non-breeders (thus, there is a SBRNB of 0.1, and a SBR of 0.86 for the ecotonal ravens). Because the habitat is not as good for breeding in ecotonal areas, we will allow breeding ravens to become non-breeders 10% of the time, but without suffering a survival cost of living in the ecotonal areas.
In the Desert sheet you'll see that ravens living in the desert, away from human-provided resources, have lower values still for hatch year survival, fecundity, and survival of breeders. Desert breeding ravens have an overall survival probability of 0.8, and have a 10% chance of becoming non-breeders.
We will estimate the growth rate and stable age distribution for the urban population first, so switch to the Urban tab.
1. Set up the model for Solver. First, we need an initial value of λ - in cell A8 write "Lambda", and in B8 enter a 1.
Next, we need to set up the formula for the characteristic equation of the matrix, det(L-λI) = 0, which we will use to find λ. As you saw in lecture, L is the matrix of demographic rates, and λI is a matrix of λ's on the main diagonal and 0's everywhere else, like this:
To keep our worksheet a little simpler, we can do the L-λI calculation in one step, without first making a λI matrix, like so:
Now we need to calculate the matrix determinant. We learned how to do these by hand in the lecture with a 2x2 matrix, but determinants are much more complicated for a 4x4 matrix - happily, there is a spreadsheet formula that will calculate it for us. In cell B15 enter =mdeterm(b10:e13). Note that this is a matrix operation, but it works on a single matrix and returns a single value, so it doesn't have to be an array formula. Label this with "Determ." in cell A15.
Your layout should look like this:
2. Use Solver to find lambda. To find the growth rate you just need to start up Solver, and have it set the determinant (in cell B15) to 0 by changing lambda (in cell B8). You should get a lambda of 1.0309, which indicates a 3.1% increase per year in the urban population. You now have an estimate of population trend that isn't based on any sort of abundance measure.
Be careful, though, because this could be an eigenvalue of the matrix but still not be the population growth rate. There are actually as many eigenvalues as there are age classes, so it's possible to end up with Solver finding one of these other eigenvalues instead. Since the growth rate is the largest positive eigenvalue, you can double-check that you have it by setting lambda to 1.5 and running Solver again (a 50% increase per year would be explosive population growth, which is not what we see - starting above any reasonable value and letting Solver work down to it ensures that Solver's solution is the largest eigenvalue).
3. Calculate the stable age distribution. Now that we know the growth rate, we can estimate the stable stage distribution. Each eigenvalue (such as λ) has an eigenvector that satisfies this relationship:
Lw = λw
where w is the eigenvector (a column vector, meaning that it is a matrix with a single column). This equation is telling us that the eigenvector, w, can be matrix multiplied by our demographic matrix, L, or can have each element multiplied by our lambda estimate, and the results will be the same.
You now have the population growth rate, and know the distribution of ages that is needed for change in population size to be smooth over time.
4. Use the Urban setup to model ecotonal and desert areas. Now that you have the setup, you can copy/paste all of the calculations to the other two sheets (don't paste special as values, you want the formulas - just a plain copy and paste will do). Make sure you paste them in the same locations in Ecotonal and Desert as they are in Urban so he cell references all point to the right places.
The other two populations are set up now, but since the demographic rates are different you need to run Solve to get the correct lambda and stable age distribution for each.
Then repeat these steps for Desert.
The population growth rates for the ecotonal and desert populations are both below 1, which means that these two populations are declining. Or at least they would be, based on the balance of births and deaths represented in the demographic matrices, but the regional population as a whole is propped up by the increasing growth rate of the urban birds. Breeding territories are limited in supply, and once they are full the surplus ravens produced in urban areas are not all able to find breeding territories there. The surplus produced by the urban population has to disperse out into ecotonal and desert sites to find territories, and even though conditions are not as good there, they at least have a chance to breed. This is a classic source-sink system, in which a large regional population is being supported by surplus production within the source.
The rest of the activity is about using these models to evaluate options for managing raven population size in the desert.
Considering the problems that unnaturally large population sizes of ravens cause for desert tortoises and other sensitive species, we might want to explore some potential ways to reduce the productivity of the urban population.
One way to evaluate the effects of various possible approaches to reducing growth rate in the urban population is to use population projection. As a baseline for comparison, we can project the population using the actual demographic rates and a starting population at stable age distribution. Then, we can see how population growth changes when we alter demographic rates, or change the numbers of ravens in each age class for the starting population. We really only need to do this for the urban birds, since urban areas are the source population for the region.
1. Duplicate the Urban worksheet. First, let's make a copy of the Urban worksheet that we can mess around with.
Switch to the Projection sheet, and delete everything except the matrix of demographic rates in A1 through E6.
In cell G2 enter "Pop. vector", and enter the numbers 27, 11, 26, and 36 - these are the stable age distribution values multiplied by 100 and rounded to a whole number, so our starting population is as close to stable age as we can get, aside from rounding error.
In cell H2 enter "t+1". You can then use the fill handle to extend to the right, which should increase the +1 part of the label - stop when you reach t+20 (column AA).
2. Project one year. This is a matrix multiplication of the demographic matrix by the population vector, so we need a 4 x 1 output range selected - select cells H3:H6, and without changing the selection type:
=mmult($b3:$e6, g3:g6)
and hit CTRL+SHIFT+ENTER to make this an array formula. The population vector for t+1 should now be in H3 through H6.
You'll see that even though we have a growth rate over 1 we still had a slight decline in numbers for juveniles - this is due to rounding error in the initial population sizes, nothing to worry about, things will settle down in another couple of years.
3. Project to year 20. Using dollar signs to identify the matrix of demographic rates makes the matrix multiplication copy and paste-able. Copy the cells in H3 to H6, and paste to i3, then continue to paste one column at a time until you get to Year 20 in column AA. You have now projected the population twenty years.
You can calculate the total population size in row 8 - label F8 "Total N", and then in G8 sum the population vector (G3 through G6). Copy and paste to the rest of the columns - these are the population sizes projected through year 20.
Confirm that lambda is predicting growth rate - in cell F9 enter "Growth rate". In cell G9 enter =h8/g8, which divides the population size at t+1 by the initial population size. Since Nt+1/Nt is what we are trying to estimate with λ, we expect the number in G9 to equal our λ for urban of 1.0309. Copy this cell to the rest of the projected years (H9 to Z9). Since rounding error prevented us from starting at exactly stable age distribution the first calculation will be a little different from 1.0309, but after a couple of years the growth rate will be 1.0309 and will stay at that number for the rest of the years. Because of this, it's possible to get a growth rate estimate by projecting a population until it reaches stable age distribution, and then using the ratio of population sizes to calculate λ.
4. Plot the numbers of individuals in each age class over time. Select cells G3 through AA6, and plot a scatter plot with connecting lines. Time will be on the x-axis and population size is on the y-axis. You should have a line for each age.
We should label the age classes - right-click and select data, and then switch row/column to get the series into the horizontal axis box. Then click the "Edit" button under horizontal axis labels and select the names of the age classes in cells A3 to A6. Once they are labeled, you can switch row/column again and the names will be retained as the series names.
Add axis labels, and use "Year" for the x-axis, and "Number" for the y-axis. Add a title and call it "Population projection".
Copy this graph by right-clicking and selecting "Copy". Then, paste a copy as a picture - right-click in a cell of the worksheet and select the right-most icon of a clipboard with a picture on it. This makes a copy of the graph that is no longer linked to the numbers in B3 through AA6, and will thus not change as you work through the rest of the activity. This will be your baseline for comparison.
We are projecting the population by multiplying a set of demographic rates by a population vector. To evaluate various strategies for reducing growth rates for urban ravens, we just need to identify which demographic rate or population vector number we would be affecting, and then change it.
For example, if we went out one year and knocked all of the nests we could find out of the trees, what would happen to the raven population? Since the ravens could build nests again in the next year, and survival isn't being affected at all, we wouldn't expect the demographic rates to change. Instead, this would be like reducing the number of hatch year ravens to 0 in the first year.
1. Project the effect of removing a year of hatch year ravens. Change the number of hatch year ravens to 0 for the first year - that is, change G3 to 0. You'll see in the graph that this causes a temporary reduction in juveniles, since there are no hatch year ravens in the first year to replace the juvenile mortalities, but this effect evens out quickly. Also, the growth rates in row 9 change for a few years while the population is re-establishing stable age distribution, but they never drop below 1.
Change the title to "No HY in year 0". Then right-click and copy the graph, then select cell A10 and right-click and paste-special as a picture (MAKE SURE you use the picture option, or this graph will change as soon as you make any other changes to the population).
2. Remove the breeding population for one year. Now we'll see what happens when we take out the breeders - set the hatch year birds back to 27 (enter 27 in G3), and set the breeders to 0. This simulates a one-time removal of the entire breeding population. You'll see the graph changes a lot - since we still only allow 10% of the non-breeding adults to enter the breeding population per year there is a slow replenishment of the breeding population. With no breeders the hatch year birds drop to zero in the second year (t+1), and with no hatch year birds the juveniles drop to 0 in the third year (t+2). This depletes the non-breeding population for several years as well, starting in year 4 (t+3). Growth rate even drops well below 1 for the first couple of years while there are too few hatch year birds produced to replace mortalities.
Even so, the distribution of age classes starts to recover and all of the stages are increasing in number again by year 10.
Change the title on the graph to "No BR in year 0", and copy and paste-special as a picture below your first graph.
There are a couple of important points to note about this part of the exercise:
This is actually to be expected - we calculated lambda from the matrix of demographic rates, without any reference to the number of individuals in the population or in each age class, so growth rate is only dependent on the demographic matrix. If we want to make the population decline, we would need to change the demographic rates.
Survival and reproductive rates are partially properties of the species and partially properties of the environment they live in, so to reduce survival or reproduction you would need to make lasting changes to one or the other of those characteristics. You could reduce reproduction by giving the birds birth control drugs, by knocking nests out of trees every year, or by preventing nesting in artificial structures. You could change survival probabilities by reducing the amount of food subsidies they get from us (e.g. preventing road kill or cleaning it up rapidly, covering trash, getting homeowners to stop feeding their pets outside), or possibly by using lethal animal control methods. To have a lasting effect on the demographic rates, these changes would have to be applied consistently year after year, or they would be like a temporary reduction in the population vector that quickly recovers.
Set the number of breeders back to 36. Now we will see if discouraging some of the breeders from breeding each year is enough to cause the population to decline.
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Imagine that we did various things to prevent nesting on artificial structures, like putting up these bird spikes on ledges. We probably can't keep every raven from nesting anywhere in town, but perhaps we could prevent 10% of the breeding population from nesting each year. Would that be enough to cause the population to decline? |
In cell E5 enter =0.1*0.96 - this will move 10% of the breeders into the non-breeding population each year, which simulates the effect or our bird spikes. Then, in cell E6 enter =0.9*0.96 - this places the remaining 90% of the surviving breeders back into the breeding population. You'll see that with a new set of rates the starting population is no longer at stable age distribution, so there is not a smooth change initially, but after about year 7 the age distribution stabilizes and the population begins to decline.
Change the title on the graph to "10% of BR become NB", and copy/paste special as a picture below the other two.
Bear in mind, our population project is a mathematical model of what would happen, and it isn't perfect. For example, if we project this population far enough it would eventually die out completely, which should make you a little skeptical. If 90% of the breeding population was still able to breed, why would the population go extinct? In fact, what would probably happen instead is that the population would decline, but when it got small enough that most breeding adults could find territories the rate of breeders moving into the non-breeder category would decline to zero again, and the population would stabilize at a smaller size. If we really wanted to cause a decline that persists over time we would need to take away additional breeding sites every year, so that 10% of the potential breeders would continue to be forced into the non-breeding population even as the population declined, until there were no more breeding sites left. Our matrix model doesn't have any way to account for these sorts of density-dependent changes, so a projection of the population assumes that all of the demographic rates stay the same as the population increases or decreases.
That's it! Save your worksheet for your report.
By the way, there are programs (like Matlab and R) that can do the eigenanalysis we need with a few commands. We will use Excel today, in part because you already know how to use it, and in part because it's easier to see what the analysis is actually doing if you work through it in Excel. If you're interested, once you're done with the analysis in Excel you can see the commands needed to do the analysis in R here.