The Tools of Demography and Population Dynamics We begin with the basic math of demography, in a situation where the dynamics are density-independent. We also assume the simplest possible situation: annual plants with no seed bank and non-overlapping generations. Why these plants? Considering annuals, we can avoid complications with year-to-year survivorship and, with no seed bank, we can evaluate the population with no lags due to later germination of banked seeds. Note, however, that weeds (frequently annuals) are very commonly major contributors to seed banks.

The basic tool is the ‘life’ table The basic tool is the ‘life’ table. For plants the question is whether analysis is more successful using age (as works for animals) or stage (potentially size). Let’s begin by going back to what you should have seen before – age-based calculations. We have a big advantage in making calculations for an annual without a seed bank, but before we come back to that special case, let’s review the more general one… The life table parameters we need are: survivorship lx - the probability of living from birth to age x age specific natality mx – the number of female young born to an average female at age x We table those values for each age from birth until all members of the cohort we follow have died.

First the general case and the calculations we can make… proportion surviving fecundity Age #alive lx mx 0 100 1.0 0.0 1 80 .8 0.2 2 60 .6 0.3 3 40 .4 1.0 4 40 .4 0.6 5 20 .2 0.1 6 0 0 ---- mx= 2.2 The mx is called the Gross Reproductive Rate.

The gross reproductive rate indicates that a mother in this population will produce 2.2 daughters if she lives to the maximum age. However, the gross reproductive rate ignores the mortality schedule evident in the life table. We know that 100% of the cohort does not survive to the maximum age. So, to determine the real contribution of an average female, we need to incorporate mortality. You do so by multiplying each mx times the corresponding lx. The summed result is called the Net Reproductive Rate, and called R0 in short form.

survivorship fecundity Age lx mx lxmx 0 1.0 0.0 0 1 .8 0.2 0.16 2 .6 0.3 0.18 3 .4 1.0 0.4 4 .4 0.6 0.24 5 .2 0.1 0.02 6 0 ---- ----- R0 =  lxmx = 1.0

The sum for this life table is 1.0. That means that an average female in this population leaves behind 1 daughter over her lifetime. (It is an assumption that there is 1 male offspring to replace the father, as well.) Since the female parent is exactly replaced by her female offspring, this population will remain constant in size from one generation to the next. Very small changes in survivorship or fecundity could shift this population to one that would grow or one that would decline over time... Try doing the calculation for this life table if you either increase or decrease l4 or m4.

Once you have calculated R0, you can use the formula for  (generation time) to approximate it, then use the values for R0 and  to calculate r. Age class lx mx lxmx xlxmx 0 1.0 0.0 0 0 1 .8 0.2 0.16 0.16 2 .6 0.3 0.18 0.36 3 .4 1.0 0.4 1.2 4 .4 0.6 0.24 0.96 5 .2 0.1 0.02 0.10 6 0 ---- ----- R0 = 1.0  = 2.78

Since the time is one generation, we know that the ratio Nt/N0 is the net replacement rate. Thus, the equation becomes: Nt/N0 = R0 = er Now take the logs of both sides of the equation… ln (R0) = r ln (1.0) = r (2.78) 0 = 2.78r r = 0 Follow your revised values through these same calculations. What are the effects on r?

The r calculated this way is a good approximation The r calculated this way is a good approximation. We can correct the approximation. The mathematical method for correction (still an approximation to using integrals), is called Euler’s equation… 1 =  e-rxlxmx Begin with the r you got from initial calculations, then adjust if necessary until the sum is within ±0.01. age lx mx e-rx e-rxlxmx 0 1.0 0.0 1.0 0 1 .8 0.2 1.0 0.16 2 .6 0.3 1.0 0.18 3 .4 1.0 1.0 0.4 4 .4 0.6 1.0 0.24 5 .2 0.1 1.0 0.02 6 0 ----  = 1.00

The e-rx calculation was artificially easy here, in the first cycle of calculation, because r = 0. Let’s make it a little more difficult and interesting by changing m4 from 0.6 to 0.8… age lx mx lxmx xlxmx e-rx e-rxlxmx 0 1.0 0.0 0 0 1.0 0 1 .8 0.2 0.16 0.16 .9735 .1557 2 .6 0.3 0.18 0.36 .9478 .1706 3 .4 1.0 0.4 1.20 .9227 .3690 4 .4 0.8 0.32 1.28 .8983 .2874 5 .2 0.1 0.02 0.10 .8746 .0174 6 0 ---- ____ ____ _____ R0=1.08 3.10 1.0001  = 3.10/1.08 = 2.87 r = (ln 1.08)/2.87 = .0268

Now we can use the correct r to calculate a number of characteristics of this life table that will reappear as matrix characteristics in the next set of lecture notes. There are two valuable characteristics: the proportions in each age class when the population has reached a stable age distribution and the reproductive value of each age class. A stable age distribution is one in which the proportions in each age class remain constant through time, even if the population is growing or declining. The reproductive value is the relative contribution a female of age x makes (through her reproduction at that age and in her remaining life) to the future population.

The proportion that is age x in a stable age distribution is called Cx, and calculated using this formula: From the earlier calculations we already have e-rx for each age. That makes calculating the proportions easy.

Age lx e-rx e-rxlx Cx 0 1 1.0 1.0 .3076 1 .8 .9735 .7788 .2395 2 .6 .9478 .5686 .1749 3 .4 .9227 .3691 .1135 4 .4 .8983 .3593 .1105 5 .2 .8746 .1749 .0538 6 0 _____  = 3.2507 These are the proportions that will eventually occur if this life table is fixed through time and independent of population size.

The reproductive value is easy to calculate in a population that is not changing in size. The formula then is: However, when a population is growing or declining, the formula and calculations are more complicated: Note that it is convention to measure reproductive value relative to that at birth. Once more earlier calculations did a lot of the work.

Age lx e-rx e-rx/lx e-rxlxmx Vx/V0 0 1 1.0 1.0 0 1.0 1.0 1 .8 .9735 1.2168 .1557 1.0 1.2168 2 .6 .9478 1.5796 .1706 .8443 1.3336 3 .4 .9227 2.3067 .3690 .6737 1.5540 4 .4 .8983 2.2457 .2874 .3047 0.6842 5 .2 .8746 4.373 .0174 .0174 0.0761 6 0 The age pattern in Vx is a simplification of a common, real-world pattern. Typically, reproductive value peaks at the onset of reproduction (at age ). All reproduction is traceable to the smaller number that survive mortality during the pre-reproductive period, therefore they are of greater reproductive value than the larger number around at birth of the cohort.

Due to this pattern in reproductive value, we can logically assume that natural selection has acted to protect the reproductive ages. That selection is evident in the pattern of qx, the death rate at different ages:

There is a better way to make these calculations There is a better way to make these calculations. The basic ‘tool’ is called the Leslie matrix. It puts the life table into a matrix form, and allows you to ‘project’ the population forward in time.