1. Demographic Calculations (cont.) To quantify population dynamics, we clearly must take into account both survivorship and natality. To start, the variable we add to the life table is the product of l x and m x. In that way we compute the expected reproduction traceable to an individual of age x. m x gives us the number of offspring the average female of age x has (if she is alive), and l x tells us the probability that she will have survived to bear those young. Summing what that proverbial average female does over her lifespan, we determine a number called the Net Replacement Rate, R 0. It represents number of offspring left behind by an average member of the initial cohort. Since we're interested in what an average female achieves, the l x has been re-computed from total numbers to fractions of the original cohort alive at age x.

2. Here are those calculations for the sample life table we’ve been using: Age x l x m x l x m x 0 1.000 0 0 1.800 0 0 2.600 1.6 3.400 2.8 4.200 0 0 5 0 - - Another value is sometimes reported; it is called the Gross Replacement Rate (or GRR). It is the sum of m x values across all ages in the table.

3. If an average female leaves more than 1 female offspring behind over her lifespan, then R 0 > 1; the population is growing. If R 0 < 1, it's declining. There are 2 minor caveats to mention: 1)We have artificially divided the age structure into distinct classes, when processes occur continuously. The proper mathematical notation is in terms of integrals rather than sums. If the age structure has been appropriately divided, the effect is unimportant. 2) Reference to the 'average' female makes it apparent that instantaneous measures of l x at the 'birthday' could be inappropriate; we are assuming that l x is an adequate representation of the 'pivotal' age within the interval. If survivorship is a smooth function (no sudden, sharp changes within an age class) we are relatively safe.

4. To this point, we have studiously avoided speaking of generations. A typical insect life history, for example, is 1 year long, and generations do not overlap, i.e. offspring do not begin reproducing during the lifespan of their parents. In many populations of larger, longer-lived organisms there are 'grandparents'; offspring begin reproduction while their parents are still alive. In such cases R 0 may represent the contribution of an average individual, but will not adequately represent the rate of population growth over time. To assess growth rate in those populations, we need to know what the generation time is within the population to go further. We can define the generation time in terms of 'averages' - T (alternatively  or G) is the time between the birth of a female and the birth of her median th offspring.

5. To be exact, this population should also be in a stable age distribution, but even when it isn't exactly in a SAD this approximation is usually within about 10% of the real generation time. We weight every offspring produced by the cohort by the age at which the reproductive event occurs, and then calculate the average age of reproducing parents. Using our sample life table, you calculate xl x m x, sum these ‘age-weighted’ births, and divide by the total number of births, R 0. The formula:

6. Age x l x m x l x m x xl x m x 0 1.000 0 0 0 1.800 0 0 0 2.600 1.6 1.2 3.400 2.8 2.4 4.200 0 0 0 5 0 - - - 1.4 3.6  = G = 3.6/1.4 = 2.57 Once we've calculated 'r', next on the list, there are methods to correct our initial estimate, and the corrections are mostly related to errors resulting from using summation approximations rather than integrals (while also fixing any error due to the approximation for generation time).

7. A useful aside: The calculation of growth rate, r, in a population with overlapping generations is analgous to calculating interest on your bank account with compounding- Amount = Principal(1+interest rate/periods per year) periods*years or N t = N 0 (1 + r/p) pt period, p, is the number of compounding intervals per unit of time (e.g. monthly compounding equals 12 periods per year, and banks normally tell you the annual interest rate), and t is the number of time units (e.g. years for banks), so that pt is the total time over which the account (or population) is growing. Daily compounding gets you more total interest (at the same interest rate) because interest dollars start earning their own interest earlier.

8. r is the interest rate when interest is compounded instantaneously. Here is the derivation: let r/p = 1/z (z is just a dummy variable) then N t = N 0 (1 + 1/z) rzt = N 0 [(1 + 1/z) z ] rt and in the limit as z approaches   (1 + 1/z) z = e thus N t = N 0 e rt This is our initial approximation of r. If the time t in this equation is set equal to the generation time G, then the population should have grown by a factor R 0.

9. After one generation time, for each 'average' female we started with, we should now have R 0 females in the population. Therefore: R 0 = e rG ln(R 0 ) = rG r = ln(R 0 )/G For our example life table: G = 2.57, R 0 = 1.4, ln(R 0 ) =.336, and r =.131 = e r = 1.14

10. Now remember that the generation time was only an approximation. The mathematical method for correction (still an approximation to using integrals, is called Euler’s equation… 1 =  e -rx l x m x For our sample life table, using the approximate ‘r’ = 0.131: Age x l x m x l x m x e -rx l x m x 0 1.000 0 0 0 1.800 0 0 0 2.600 1.600.462 3.400 2.800.540 4.200 0 0 0 5 0 - _____ 1.002

11. If you increase the value for r, the e -rx multiplier gets smaller, and will decrease the sum. So, try r =.132… Now the sum is 0.999. So try a value between those – set r = 0.1315… Then the sum is 1.000. This is the corrected value for r. One of the other things we'd like to know is the age structure of our population when a stable age distribution has been reached. Once the corrected r has been determined, a bit of algebra using Fisher's (or Euler's) equation can give us a formula for the proportions in each age class. That formula is: C x = (e -rx l x )/[  e -rx l x ]

12. for the sample life table… Age x l x e -rx l x C x 0 1.000 1.392 1.800.7014.275 2.600.4612.181 3.400.2696.106 4.200.1182.046 5 0 __0__  = 2.5504 These proportions can be turned into numbers corresponding to any starting population you like, e.g. fractions of a cohort of 1000.

13. Sharpe and Lotka (1911), and later Lotka (1921) were the ones who proved that with a constant life table (i.e. l x and m x ) a population inevitably reaches a stable age distribution. (An aside: when such a population reaches equilibrium total size, i.e. K, the population is then said to be in a stationary age distribution.)

14. To project populations forward in time, we could use r, but there is a simple, ‘brute force’ method that is straightforward. To use it, we need to calculate one more variable from the life table, p x, called proportional survivorship: p x = l x+1 /l x = 1 - q x the proportional survivorship for the sample life table: Age x l x m x p x F x =p x m x N t+1 0 1.000 0.8  0  n t F x 1.800 0.75 0 n 0 p 0 2.600 1.66.66 n 1 p 1 3.400 2.50 1 n 2 p 2 4.200 0 0 0 n 3 p 3 5 0 - -

15. Explaining the population in the next time period: The number of newborns is the product of the number of organisms of each age times the fecundity. Fecundity is the number of babies each age has discounted by the probability that the female survives the age interval. The number in each other age class is the number one time unit younger one time unit ago times the proportion of them that survive.

16. We can anticipate the matrix approach to calculations. In it we first calculate a fertility for each age class, F i, which is, for age class i, the product of the m x and p x. This product is the number of offspring produced by a female which survived throughout the period, discounted by the probability of survival through the period. Note that this refers to age class, rather than age. The total number of newborns in the next period is the sum of fertilities for all age classes: n 0,t+1 =  F x n x,t Now we are at a point where introduction of the Leslie matrix can, if you are willing to work with matrices, simplify calculations…

17. This matrix has the fertilities (F x ) for age classes as its first row and the proportional survivorship (p x ) as subdiagonal elements. If, alternatively, data was collected in the form of the stages of organisms in the population, the probabilities of transition between stages (or of remaining in the same stage into an additional time period) are recorded in the matrix, which is then properly a transition (or Lefkovich) matrix. Before even beginning to set up and use the Leslie matrix, you need to understand the basics of matrix algebra…

18. An Introduction to Matrix Algebra Matrices are simply rectangular arrays of numbers or variables. The size of a matrix is important. Size is indicated by dimensions (m x n) which are the number of rows (m) and the number of columns (n). Matrices can take the form of column vectors (m x 1) or row vectors (1 x n), as well. Matrices can be added, subtracted, multiplied by a constant (a `scalar`), or by other matrices (as long as they have appropriate numbers of rows and columns), they may have an inverse, and they can be transposed. There are, additionally, certain characteristics associated with matrices: determinants, eigenvalues and eigenvectors. Addition and subtraction of matrices is straightforward. Corresponding elements (the value in a given row x column position) are added or subtracted.

19. Addition and subtraction of matrices, like addition and subtraction of numbers, is transitive, i.e. A + B = B + A. |a 11 a 12 | + |b 11 b 12 | = |a 11 +b 11 a 12 +b 12 | |a 21 a 22 | |b 21 b 22 | |a 21 +b 21 a 22 +b 22 | The matrix sum is the sum of the corresponding elements in the individual matrices. Subtraction of matrices occurs the same way. To multiply a matrix by a scalar, each element in the matrix is multiplied by that scalar value: X x |a 11 a 12 | = |xa 11 xa 12 | |a 21 a 22 | |xa 21 xa 22 |

20. The transpose of a matrix A, designated A`, is constructed by switching the rows and columns of it. The elements in the matrices above have subscripts. By reversing the subscripts for each element, and placing the modified elements in the positions then appropriate, the transpose has been constructed. |a 11 a 12 | T = |a 11 a 21 | |a 21 a 22 | |a 12 a 22 | Multiplication of two matrices is slightly more complicated. The product only exists if the number of columns in the first matrix equals the number of rows in the second. The process is also intransitive, i.e. A x B is not equal to B x A. Each element in the matrix which results from multiplication is the sum of the products of corresponding elements in the first row of matrix A and the first column of matrix B, the second row of A with the second column of B, etc…

21. In a standard notation: c i,j =  k=1 to n [a i,k b k,j ] A sample multiplication: |1 2| x |5 6| = |5+14 6+16 | = |19 22| |3 4| |7 8| |15+28 18+32| |43 50| The inverse of a matrix corresponds in a general way to the inverse of a number or function in ordinary algebra, i.e. if y = ax then a -1 y = x

22. The parallel matrix equation, which would be equivalent to a set of linear equations, would be: y = Ax and the 'solution' would involve the use of the inverse matrix, i.e.: A -1 y = x Actual calculation of the inverse of a matrix is best left to computer programmes, and inverses only exist for square matrices (those in which the numbers of rows and columns are equal). For a 2 x 2 matrix, the calculations are readily done by hand. Here is the abstract ‘formula’ to calculate that inverse: |a b| -1 = 1 |d -b| |c d| ad-bc |-c a|

23. The determinant of a matrix is another important characteristic. It is straightforward to calculate for a 2 x 2 matrix, but best left to computer programmes for larger matrices. For that 2 x 2: det |a b| = ad - bc |c d| The determinant of a matrix has a geometrical meaning; it is the volume of the n-dimensional parallelogram formed by the vectors represented by each of the n rows of the matrix. (In a simple expression for what that means, Oy vey!!)

24. The eigenvector(s) of a matrix are vectors which remain the same when multiplied by the matrix, or are changed only by scalar multiplication. That is, x is an eigenvector of the matrix L, specifically a right eigenvector (note that it is written to the right of the matrix) if: Lx = x In this equation is an eigenvalue of the matrix. There are also left eigenvectors for a matrix. They satisfy the equation: y'L = y'

25. It would be possible now to calculate eigenvalues and eigenvectors of a matrix. They have meaning when they are determined for a Leslie matrix: is the dominant eigenvalue of the matrix. The left and right eigenvectors are the stable age structure (C x ) and reproductive values (V x ) for the life table. We won’t even try that for any matrix derived from a life table. However, like other calculations, it can be done without too much hassle for a 2 x 2 matrix… Eigenvalues and eigenvectors are calculated by using the characteristic equation. If Lx = x, then: Lx - x = 0

26. To reorganize this equation, we need to use the 'identity matrix'. This special matrix has 1s along the diagonal, and 0s elsewhere. It is a matrix version of the number 1, i.e. IL = LI = L. Using the identity matrix: (L - I)x = 0 For this equation to hold and have a non-zero solution for x, det(L - I) = 0 Making this equation hold involves solving an n th degree equation in for its roots, which are the eigenvalues of the matrix.

27. For a simple 2x2 example, the eigenvalues of the matrix |2 1| |3 4| det (L - I) = det |2- 1| | 3 4- | s are solutions of the parabolic equation: 2 - 6 + 5 = 0 ( - 5)( - 1) = 0 and the eigenvalues are lambda 1 = 1 and lambda 2 = 5.

28. By substituting the eigenvalues into the characteristic equation, the eigenvectors can be determined. Substitute = 5 into the matrix multiplication below, and try x 1 = 1. You will then find that x 2 = 3, so that one eigenvector is (1,3)'. Substituting the other eigenvalue ( = 1), then if x 1 = 1, x 2 = -1, and the eigenvector for the eigenvalue = 1 is (1,-1)'. Thus, |2- 1 | |x 1 | = |0| | 3 4- | |x 2 | |0| Assuming you are now mostly confused, recognize that this is not a course in linear algebra, and you do not have to understand all this. It is time to move on to… The Leslie Matrix