1. Deme, demography, vital statistics of populations Population parameters, mean and variance “Life” Tables: Cohort vs. Segment Samples Age and sex specificity Homocide example: Chicago vs. England Numbers dying in each age interval Discrete vs. continuous approaches Force of Mortality q x Age-specific survivorship l x Type I, II, III survivorship (rectangular, diagonal, inverse hyperbolic)

2. Expectation of further life, Age-specific fecundity, m x Age of first reproduction, alpha,  — menarche Age of last reproduction, omega,  Realized fecundity at age x, l x m x Net Reproductive rate Human body louse, R 0 = 31 Generation Time, T =  xl x m x Reproductive value, v x Stable vs. changing populations Residual reproductive value

3. Age of first reproduction, alpha,  — menarche Age of last reproduction, omega,  Reproductive value v x, Expectation of future offspring Stable vs. changing populations Present value of all expected future progeny Residual reproductive value Intrinsic rate of increase (little r, per capita = b - d) J-shaped exponential runaway population growth Differential equation: dN/dt = rN = (b - d)N, N t = N 0 e rt Demographic and Environmental Stochasticity

4. T, Generation time = average time from one gener- ation to the next (average time from egg to egg) v x = Reproductive Value = Age-specific expectation of all future offspring p.143, right hand equation “dx” should be “dt”

5. In populations that are expanding or contracting, reproductive value is more complicated. Must weight progeny produced earlier as being worth more in expanding populations, but worth less in declining populations. The verbal definition is also changed to “the present value of all future offspring” p.146, left hand equation left out e -rt term

6. v x = m x +  (l t / l x ) m t Residual reproductive value = age-specific expectation of offspring in distant future v x * = (l x+1 / l x ) v x+1

7. Intrinsic rate of increase (per capita, instantaneous) r = b - d r max and r actual — l x varies inversely with m x Stable (stationary) age distributions Leslie Matrices (Projection Matrix) Dominant Eigenvalue = Finite rate of increase

8. Illustration of Calculation of E x, T, R 0, and v x in a Stable Population with Discrete Age Classes _____________________________________________________________________ AgeExpectation Reproductive Weighted of Life Value Survivor-Realizedby Realized E x v x Age (x) shipFecundityFecundityFecundity l x m x l x m x x l x m x _____________________________________________________________________ 0 1.0 0.0 0.00 0.00 3.40 1.00 1 0.8 0.2 0.16 0.16 3.00 1.25 2 0.6 0.3 0.18 0.36 2.67 1.40 3 0.4 1.0 0.40 1.20 2.50 1.65 4 0.4 0.6 0.24 0.96 1.50 0.65 5 0.2 0.1 0.02 0.10 1.00 0.10 6 0.0 0.0 0.00 0.00 0.00 0.00 Sums2.2 (GRR) 1.00 (R 0 ) 2.78 (T) _____________________________________________________________________ E 0 = (l 0 + l 1 + l 2 + l 3 + l 4 + l 5 )/l 0 = (1.0 + 0.8 + 0.6 + 0.4 + 0.4 + 0.2) / 1.0 = 3.4 / 1.0 E 1 = (l 1 + l 2 + l 3 + l 4 + l 5 )/l 1 = (0.8 + 0.6 + 0.4 + 0.4 + 0.2) / 0.8 = 2.4 / 0.8 = 3.0 E 2 = (l 2 + l 3 + l 4 + l 5 )/l 2 = (0.6 + 0.4 + 0.4 + 0.2) / 0.6 = 1.6 / 0.6 = 2.67 E 3 = (l 3 + l 4 + l 5 )/l 3 = (error: extra terms) 0.4 + 0.4 + 0.2) /0.4 = 1.0 / 0.4 = 2.5 E 4 = (l 4 + l 5 )/l 4 = (error: extra terms) 0.4 + 0.2) /0.4 = 0.6 / 0.4 = 1.5 E 5 = (l 5 ) /l 5 = 0.2 /0.2 = 1.0 v 1 = (l 1 /l 1 )m 1 +(l 2 /l 1 )m 2 +(l 3 /l 1 )m 3 +(l 4 /l 1 )m 4 +(l 5 /l 1 )m 5 = 0.2+0.225+0.50+0.3+0.025 = 1.25 v 2 = (l 2 /l 2 )m 2 + (l 3 /l 2 )m 3 + (l 4 /l 2 )m 4 + (l 5 /l 2 )m 5 = 0.30+0.67+0.40+ 0.03 = 1.40 v 3 = (l 3 /l 3 )m 3 + (l 4 /l 3 )m 4 + (l 5 /l 3 )m 5 = 1.0 + 0.6 + 0.05 = 1.65 v 4 = (l 4 /l 4 )m 4 + (l 5 /l 4 )m 5 = 0.60 + 0.05 = 0.65 v 5 = (l 5 /l 5 )m 5 = 0.1 ___________________________________________________________________________ p. 144 delete extra terms (red)

11. Stable age distribution Stationary age distribution

13. Leslie Matrix (a projection matrix) Assume l x and m x values are fixed and independent of population size. p x = l x+1 /l x Mortality precedes reproduction.

15. Leslie Matrix (a projection matrix) Assume l x and m x values are fixed and independent of population size. p x = l x+1 /l x Mortality precedes reproduction.

16. n (t +1) = L n(t ) n (t +2) = L n(t +1) = L [Ln(t)] = L 2 n(t ) n (t +k) = L k n(t ) With a fixed Leslie matrix, any age distribution converges on the stable age distribution in a few generations. When this distribution is reached, each age class changes at the same rate and n(t +1) = n(t). is the finite rate of increase, the real part of the dominant root or the eigenvalue of the Leslie matrix (an amplification factor). See Handout No. 1.

17. Reproductive value, intrinsic rate of increase (little r, per capita) J-shaped exponential runaway population growth Differential equation: dN/dt = rN = (b - d)N, N t = N 0 e rt Demographic and Environmental Stochasticity Evolution of Reproductive Tactics: semelparous versus iteroparous Reproductive effort (parental investment)

18. Estimated Maximal Instantaneous Rates of Increase (r max, Per Capita Per Day) and Mean Generation Times ( in Days) for a Variety of Organisms ____________________________________________________________________________ TaxonSpecies r max Generation Time (T) ------------------------------------------------------------------------------------------------------------------ BacteriumEscherichia coli ca. 60.00.014 ProtozoaParamecium aurelia1.24 0.33–0.50 ProtozoaParamecium caudatum0.94 0.10–0.50 InsectTribolium confusum 0.120 ca. 80 InsectCalandra oryzae0.110(.08–.11) 58 InsectRhizopertha dominica0.085(.07–.10) ca. 100 InsectPtinus tectus0.057 102 InsectGibbum psylloides0.034 129 InsectTrigonogenius globulosus0.032 119 InsectStethomezium squamosum0.025 147 InsectMezium affine0.022 183 InsectPtinus fur0.014 179 InsectEurostus hilleri0.010 110 InsectPtinus sexpunctatus0.006 215 InsectNiptus hololeucus0.006 154 MammalRattus norwegicus0.015 150 MammalMicrotus aggrestis0.013 171 MammalCanis domesticus0.009 ca. 1000 InsectMagicicada septendcim0.001 6050 MammalHomo sapiens0.0003 ca. 7000 _____________________________________________________

19. J - shaped exponential population growth http://www.zo.utexas.edu/courses/THOC/exponential.growth.html

20. Instantaneous rate of change of N at time t is total births minus total deaths dN/dt = bN – dN = (b – d )N = rN N t = N 0 e rt log N t = log N 0 + log e rt = log N 0 + rt log R 0 = log 1 + rt r = log R 0 / T r = log or = e r ~

21. Demographic and Environmental Stochasticity random walks, especially important in small populations Evolution of Reproductive Tactics Semelparous versus Interoparous Big Bang versus Repeated Reproduction Reproductive Effort (parental investment) Age of First Reproduction, alpha,  Age of Last Reproduction, omega, 

22. Mola mola (“Ocean Sunfish”) 200 million eggs! Poppy (Papaver rhoeas) produces only 4 seeds when stressed, but as many as 330,000 under ideal conditions

24. How much should an organism invest in any given act of reproduction? R. A. Fisher (1930) anticipated this question long ago: “It would be instructive to know not only by what physiological mechanism a just apportionment is made between the nutriment devoted to the gonads and that devoted to the rest of the parental organism, but also what circumstances in the life history and environment would render profitable the diversion of a greater or lesser share of available resources towards reproduction.” [Italics added for emphasis.] Reproductive Effort R. A. Fisher

25. Joint Evolution of Rates of Reproduction and Mortality Donald Tinkle Xantusia

27. Inverse relationship between r max and generation time, T

29. Asplanchna (Rotifer)

30. Optimal Reproductive Tactics Trade-offs between present progeny and expectation of future offspring

31. Iteroparous organism

32. Semelparous organism