v x = Reproductive Value = Age-specific expectation of all future offspring

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 qx Age-specific survivorship lxType I, II, III survivorship (rectangular, diagonal, inverse hyperbolic)

2. Expectation of further life, Age-specific fecundity, mxAge of first reproduction, alpha,  — menarche Age of last reproduction, omega,  Realized fecundity at age x, lxmxNet Reproductive rate Human body louse, R0 = 31Generation Time, T = xlxmxReproductive value, vxStable vs. changing populations Residual reproductive value

3. Age of first reproduction, alpha,  — menarche Age of last reproduction, omega, Reproductive value vx , 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, Nt = N0 ertDemographic and Environmental Stochasticity

4. T, Generation time = average time from one gener- ation to the next (average time from egg to egg) vx= 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. vx = mx + S (lt / lx ) mt Residual reproductive value = age-specific expectation of offspring in distant futurevx* = (lx+1 / lx ) vx+1

7. Intrinsic rate of increase (per capita, instantaneous) r = b - d rmax and ractual — lx varies inversely with mx Stable (stationary) age distributions Leslie Matrices (Projection Matrix)Dominant Eigenvalue = Finite rate of increase 

8. Illustration of Calculation of Ex, T, R0, and vx in a Stable Population with Discrete Age Classes _____________________________________________________________________ Age Expectation Reproductive Weighted of Life Value Survivor- Realized by Realized Exvx Age (x) ship Fecundity Fecundity Fecundity lx mx lxmx x lxmx _____________________________________________________________________ 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 Sums 2.2 (GRR) 1.00 (R0) 2.78 (T) _____________________________________________________________________ E0 = (l0 + l1 + l2 + l3 + l4 + l5)/l0 = (1.0 + 0.8 + 0.6 + 0.4 + 0.4 + 0.2) / 1.0 = 3.4 / 1.0 E1 = (l1 + l2 + l3 + l4 + l5)/l1 = (0.8 + 0.6 + 0.4 + 0.4 + 0.2) / 0.8 = 2.4 / 0.8 = 3.0 E2 = (l2 + l3 + l4 + l5)/l2 = (0.6 + 0.4 + 0.4 + 0.2) / 0.6 = 1.6 / 0.6 = 2.67 E3 = (l3 + l4 + l5)/l3 = (error: extra terms) 0.4 + 0.4 + 0.2) /0.4 = 1.0 / 0.4 = 2.5 E4 = (l4 + l5)/l4 = (error: extra terms) 0.4 + 0.2) /0.4 = 0.6 / 0.4 = 1.5 E5 = (l5) /l5 = 0.2 /0.2 = 1.0 v1 = (l1/l1)m1+(l2/l1)m2+(l3/l1)m3+(l4/l1)m4+(l5/l1)m5 = 0.2+0.225+0.50+0.3+0.025 = 1.25 v2 = (l2/l2)m2 + (l3/l2)m3 + (l4/l2)m4 + (l5/l2)m5 = 0.30+0.67+0.40+ 0.03 = 1.40 v3 = (l3/l3)m3 + (l4/l3)m4 + (l5/l3)m5 = 1.0 + 0.6 + 0.05 = 1.65 v4 = (l4/l4)m4 + (l5/l4)m5 = 0.60 + 0.05 = 0.65 v5 = (l5/l5)m5 = 0.1 ___________________________________________________________________________ p. 144 delete extra terms (red)

11. Stable age distribution Stationary age distribution

13. Leslie Matrix (a projection matrix) Assume lx and mxvalues are fixed and independent of population size. px = lx+1 /lx Mortality precedes reproduction.

15. Leslie Matrix (a projection matrix) Assume lx and mxvalues are fixed and independent of population size. px = lx+1 /lx Mortality precedes reproduction.

16. n (t +1) = L n(t ) n (t +2) = L n(t +1) = L [Ln(t)] = L2 n(t ) n (t +k) = Lk 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) = ln(t). lis 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, Nt = N0 ertDemographic and Environmental Stochasticity Evolution of Reproductive Tactics: semelparous versus iteroparousReproductive effort (parental investment)

18. Estimated Maximal Instantaneous Rates of Increase (rmax, Per Capita Per Day) and Mean Generation Times ( in Days) for a Variety of Organisms ____________________________________________________________________________ Taxon Species rmax Generation Time (T) ------------------------------------------------------------------------------------------------------------------ Bacterium Escherichia coli ca. 60.0 0.014 Protozoa Paramecium aurelia 1.24 0.33–0.50 Protozoa Paramecium caudatum 0.94 0.10–0.50 Insect Tribolium confusum 0.120 ca. 80 Insect Calandra oryzae 0.110(.08–.11) 58 Insect Rhizopertha dominica 0.085(.07–.10) ca. 100 Insect Ptinus tectus 0.057 102 Insect Gibbum psylloides 0.034 129 Insect Trigonogenius globulosus 0.032 119 Insect Stethomezium squamosum 0.025 147 Insect Mezium affine 0.022 183 Insect Ptinus fur 0.014 179 Insect Eurostus hilleri 0.010 110 Insect Ptinus sexpunctatus 0.006 215 Insect Niptus hololeucus 0.006 154 Mammal Rattus norwegicus 0.015 150 Mammal Microtus aggrestis 0.013 171 Mammal Canis domesticus 0.009 ca. 1000 Insect Magicicada septendcim 0.001 6050 Mammal Homo sapiens 0.0003 ca. 7000 _____________________________________________________

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

20. Instantaneous rate of change of Nat time t is total births minus total deathsdN/dt = bN – dN = (b – d )N = rNNt = N0 ertlog Nt = log N0 + log ert = log N0 + rtlog R0 = log 1 + rt r = log R0 / T r = log l or l = er ~

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, a 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. Reproductive Effort 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.] R. A. Fisher

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

27. Inverse relationship between rmax 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