Life tables Age-specific probability statistics Force of mortality q x Survivorship l x Fecundity m x Realized fecundity l x m x Net reproductive rate R 0 Generation time T Reproductive value v x

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

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 _____________________________________________________________________ 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 = 3.4 / 1.0 E 1 = (l 1 + l 2 + l 3 + l 4 + l 5 )/l 1 = ( ) / 0.8 = 2.4 / 0.8 = 3.0 E 2 = (l 2 + l 3 + l 4 + l 5 )/l 2 = ( ) / 0.6 = 1.6 / 0.6 = 2.67 E 3 = (l 3 + l 4 + l 5 )/l 3 = ( ) /0.4 = 1.0 / 0.4 = 2.5 E 4 = (l 4 + l 5 )/l 4 = ( ) /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 = = 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 = = 1.40 v 3 = (l 3 /l 3 )m 3 + (l 4 /l 3 )m 4 + (l 5 /l 3 )m 5 = = 1.65 v 4 = (l 4 /l 4 )m 4 + (l 5 /l 4 )m 5 = = 0.65 v 5 = (l 5 /l 5 )m 5 = 0.1 ___________________________________________________________________________

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 _____________________________________________________________________ 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 = 3.4 / 1.0 E 1 = (l 1 + l 2 + l 3 + l 4 + l 5 )/l 1 = ( ) / 0.8 = 2.4 / 0.8 = 3.0 E 2 = (l 2 + l 3 + l 4 + l 5 )/l 2 = ( ) / 0.6 = 1.6 / 0.6 = 2.67 E 3 = (l 3 + l 4 + l 5 )/l 3 = (error: extra terms) ) /0.4 = 1.0 / 0.4 = 2.5 E 4 = (l 4 + l 5 )/l 4 = (error: extra terms) ) /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 = = 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 = = 1.40 v 3 = (l 3 /l 3 )m 3 + (l 4 /l 3 )m 4 + (l 5 /l 3 )m 5 = = 1.65 v 4 = (l 4 /l 4 )m 4 + (l 5 /l 4 )m 5 = = 0.65 v 5 = (l 5 /l 5 )m 5 = 0.1 ___________________________________________________________________________

Stable age distribution Stationary age distribution

Intrinsic rate of natural increase r = b – d when birth rate exceeds death rate (b > d), r is positive when death rate exceeds birth rate (d > b), r is negative Euler’s implicit equation:  e -rx l x m x = 1 (solved by iteration) If the net reproductive rate R 0 is near one, r ≈ log e R 0 /T

When R 0 equals one, r is zero When R 0 is greater than one, r is positive When R 0 is less than one, r is negative Maximal rate of natural increase, r max

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 ProtozoaParamecium aurelia –0.50 ProtozoaParamecium caudatum –0.50 InsectTribolium confusum ca. 80 InsectCalandra oryzae0.110(.08–.11) 58 InsectRhizopertha dominica0.085(.07–.10) ca. 100 InsectPtinus tectus InsectGibbum psylloides InsectTrigonogenius globulosus InsectStethomezium squamosum InsectMezium affine InsectPtinus fur InsectEurostus hilleri InsectPtinus sexpunctatus InsectNiptus hololeucus MammalRattus norwegicus MammalMicrotus aggrestis MammalCanis domesticus0.009 ca InsectMagicicada septendcim MammalHomo sapiens ca __________________________________________________________________ _

J - shaped exponential population growth

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 or = e r

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, 

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

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

Asplanchna (Rotifer)

Trade-offs between present progeny and expectation of future offspring

Iteroparous organism

Semelparous organism

Patterns in Avian Clutch Sizes Altrical versus Precocial Nidicolous vs. Nidifugous Determinant vs. Indeterminant Layers Classic Experiment: Flickers usually lay 7-8 eggs, but in an egg removal experiment, a female laid 61 eggs in 63 days

Great Tit Parus major David Lack

European Starling, Sturnus vulgaris