Life tables Age-specific probability statistics Force of mortality q x Survivorship l x l y / l x = probability of living from age x to age y Fecundity m x Realized fecundity at age x = l x m x Net reproductive rate R 0 =  l x m x Generation time T =  xl x m x Reproductive value v x =  (l t / l x ) m t E x = Expectation of further life

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 ”

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 _____________________________________________________________________ Sums 2.2 (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 _____________________________________________________________________ Sums 2.2 (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 (per capita) 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

J - shaped exponential population growth

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

Instantaneous rate of change of N at time t is total births (bN) minus total deaths (dN) dN/dt = bN – dN = (b – d )N = rN N t = N 0 e rt (integrated version of dN/dt = rN) log N t = log N 0 + log e rt = log N 0 + rt log R 0 = log 1 + rt (make t = T) r = log or = e r (  is the finite rate of increase)

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 septendecim MammalHomo “sapiens” (the sap) ca __________________________________________________________________ _

Inverse relationship between r max and generation time, T

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

Indeterminant Layers

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 Ronald A. Fisher

Asplanchna (Rotifer)

Trade-offs between present progeny and expectation of future offspring

Iteroparous organism

Semelparous organism

Patterns in Avian Clutch Sizes Altrical versus Precocial

Patterns in Avian Clutch Sizes Altrical versus Precocial Nidicolous vs. Nidifugous Determinant vs. Indeterminant Layers N = 5290 Species

Patterns in Avian Clutch Sizes Open Ground Nesters Open Bush Nesters Open Tree Nesters Hole Nesters MALE (From: Martin and Ghalambor 1999)

Patterns in Avian Clutch Sizes Classic Experiment: Flickers usually lay 7-8 eggs, but in an egg removal experiment, a female flicker laid 61 eggs in 63 days

Great Tit Parus major David Lack

Parus major

European Starling, Sturnus vulgaris

Chimney Swift, Apus apus

Seabirds (N. Philip Ashmole) Boobies, Gannets, Gulls, Petrels, Skuas, Terns, Albatrosses Delayed sexual maturity, Small clutch size, Parental care

Albatross Egg Addition Experiment Diomedea immutabilis An extra chick added to each of 18 nests a few days after hatching. These nests with two chicks were compared to 18 other natural “ control ” nests with only one chick. Three months later, only 5 of the 36 experimental chicks survived from the nests with 2 chicks, whereas 12 of the 18 chicks from single chick nests were still alive. Parents could not find food enough to feed two chicks and most starved to death.

Latitudinal Gradients in Avian Clutch Size

Daylength Hypothesis Prey Diversity Hypothesis Spring Bloom or Competition Hypothesis

Latitudinal Gradients in Avian Clutch Size Nest Predation Hypothesis Alexander Skutch ––––––>

Latitudinal Gradients in Avian Clutch Size Hazards of Migration Hypothesis Falco eleonora

Evolution of Death Rates Senescence, old age, genetic dustbin Medawar ’ s Test Tube Model p (surviving one month) = 0.9 p (surviving two months) = p (surviving x months) = 0.9 x recession of time of expression of the overt effects of a detrimental allele precession of time of expression of the effects of a beneficial allele Peter Medawar

Age Distribution of Medawar ’ s test tubes

Percentages of people with lactose intolerance

Joint Evolution of Rates of Reproduction and Mortality Donald Tinkle Sceloporus

J - shaped exponential population growth

Instantaneous rate of change of N at time t is total births (bN) minus total deaths (dN) dN/dt = bN – dN = (b – d )N = rN N t = N 0 e rt (integrated version of dN/dt = rN) log N t = log N 0 + log e rt = log N 0 + rt log R 0 = log 1 + rt (make t = T) r = log or = e r (  is the finite rate of increase)

Once, we were surrounded by wilderness and wild animals, But now, we surround them.