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 lx

Type I, II, III survivorship

(rectangular, diagonal, inverse hyperbolic)

Expectation of further life,

Age-specific fecundity, mx

Age of first reproduction, alpha, —

menarche

Age of last reproduction, omega,

Realized fecundity at age x, lxmx

Net Reproductive rate

Human body louse, R0 = 31

Generation Time, T = xlxmx

Reproductive value, vx

Stable vs. changing populations

Residual reproductive value

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 ert

Demographic and Environmental Stochasticity

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”

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 handequation left oute-rt term

QuickTime™ and a decompressor

are needed to see this picture.

vx = mx + (lt / lx ) mt

Residual reproductive value =

age-specific expectation of

offspring in distant future

vx* = (lx+1 / lx ) vx+1

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

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 Ex vx

Age (x) ship Fecundity Fecundity Fecundity lx mx lxmx x lxmx

_____________________________________________________________________0 1.0 0.0 0.00 0.00 3.40 1.001 0.8 0.2 0.16 0.16 3.00 1.252 0.6 0.3 0.18 0.36 2.67 1.403 0.4 1.0 0.40 1.20 2.50 1.654 0.4 0.6 0.24 0.96 1.50 0.655 0.2 0.1 0.02 0.10 1.00 0.106 0.0 0.0 0.00 0.00 0.00 0.00Sums 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 / 1.0E1 = (l1 + l2 + l3 + l4 + l5)/l1 = (0.8 + 0.6 + 0.4 + 0.4 + 0.2) / 0.8 = 2.4 / 0.8 = 3.0E2 = (l2 + l3 + l4 + l5)/l2 = (0.6 + 0.4 + 0.4 + 0.2) / 0.6 = 1.6 / 0.6 = 2.67E3 = (l3 + l4 + l5)/l3 = (error: extra terms) 0.4 + 0.4 + 0.2) /0.4 = 1.0 / 0.4 = 2.5E4 = (l4 + l5)/l4 = (error: extra terms) 0.4 + 0.2) /0.4 = 0.6 / 0.4 = 1.5E5 = (l5) /l5 = 0.2 /0.2 = 1.0v1 = (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 = 0.65v5 = (l5/l5)m5 = 0.1

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p. 144 deleteextra terms (red)

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

Stable age distribution

Stationary age distribution

Leslie Matrix (a projection matrix)

Assume lx and mx values are fixed and independent ofpopulation size. px = lx+1 /lx Mortality precedes reproduction.

Leslie Matrix (a projection matrix)

Assume lx and mx values are fixed and independent ofpopulation size. px = lx+1 /lx Mortality precedes reproduction.

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) = 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.

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 ert

Demographic and Environmental Stochasticity

Evolution of Reproductive Tactics: semelparous versus iteroparous

Reproductive effort (parental investment)

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 0.014Protozoa Paramecium aurelia 1.24 0.33–0.50Protozoa Paramecium caudatum 0.94 0.10–0.50Insect Tribolium confusum 0.120 ca. 80 Insect Calandra oryzae 0.110(.08–.11) 58Insect Rhizopertha dominica 0.085(.07–.10) ca. 100Insect Ptinus tectus 0.057 102Insect Gibbum psylloides 0.034 129Insect Trigonogenius globulosus 0.032 119Insect Stethomezium squamosum 0.025 147Insect Mezium affine 0.022 183Insect Ptinus fur 0.014 179Insect Eurostus hilleri 0.010 110Insect Ptinus sexpunctatus 0.006 215Insect Niptus hololeucus 0.006 154Mammal Rattus norwegicus 0.015 150Mammal Microtus aggrestis 0.013 171Mammal Canis domesticus 0.009 ca. 1000Insect Magicicada septendcim 0.001 6050Mammal Homo sapiens 0.0003 ca. 7000

_____________________________________________________

Exponential population growth under the assumption that the rate of increase per individual, r, remains constant with changes in population density. Note that a straight-line estimate of the rate of population growth at time t becomes more and more accurate as t1 and t2 converge; in the limit, as t1 and t2 approach t, or ∆t —> 0, the rate of population growth equals the slope of a line tangent to the curve at time t (open circle).

J - shaped exponential population growth

http://www.zo.utexas.edu/courses/THOC/exponential.growth.html

Instantaneous rate of change of N

at time t is total births minus

total deaths

dN/dt = bN – dN = (b – d )N = rN

Nt = N0 ert

log Nt = log N0 + log ert = log N0

+ rt

log R0 = log 1 + rt r = log

R0 / T

r = log or = er

~

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

R. A. Fisher

Joint Evolution of Rates of Reproduction and Mortality

Donald Tinkle

Xantusia

Inverserelationshipbetween rmax

and generation time, T

Asplanchna (Rotifer)

Optimal Reproductive TacticsTrade-offs between present progenyand expectation of future offspring

Iteroparous organism

Semelparous organism