Deme, demography, vital statistics of populations Population parameters, mean and variance

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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)
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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
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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
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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
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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
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vx mx S (lt / lx ) mt Residual
reproductive value age-specific
expectation of offspring in distant
future vx (lx1 / lx ) vx1
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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 ?
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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 1.00 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
0.20.2250.500.30.025 1.25 v2
(l2/l2)m2 (l3/l2)m3 (l4/l2)m4 (l5/l2)m5
0.300.670.40 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)
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Stable age distribution Stationary age
distribution
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Leslie Matrix (a projection matrix)
Assume lx and mx values are fixed and independent
of population size. px lx1 /lx Mortality
precedes reproduction.
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Leslie Matrix (a projection matrix)
Assume lx and mx values are fixed and independent
of population size. px lx1 /lx Mortality
precedes reproduction.
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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) l n(t). l 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.
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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)
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Estimated Maximal Instantaneous Rates of
Increase (rmax, Per Capita Per Day) and Mean
Generation Times ( in Days) for a Variety of
Organisms ________________________________________
____________________________________ Taxon Specie
s rmax Generation Time (T) ------------------
--------------------------------------------------
---------------------------------------------- Bac
terium Escherichia coli ca.
60.0 0.014 Protozoa Paramecium aurelia 1.24
0.330.50 Protozoa Paramecium
caudatum 0.94 0.100.50 Insect Triboli
um 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 ________________________________
_____________________
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J - shaped exponential population growth
http//www.zo.utexas.edu/courses/THOC/exponential.
growth.html
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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

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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, ?
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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
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How much should an organism invest in any given
act of reproduction? R. A. Fisher (1930)
anticipated this question long agoIt 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
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Joint Evolution of Rates of Reproduction and
Mortality
Xantusia
Donald Tinkle
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Inverse relationship between rmax and generation
time, T
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Asplanchna (Rotifer)
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Optimal Reproductive Tactics Trade-offs between
present progeny and expectation of future
offspring
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Iteroparous organism
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Semelparous organism