Life Tables Discrete versus Continuous Age Classes Pivotal Age assumption (age classes) qx = force of mortality (fraction dying during age interval) qx = age-specific death rate Survivorship curves lx = fraction of initial cohort that

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Life Tables Discrete versus
Continuous Age Classes Pivotal Age assumption
(age classes) qx force of mortality
(fraction dying during age interval) qx
age-specific death rate Survivorship curves
lx fraction of initial cohort that survives to
age x ly / lx probability of living from age
x to age y Ex Expectation of further
life Realized fecundity lxmx Net reproductive
rate R0 S lxmx Generation time T S xlxmx
Reproductive value vx (stable, non-stable
poplns) Residual reproductive value vx
Stable age distributions
Lecture 13 5th March 2020
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Instantaneous rate of natural increase
(per capita) r b dwhen birth rate exceeds
death rate (b gt d), r is positivewhen death
rate exceeds birth rate (d gt b), r is
negativeEulers implicit equation S?e-rx lxmx
1(solved by iteration)If the Net
Reproductive Rate R0 is near one, r loge R0
/T
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When R0 equals one, r is zero When R0 is
greater than one, r is positiveWhen R0 is less
than one, r is negative Maximal rate of
natural increase, rmax
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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 Species
rmax Generation Time (T) ---------------------
--------------------------------------------------
------------------------------------- Bacterium E
scherichia 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 Inse
ct Gibbum psylloides 0.034 129 Insect Trigonog
enius 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 septendecim 0.001
6050 Mammal Homo sapiens 0.0003
ca. 7000 _______________________________________
__________
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Inverse relationship between rmax and generation
time, T Threshold of Annuality
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J - shaped exponential population growth
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y c mx
log Nt log N0 rt

NT
Slope r
Log N
N0
t
Time
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Instantaneous rate of change of N at time t is
total births (bN) minus total deaths (dN)dN/dt
bN dN (b d )N rNNt N0 ert
(integrated version of dN/dt rN)log Nt log
N0 log ert log N0 rtlog R0 log 1 rtr
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, v
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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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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 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
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Asplanchna (Rotifer) Microscopic multicellular
animal
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Trade-offs between present progeny and
expectation of future offspring
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Iteroparous organism
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Iteroparous organism
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Iteroparous organism
a
w
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http//www.commondreams.org/view/2011/03/07-0
Microtus
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Chittys Genetic Control Hypothesis
Could optimal reproductive tactics be involved in
driving population cycles?
Dennis Chitty
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Semelparous organism
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Semelparous organism
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Semelparous organism
mx
Age, x
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Patterns in Avian Clutch SizesAltrical versus
Precocial
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Patterns in Avian Clutch SizesAltrical versus
PrecocialNidicolous vs. NidifugousDeterminant
vs. Indeterminant Layers
N 5290 Species
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Patterns in Avian Clutch SizesOpen Ground
Nesters Open Bush Nesters Open Tree Nesters
Hole Nesters
MALE
(From Martin and Ghalambor 1999)
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Patterns in Avian Clutch SizesClassic
Experiment Flickers usually lay 7-8 eggs, but
in an egg removal experiment, a female flicker
laid 61 eggs in 63 days
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Great Tit Parus major
David Lack
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Parus major
Christopher Perrins
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European Starling, Sturnus vulgaris
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Chimney Swift, Apus apus
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Seabirds (N. Philip Ashmole) Albatrosses,
Boobies, Gannets, Gulls, Petrels, Skuas,
Terns Delayed sexual maturity, Small clutch size,
Parental care
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Albatross Egg Addition Experiment
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.
Diomedea immutabilis
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Latitudinal Gradients in Avian Clutch Size
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Latitudinal Gradients in Avian Clutch Size
Daylength Hypothesis Prey Diversity
Hypothesis Spring Bloom or Competition Hypothesis
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Latitudinal Gradients in Avian Clutch Size
Nest Predation Hypothesis Alexander Skutch
gt
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Latitudinal Gradients in Avian Clutch Size
Hazards of Migration Hypothesis
Falco eleonora
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Latitudinal Gradients in Avian Clutch Size
Hazards of Migration Hypothesis
Cyprus
Falco eleonora
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Latitudinal Gradients in Avian Clutch Size
Hazards of Migration Hypothesis
Illegal Slaughter of Songbirds in Cyprus
Ambelopouliahttps//thebirdersreport.com/conserv
ation/stop-the-illegal-slaughter-of-songbirds-in-c
yprushttp//www.10000birds.com/why-ambelopoulia-i
s-a-problem-for-europe.htmhttps//en.wikipedia.or
g/wiki/Ambelopoulia
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Latitudinal Gradients in Avian Clutch Size

• Daylength Hypothesis
• Prey Diversity Hypothesis
• Spring Bloom or Competition Hypothesis
• Nest Predation Hypothesis (Skutch)
• Hazards of Migration Hypothesis

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Evolution of Death Rates Senescence, old age,
genetic dustbin Medawars Test Tube Model
p(surviving one month) 0.9 p(surviving
two months) 0.92 p(surviving x months)
0.9xRecession of time of expression of the
overt effects of a detrimental allelePrecession
of time of expression of the effects of a
beneficial allele
Peter Medawar
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Age Distribution of Medawars test tubes
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Percentages of people with lactose intolerance
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Daylength HypothesisPrey Diversity
HypothesisSpring Bloom or Competition
HypothesisNest Predation Hypothesis
(Skutch)Hazards of Migration HypothesisEvolutio
n of Senescence

• Recession of time of expression of the overt
  effects of a detrimental allele
• Precession of time of expression of the effects
  of a
• beneficial allelle

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Joint Evolution of Rates of Reproduction and
Mortality
Sceloporus
Donald Tinkle
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J - shaped exponential population growth
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Once, we were surrounded by wilderness and wild
animals, But now, we surround them.

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S - shaped sigmoidal population growth
Pearl-Verhulst Logistic Equation
(
K N K K
(

(
(
N K
1

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Verhulst-Pearl Logistic Equation dN/dt rN
rN (N/K) rN (rN2)/K dN/dt rN 1
(N/K) rN (K N)/K dN/dt 0 when (K
N)/K 0 (K N)/K 0 when N K dN/dt
rN (1 N/K) rN (r/K)N2
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Inhibitory effect of each individual on its own
population growth is 1/K
ra rmax rmax
/
(
K)N
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bN
V
rN
V
dN
N
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Derivation of VerhulstPearl logistic equation
At equilibrium, birth rate must equal death
rate, bN dN bN b0 x N dN
d0 y N b0 x N d0 y N
Substituting K for N at equilibrium and r for b0
d0 r (x y) K or K
r/(x y)
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bN
V
rN
V
dN
N
r/(xy)
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Derivation of the Logistic Equation Derivation
of the VerhulstPearl logistic equation is easy.
Write an equation for population growth using
the actual rate of increase rN
dN/dt rN N (bN dN) N
Substitute the equations for bN and dN into
this equation dN/dt
(b0 xN) (d0 yN) N
Rearrange terms,
dN/dt (b0 d0 ) (x y)N) N
Substituting r for (b d)
and, from before, r/K for (x y), multiplying
through by N, and rearranging terms,
dN/dt rN
(r/K)N2
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Density Dependence versus Density Independence
Dramatic Fish Kills, Illustrating
Density-Independent Mortality ____________________
_______________________________


Commercial Catch Percent
Locality Before After Decline ________________
___________________________________ Matagorda 16,
919 1,089 93.6 Aransas 55,224 2,552
95.4 Laguna Madre 12,016 149
92.6 _____________________________________________
______ Note These fish kills resulted from
severe cold weather on the Texas Gulf Coast in
the winter of 1940.
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Fugitive Species
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Some of the Correlates of r- and K-Selection
____________________________________
__________________________________________________
_


r-selection K-selection
_________________________________________________
_______________________________________


Climate Variable and unpredictable
uncertain Fairly constant or predictable more
certain Mortality Often catastrophic,
nondirected, More directed, density
dependent density independent
Survivorship Often Type III Usually Types I
and II Population size Variable in time,
nonequil- Fairly constant in time, ibrium
usually well below equilibrium at or
near carrying capacity of envi- carrying
capacity of the ronment unsaturated
com- environment saturated munities or
portions thereof communities no
recolon- ecologic vacuums recolon- ization
necessary ization each year Intra- and
inter- Variable, often lax Usually
keen specific competition Selection favors 1.
Rapid development 1. Slower development 2.
High maximal rate of 2. Greater competitive
ability increase, rmax 3. Early
reproduction 3. Delayed reproduction 4. Small
body size 4. Larger body size 5. Single
reproduction 5. Repeated reproduction 6. Many
small offspring 6. Fewer, larger progeny Length
of life Short, usually less than a year Longer,
usually more than a year Leads
to Productivity Efficiency Stage in
succession Early Late, climax _________________
_________________________________________________
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Mola mola
Dr. Kirk Winemiller Texas A M. Univ.
Sturgeon
Gambusia
Sharks, skates, and Rays
Mosquito Fish
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Sequoia Tree
Dr. Kirk Winemiller Texas A M. Univ.
Dandelion
Cocoa Nut Tree