Life Histories
• What is a life history?
Stearns 1992
http://ic.ucsc.edu/~whs68/bio150/07LifeTableLifeHistory.pdf#search=%22life%20history%20strategies%20age%20size%20at%20maturity%22
Life History Traits
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Size at birth
Growth patterns
Age at maturity
Size at maturity
#, size, sex ratio of offspring
Age & size-specific reproductive investment
Age & size-specific mortality schedules
Length of life
Life Histories
Life Histories
Two views of the evolution of life histories, both of which recognize a primary (first) axis based on body size. (A)
The slow–fast continuum, where the second axis of life history reflects long-lived species (slow life histories)
compared with short-lived species (high mortality, particularly for juveniles). A tradeoff of survival and
reproduction produces low and high levels of reproductive success, respectively. (B) The second axis of life
histories reflects different “lifestyles” based on ease of resource acquisition from herbivores and marine
mammals (with high production of offspring) to alternative lifestyles (see text) in which mortality is reduced. In
the latter, production of offspring is reduced by the tradeoff of survival and reproduction.
Dobson 2007, PNAS
Life Histories
Ryan and Semlitsch 1998, PNAS
Life History Evolution
• Phenotype consists of demographic traits
connected by constraints (trade-offs) and
interact to to determine individual fitness.
• Trade-offs:
– Current reproduction & survival
– Current reproduction & future reproduction
– #, size, sex of offspring
Demography
• What does demography study?
Demography
• Originally to forecast population growth.
• Allows to calculate strength of selection on
life history traits for many conditions.
• Uses (st)age, size schedules of
reproduction and mortality.
• Assumes changes are due to gene
substitutions
– Explores fate of gene substitutions with agespecific effects
Reproduction
Discrete vs. Overlapping Generations
a. Discrete generations – a “female” reproduces once and dies so only
a single generation exists at any one time – e.g., annual plants,
grasshoppers
b. Overlapping generations – a female can reproduce multiple times, so
more than one generation exists at the same time – e.g., perennial
polycarpic plants, rodents
Growth equation for discrete generations
Simplest model of population growth is exponential:
Where
Nt+1 = R0Nt
Nt is the number of individuals at time t
N0 is the number of individuals at time 0
R0 is the net reproductive rate i.e.number of female
offspring per female per generation (averaged across
all ages)
There is no density dependence, frequency
dependence, genetic variability, or age structure
- Adding density dependence leads to logistic growth
Constant multiplication rate
density-independent
Geometric or exponential population growth, discrete generations,
reproductive rates constant. Starting population size, N =10.
If Ro < 1 population size decreases
Population size (N)
Time (t)
Population size (N)
K
Time (t)
http://www.marine.rutgers.edu/pinelands/Course%20material/ESO/18_Population%20Growth.pdf#search=%2218_Population%20Growth.pdf%22
http://www.marine.rutgers.edu/pinelands/Course%20material/ESO/18_Population%20Growth.pdf#search=%2218_Population%20Growth.pdf%22
Life Tables
• Under overlapping generations
• Keep track of females (since their fecundity limits pop
growth)
• LT represent birth and death probabilities
• Cohort life table: follow individuals born at = time through
life
• Horizontal life table: measure individuals of ≠ age alive at
= time
• C & H are = if birth and death rates are constant
http://ic.ucsc.edu/~whs68/bio150/07LifeTableLifeHistory.pdf#search=%22life%20history%20strategies%20age%20size%20at%20maturity%22
Population Growth
• Euler (1760)
• Lotka (1913)
• There is a certain age distribution to which a population
will tend,
• given a fixed life table,
• a fixed sex ratio,
• and a fixed schedule of age-specific fecundity
• At that stable age distribution the pop will grow
exponentially at constant rate
Euler-Lotka equation
Specifies the relationship of
• Age at maturity
• Age at last reproduction
• Probability of survival to a given age class
to the rate of growth of the population (r)
• Can think of it as comparing growth rates (fitness) of clones with ≠ LH
• Analogous to analysing the marginal effect of a gene substitution with
impact on birth and death rates in sexually ourcrossing pop.
Instantaneous rate of natural increase
∫
∞
0
− rx
e l x mx d x = 1
Survivorship Fecundity Mortality
to age x
@x
@x
Euler-Lotka equation
Instantaneous rate of natural increase
∫
∞
0
− rx
e l x mx d x = 1
Survivorship Fecundity Mortality
to age x
@x
@x
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Pop may not grow at that rate if age distribution is not stable but it tends
towards that rate
Exponential growth includes positive, zero and negative rates
It describes the quality of growth: smooth, differentiable, or monotonic
Describes the spread of an allele that determines if it gets fixed or
eliminated
Stable Age Distribution
• SAD: Proportion of individuals in each age
class is stable
• Pop can increase, not grow, or decrease
• When r = 0, stationary age distribution
• Generation time: avg age of mothers of
newborns in a pop with SAD.
Rates of Increase
• r : intrinsic rate of natural increase = per capita
instantaneous rate of increase of a pop in SAD
• Ro : per-generation rate of multiplication of a
population = # daughters expected per ♀ per
lifetime
• λ : per-time-unit rate of multiplication
• λ = er and r = ln(λ) when r = 0, λ = 1, Ro = 1 and
pop growth is 0.
Reproductive Value
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Expectation of offspring from a female of age A to the end of
her life
RV weighs contribution of individuals of different age to pop growth
beyond that age
1.
•
Has into account having survived to age A to reproduce, prob of
surviving beyond A and associated fecundity
2.
RV compares sensitivity of fitness to events at different ages
3.
When pop grows, offspring produced later contribute less to fitness,
because older offspring may be reproducing already (new offspring
is discounted by rate of pop growth and delay of their birth)
RV is defined relative to newborn females which is 1
4.
Residual Reproductive Value
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RRV = RV at next age class – births at
current age
RRV ≠ RV of the next age class
Cost of [current] reproduction:
1. Decrease survival to next age class
2. RV of next age class
3. Both
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LH evolution: RRV per age class should be
balanced to maximize fitness
http://ic.ucsc.edu/~whs68/bio150/07LifeTableLifeHistory.pdf#search=%22life%20history%20strategies%20age%20size%20at%20maturity%22
Leslie Matrix models
(Population Projection Matrices)
• Special case of a projection matrix for an age-classified population
• Leslie Matrix models allow age and size to be taken into account.
Effective fecundities:
Fi = pi-1mi
⎡ N o (t + 1) ⎤ ⎡ Fo F1 F2
⎢ N (t + 1) ⎥ ⎢ S 0 0
⎢ 1
⎥=⎢ 0
⎢ N 2 (t + 1) ⎥ ⎢0 S1 0
⎢
⎥ ⎢
⎢⎣ N 3 (t + 1) ⎥⎦ ⎢⎣0 0 S 2
Nt+1 =
L
F3 ⎤ ⎡ N 0 (t ) ⎤
⎥ ⎢
0 ⎥ N1 (t ) ⎥
⎥
•⎢
0 ⎥ ⎢ N 2 (t ) ⎥
⎥ ⎢
⎥
0 ⎥⎦ ⎢⎣ N 3 (t ) ⎥⎦
x
Nt
Population density of youngest class at t+1
= FoNo+F1N1+F2N2+F3N3……..FkNk
Population density of all others at t+1
= PoNo+P1N1+P2N2+P3N3……..PkNk
⎡ N o (t + 1) ⎤ ⎡ Fo F1 F2
⎢ N (t + 1) ⎥ ⎢ S 0 0
⎢ 1
⎥=⎢ 0
⎢ N 2 (t + 1) ⎥ ⎢0 S1 0
⎢
⎥ ⎢
⎢⎣ N 3 (t + 1) ⎥⎦ ⎢⎣0 0 S2
F3 ⎤ ⎡ N 0 (t ) ⎤
⎥ ⎢
0 ⎥ N1 (t ) ⎥
⎥
•⎢
0 ⎥ ⎢ N 2 (t ) ⎥
⎥ ⎢
⎥
0 ⎥⎦ ⎢⎣ N 3 (t ) ⎥⎦
With age-structure, the only transitions that can happen are from one
age to the next and from adult ages back to the first age class
Stage or class number – generation t+1
1
1
2
3
4
5
6
7
2
Stage or class number – generation t
3
4
5
6
7
http://ic.ucsc.edu/~whs68/bio150/07LifeTableLifeHistory.pdf#search=%22life%20history%20strategies%20age%20size%20at%20maturity%22
Fitness Measures
Ro
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Assumes variation in G does not affect outcome of selection
Only # of offspring per lifetime matters no matter how long it
takes to produce them (i.e. assumes r near 0)
•Ro cannot be compared among populations or clones because G differs
and Ro is a per generation rate of increase
•Assumes stable births and deaths rates
r
• r captures better the age distribution of selection pressures on traits
than Ro
•Fisher (1930) developed r to measure fitness of genotypes
•Used by Cole (1954) and Lewontin (1965) to evolution of age at
maturity and age distribution of reproductive effort
•Problem: r holds in stable homogeneous environments with no density
dependent effects
•Assumes stable births and deaths rates
Deterministic versus stochastic models
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In deterministic models, there is
one outcome based on the
input parameters
In reality, biological systems
are probabilistic, e.g., there are
varying degrees of probability
of a specific outcome
Probabilities of biological
outcomes, in turn, are driven by
the various environmental
factors that influence specific
processes
Because of this, stochastic or
probabilistic models most
commonly used
a = stochastic growth rate:
Use coin toss to decide whether 1 offspring (h) or 3 (t). N0 = 6, average
birth rate = 2
Multiplication rate density-dependent
In most populations, growth rate
decreases as population increases
–Competition for
food/nutrients/water/light increases
–Disease increases
–Predation increases
–Birth rate falls
–Death rate rises
Density dependence of birth and death rates
Geometric: 1.
dN / dt = r N
Logistic:
dN / dt = r N [(K – N)/K]
2.
Where K is the upper asymptote or maximum
value of N, the i.e. “Carrying capacity”
Note (1 – N/K) = (K – N) / K
(K-N)/K is the unutilized opportunity for
population growth.
This is the differential form of the logistic curve
(cannot be used to calculate N in this form).
Sensitivity & Elasticity
• Fitness = survival + reproduction
• Fitness of a LH variant with a particular life
table: λ = er
• How sensitive is λ = er to changes in each
trait in the matrix while holding others
constant?
• Sensitivity tells us how strong selection is
on those traits.
Sensitivity & Elasticity
• Problem: fecundity and survival measured
in different units and scales
• Elasticities: dimensionless (i.e.
comparable) sensitivity coefficients
• Elasticities = percentage change in λ
caused by % change in mortality or
fecundity
Sensitivity & Elasticity
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Sensitivity: absolute rate of change of λ1 with respect to
absolute change in a matrix element
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Elasticity: relative rate of change of λ1 with respect to relative
change in a matrix element
1
∂λ1
aij ∂λ1 aij
=
= Sij
Eij =
1
∂aij λ1 ∂aij λ1
aij
λ1
•
∂λ1
Sij =
∂aij
Sensitivity & elasticity of vital rates
s
s
S rk = ∑∑ Sij
i =1 j =1
∂aij
∂rk
rk ∂aij
Erk = ∑∑ Eij
aij ∂rk
i =1 j =1
s
s
Sensitivity & Elasticity
• Sensitivities are equivalent to selective pressures and
selective gradients
• Fitness is more sensitive to changes in LH traits in
younger than older individuals
• Particular sensitivities depend on life table structure, i.e.
context dependent or situational
• E.g.: how much change in age at maturity, survival terms
or feculdity terms to produce a 10% increase in r?
Answer differs for different life tables.
• When r near 0, fitness sensitive to survival and fecundity
but not to age at maturity
• When r high, sensitive to changes in all traits for early
maturity
• When maturity is delayed, r sensitive to survival and
fecundity changes.
Stage-structured sea turtle model
Sea turtle elasticities
Predicted growth rates
Effective population size
(Wright 1938)
The effective population size of a population is the size of an ideal population
which behaves the same as the real population in question.
Sex Ratio
Population size
Reproduction
Allele freq
Inbreeding
•Ne is the effective population size
•Nf is the number of females
•Nm is the number of males
•Nt is the population size at a particular time
•t is the number of times the population is sampled
Ne is related to the rate of heterozygosity loss per generation from the
population. The proportion of genetic variation remaining in the population per
generation is:
(1 - 1/2Ne)
http://www.zoology.ubc.ca/~whitlock/bio434/LectureNotes/05.EffectiveSize/EffectiveSize.html
Probability of Extinction
⎛d ⎞
P0,t = ⎜ ⎟
⎝b⎠
N0
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Pielou (1977)
d = per capita death rate
b = per capita birth rate
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Allee effect