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Stochastic age-structured modelling: dynamics,
genetics and estimation
Steinar Engen, Norwegian University of Science and Technology
Abstract
In his book published in 1930 R.A.Fisher introduced the concept of reproductive value in relation to his
fundamental theorem of natural selection, claiming that the theorem is valid also for age-structured
populations provided that individuals are weighted by their reproductive value rather than using the
individual numbers to define gene frequencies. This was based on the fact that the total reproductive
value always grows exactly exponential in the absence of density regulation. Although Fisher defined this
concept for deterministic models only, it has proved very useful also in stochastic modeling.
Reproductive values can be used to define stochastic dynamics of age-structured populations and point
toward simple definitions of a few main parameters, the deterministic growth rate and the demographic
and environmental variances, sufficient for accurate description of the dynamics. These parameters can
be estimated from individual recordings of vital rates and applied in diffusion approximations for the
population size. We show that there is a simple relation between the demographic variance and genetic
drift in age-structured populations and how to use this to estimate effective population size and its agespecific components from individual data of age, survival and fecundity. Reproductive values are also
useful in describing stabilizing fluctuating selection on quantitative characters leading to a rather simple
time series model for the stochastic evolution of mean phenotype. We also demonstrate how Fisher's
concept can be used to estimate age-specific components of fluctuating selection by rather simple
regression models.
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Fisher’s reproductive value. Demographic and environmental
stochasticity. Reproductive value dynamics.
Individual reproductive value. Estimation.
Genetic drift. Effective population size. Relations to reproductive
value demographic variance. Estimation of genetic drift.
Fixation of mutations in age-structured populations.
Fluctuating stabilizing selection in age-structured populations
Estimation of directional selection in age-structured populations.
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Between years
Among individuals,
within years
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Distribution of
individual
fitness W
for two bird
species
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The rate of increase in fitness of any
organism at any time is equal to its genetic
variance in fitness at that time
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The rate of increase in fitness of any
organism at any time is equal to its genetic
variance in fitness at that time
The rigour of its demonstration requires that the terms
employed should be used stricktly as defined; the ease
of its interpretation may be increased by appropriate
conventions of measurement. For example, the ratio p:q
should stricktly be evaluated at any instant by
enumeration, not necessarily of the census population,
but of all individuals having reproductive value, weighted
according to the reproductive value of each.
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The total reproductive value V
of the population growth
exactly exponential, and lnV
has exactly linear growth.
R. A. Fisher
The age distribution
approaches the stable age
distribution
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Log of reproductive value have
white noise if the matrices are
temporally uncorrelated
Generalization of result of Tuljapurkar
1982
Engen et al. 2005, 2007
Population size
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diffusion approximation
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0
0
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Time in years
100
150
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Age-structured population, no density regulation.
(The total reproductive value serves as a filter)
Population size
Total reproductive
value
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The distribution of the bootstrap replicates (n=1000) of the
population growth rate, the demographic variance, and the
environmental variance, for the population of Columbian
ground squirrel. The black lines denote the estimated values
and the 95 % confidence intervals are 1.12-1.22, 0.28-0.37
and 0.004-0.045.
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Vindenes, Y., Engen, S. and Sæther, B.-E. 2008. Individual
heterogeneity in vital parameters and demographic stochasticity.
American naturalist 171:455-467.
Vindenes, Y., Engen, S. and Sæther, B.E. 2011. Integral
projection models for finite populations in a stochastic
environment. Ecology 92: 1146-1156.
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Effective population size
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Effective population size
determines
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•
•
•
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Expected rate of random genetic drift
Increase in inbreeding
Loss of selectively neutral heterozygosity
Loss of genes (loss og genetic diversity)
Rate of fixation of beneficial mutations
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Allele frequeny weighted by
reproductive value as
proposed by Fisher in 1930.
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According to
Wright’s
definition of
variance
effective
population size
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For details see
Engen et al. 2005,
Genetics
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Bootstrap replicates
The Siberian Jay
Data from Phillip
Gienapp, University of
Helsinki, recorded
1987-2005, analysed
by Bernt-Erik Sæther,
NTNU.
Environmental
stochasticity is also
included in this
analysis.
(Ratio of effective and actual population size)
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Lee, A.M. Engen, S. and Sæther, B.E. 2011. The influence of
persistent individual differences and age at maturity on effective
population size. Proceedings of the Royal Society, B.
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From modern stochastic
population dynamics
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Vindenes,Y., Lee, A.M., Engen, S. and Sæther, B.-E. 2010.
Fixation of slightly beneficial mutations: Effects of life history.
Evolution 64: 1063-1075.
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One-dimensional
character, stabilizing
fluctuating selection
For details see
Engen et al. 2011,
Evolution.
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For papers by our group on this
subject 2005- 2012 see
http://www.math.ntnu.no/~steinaen/
The rigour of its demonstration requires
that the terms employed should be used
stricktly as defined; the ease of its
interpretation may be increased by
appropriate conventions of measurement.
For example, the ratio p:q should stricktly
be evaluated at any instant by
enumeration, not necessarily of the census
population, but of all individuals having
reproductive value, weighted according to
the reproductive value of each.