Publiée 29 juillet 2026
PhD Position F/M Bias in evolutionary studies induced by rare observations and applications to paleontological records
Inria
Nancy, Grand-Est 54000, France
CDI
Contexte et atouts du poste
Biological and mathematical context
Observation and sampling bias are known to play a key role in statistical studies in ecology,
evolution and epidemiology. In paleontology, the rareness and time inhomogeneity of ob-
servations introduce important bias in the estimation of extinction rates and times [SJL82],
taxonomic duration [SS97] or diversity indices [MBB13]. For example, the rareness of
observations is known to introduce a strong bias in measurements of species abundances,
which is modified according to the so-called "size-biased distribution". Specific statistical
methods are being developed, either based on subsampling, on maximum likelihood in
statistical models (for example for occupancy statistics [Fo16]), or on stochastic models
representing jointly the dynamics of species and the process of observation. For examples,
birth-death models are commonly used in paleo-phylogenetics [St10] and the observation
process is usually represented as a Poisson process [SL16,St10]. In many models, the
Poisson process of observation is assumed to be independent of the population dynamics,
which is often not represesented, but in general the rate of obsevration of a species is
proportional to its size.
The above-mentionned biases are now taken into account in most of the empirical
works in biology and, in particular, it is well-known that empirical abundance estimates are
biased toward larger values than expected, particularly in the case of rare obsevrations.
However, even though questions of evolution are central in paleontology, very few works
(if none) address the possibility of a bias induced by the rareness of observations on the
adaptation itself of observed species. Indeed, it can be expected that, due to the fact
that the observation rate is proportional to a species abundance, rare observations will
favor species which have undergone a specific evolutionary route, leading them to higher
abundance and longer survival than normaly expected. This bias may have strong influence
on the paleontological patterns, but is hard to describe. We propose an approach based on
stochastic models from the theory of adaptive dynamics.
The theory of adaptive dynamics [MGM+96] aims at studying the long term evolution
of biological populations taking explicitly into account ecological interactions which drive
the selection process. The success of this theory relies on the canonical equation of
adaptive dynamics (CEAD) [DL96], a differential equation which describes the approximate
long term dynamics of the dominant trait in the population as driven by the gradient
of the invasion fitness of mutants, and the evolutionary branching criterion derived in
[MGM+96] which gives conditions on the fitness function under which, after reaching
a steady state of the CEAD, the evolutionary process drives the population to diversify
from a state with a single dominant trait to states with two (or more) coexisting dominant
traits which evolve in opposite directions (Fig. 1). The mathematical analysis of these
phenomena relies on parameter scalings combining large population, rare mutations and
small mutations asymptotics, applied to stochastic individual-based models (IBM), i.e.
models which describe each individual event in the population, including births, deaths and mutations. Under a joint scaling of large population and rare mutations, the process
converges to the so-called trait substitution sequence (TSS) [MGM+96,CM11], which
describes evolution as a succession of random mutant invasions followed by deterministic
fast competition phases where unfit traits are eliminated. Applying a scaling of small
mutation to the TSS then allows to recover the CEAD and the evolutionary branching
criterion [CM11].
Mission confiée
Project description
This PhD project proposes to use an approach based on stochastic models related to adaptive
dynamics, specifically simplified versions of the IBM, the TSS and the CEAD, combined with
a Poisson observation process with a rate proportional to the population size, to analyse the
dynamics of the process conditionally on being observed before its extinction, in a regime
of rare observations. The goal is to evaluate the bias introduced by this conditioning on the
evolutionary dynamics and to examine if it can explain features of paleontological records
such as the presence of periods of long stasis or the so-called "lilliput effect".
A key ingredient in this analysis will be played by parameters scaling: in addition to
the rate of observation which will converge to zero, we also need to consider evolutionary
models where extinction is certain. This is the case for standard IBM, but simplified models
of evolution like TSS do not allow for extinction. Hence, we will need to consider a
modified version of the TSS where extinction is possible, considering for example a similar
process as the one of [CL07]. The asymptotic analysis then requires to tune carefully the
rate of observation with respect to the time to extinction of the population. Two main
mathematical question will be considered: 1) in a regime of large population, this will
require to use tools from the theory of large deviations; 2) in a small population regime,
we will need to use spectral theoretic tools from the theory of quasi-stationary distributions
(QSD) [MV12,KS+11] to characterize the population state before extinction and the rate
of extinction.
In the simplest models, the PhD student will seek for results as explicit as possible. In
more complex models like structured population models or the TSS, the main part of the
analysis will be numerical.
Principales activités
Tasks
The first task of the PhD student will be to get familiar with the biological and statistical
methods to account for observation or sampling bias in ecology, and more specifically in
paleontology. The PhD student will also learn the basic tools and methods from the theory
of large deviations and of quasi-stationary distributions.
The second task of the PhD student will be to study the general question of the project
on very simple models of evolution with explicit population dynamics. The first model
will deal with a species with only two phenotypes, the initial one being disadvantaged in
terms of survival. The main question will be to characterize the dynamics of the model
conditionally on being observed under the two analytical frameworks 1) (large deviations)
and 2) (QSD) described above.
The PhD student will progressively increase the complexity of the models, from models
with linear advantageous mutations, in the sense that each possible mutation can only
improve the survival properties of the population, to TSS models with extinction. When
the mathematical analysis of the model will not be possible, the PhD student will study nu-
merically the evolutionary dynamics conditionally on observation. The relevant numerical
method for such a study is unkown, as far as we know, so part of the work will consist in
developing appropriate numerical tools.
In parallel, biological questions will be addressed in the light of the obtained mathe-
matical results. For example, can the evolutionary observation bias studied above explain
the stasis periods often observed in paleontological records? We can expect that rare obser-
vations bias the evolutionary dynamics towards faster adaptation. One way to answer this
question is to study how the speed of evolution in the CEAD is modified when conditioned
on rare observations. Another question is the lilliput effect, i.e. the observed decrease
in animal body size in genera that have survived a major extinction. This phenomenon
could be explained by incorporating in the model known allometries between body size
and population size and see how, in a context of relatively fast extinction, rare observations
biases population size. During his exploration of the literature, the PhD student may find
other general observations in paleontology to which his mathematical findings will be
applied.
A final ambitious question could also be considered during the PhD: how is the phe-
nomenon of evolutionary branching influenced by conditioning on rare observations? This
is probably a difficult mathematical questions, that can be tackled only numerically, but this
could help understand why species radiations are empirically faster than observed today
paleontological records.
Bibliography
[CL07] Nicolas Champagnat. Amaury Lambert. Evolution of discrete populations and the canonical diffusion of adaptive dynamics. Ann. Appl. Probab. 17(1), 102-155, 2007.
[CM11] N. Champagnat and S. Méléard. Polymorphic evolution sequence and evolutionary branching. Probab. Theory Relat. Fields, 151:45-94, 2011.
[DL96] U. Dieckmann and R. Law. The dynamical theory of coevolution: a derivation from stochastic ecological processes. J. Math. Biol., 34(5-6):579-612, 1996.
[Fo16] Michael Foote; On the measurement of occupancy in ecology and paleontology. Paleobiology 42(4), 707-729, 2016. doi: https://doi.org/10.1017/pab.2016.24
[MBB13] Philip D. Mannion, Roger B.J. Benson, Richard J. Butler, Vertebrate palaeobiodiversity patterns and the impact of sampling bias, Palaeogeography, Palaeoclimatology, Palaeoecology 372, 1-4, 2013.
[MV12] Sylvie Méléard, Denis Villemonais. Quasi-stationary distributions and population processes, Probability Surveys 9, 340-410, 2012.
[MGM+96] J.A.J. Metz, S.A.H. Geritz, G. Meszéena, F.J.A. Jacobs and J.S. van Heerwaarden. Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In
Stochastic and spatial structures of dynamical systems (Amsterdam, 1995), Konink. Nederl.
Akad. Wetensch. Verh. Afd. Natuurk. Eerste Reeks, 45, pp. 183-231. North-Holland,
Amsterdam, 1996.
[KS+11] Klebaner, F. C., Sagitov, S., Vatutin, V. A., Haccou, P., Jagers, P. Stochasticity in the adaptive dynamics of evolution: the bare bones. Journal of biological dynamics 5(2), 147-162, 2011.
[SJL82] Philip W. Signor, III, Jere H. Lipps, 1982. Sampling bias, gradual extinction patterns and
catastrophes in the fossil record. Geological Implications of Impacts of Large Asteroids and
Comets on the Earth, Leon T. Silver, Peter H. Schultz
[SS97] Solow AR, Smith W. On fossil preservation and the stratigraphic ranges of taxa. Paleobiology 23(3), 271-277, 1997. doi:10.1017/S0094837300019680
[St10] Tanja Stadler, Sampling-through-time in birth-death trees, Journal of Theoretical Biology
267(3), 396-404, 2010.
[SL16] Starrfelt J, Liow LH. How many dinosaur species were there? Fossil bias and true richness estimated using a Poisson sampling model. Philos Trans R Soc Lond B Biol Sci. 371(1691):20150219, 2016. doi: 10.1098/rstb.2015.0219.
Compétences
Skills
The candidate should have skills and knowledge in theoretical ecology, in particular
adaptive dynamics, and in stochastic modeling of population dynamics and the basic related
probabilitic tools. An experience, or a willingness to develop skills, in numerical analysis of
stochastic models will be highly appreciated. An interest in pluridisciplinary projects and
willingness to learn the scientific language of another discipline is also welcome.
Supervision
The PhD thesis will be co-supervised by Nicolas Champagnat and Sylvain Billiard
Contact
[email protected]; [email protected]
Avantages
Rémunération
2300 € gross/month
Biological and mathematical context
Observation and sampling bias are known to play a key role in statistical studies in ecology,
evolution and epidemiology. In paleontology, the rareness and time inhomogeneity of ob-
servations introduce important bias in the estimation of extinction rates and times [SJL82],
taxonomic duration [SS97] or diversity indices [MBB13]. For example, the rareness of
observations is known to introduce a strong bias in measurements of species abundances,
which is modified according to the so-called "size-biased distribution". Specific statistical
methods are being developed, either based on subsampling, on maximum likelihood in
statistical models (for example for occupancy statistics [Fo16]), or on stochastic models
representing jointly the dynamics of species and the process of observation. For examples,
birth-death models are commonly used in paleo-phylogenetics [St10] and the observation
process is usually represented as a Poisson process [SL16,St10]. In many models, the
Poisson process of observation is assumed to be independent of the population dynamics,
which is often not represesented, but in general the rate of obsevration of a species is
proportional to its size.
The above-mentionned biases are now taken into account in most of the empirical
works in biology and, in particular, it is well-known that empirical abundance estimates are
biased toward larger values than expected, particularly in the case of rare obsevrations.
However, even though questions of evolution are central in paleontology, very few works
(if none) address the possibility of a bias induced by the rareness of observations on the
adaptation itself of observed species. Indeed, it can be expected that, due to the fact
that the observation rate is proportional to a species abundance, rare observations will
favor species which have undergone a specific evolutionary route, leading them to higher
abundance and longer survival than normaly expected. This bias may have strong influence
on the paleontological patterns, but is hard to describe. We propose an approach based on
stochastic models from the theory of adaptive dynamics.
The theory of adaptive dynamics [MGM+96] aims at studying the long term evolution
of biological populations taking explicitly into account ecological interactions which drive
the selection process. The success of this theory relies on the canonical equation of
adaptive dynamics (CEAD) [DL96], a differential equation which describes the approximate
long term dynamics of the dominant trait in the population as driven by the gradient
of the invasion fitness of mutants, and the evolutionary branching criterion derived in
[MGM+96] which gives conditions on the fitness function under which, after reaching
a steady state of the CEAD, the evolutionary process drives the population to diversify
from a state with a single dominant trait to states with two (or more) coexisting dominant
traits which evolve in opposite directions (Fig. 1). The mathematical analysis of these
phenomena relies on parameter scalings combining large population, rare mutations and
small mutations asymptotics, applied to stochastic individual-based models (IBM), i.e.
models which describe each individual event in the population, including births, deaths and mutations. Under a joint scaling of large population and rare mutations, the process
converges to the so-called trait substitution sequence (TSS) [MGM+96,CM11], which
describes evolution as a succession of random mutant invasions followed by deterministic
fast competition phases where unfit traits are eliminated. Applying a scaling of small
mutation to the TSS then allows to recover the CEAD and the evolutionary branching
criterion [CM11].
Mission confiée
Project description
This PhD project proposes to use an approach based on stochastic models related to adaptive
dynamics, specifically simplified versions of the IBM, the TSS and the CEAD, combined with
a Poisson observation process with a rate proportional to the population size, to analyse the
dynamics of the process conditionally on being observed before its extinction, in a regime
of rare observations. The goal is to evaluate the bias introduced by this conditioning on the
evolutionary dynamics and to examine if it can explain features of paleontological records
such as the presence of periods of long stasis or the so-called "lilliput effect".
A key ingredient in this analysis will be played by parameters scaling: in addition to
the rate of observation which will converge to zero, we also need to consider evolutionary
models where extinction is certain. This is the case for standard IBM, but simplified models
of evolution like TSS do not allow for extinction. Hence, we will need to consider a
modified version of the TSS where extinction is possible, considering for example a similar
process as the one of [CL07]. The asymptotic analysis then requires to tune carefully the
rate of observation with respect to the time to extinction of the population. Two main
mathematical question will be considered: 1) in a regime of large population, this will
require to use tools from the theory of large deviations; 2) in a small population regime,
we will need to use spectral theoretic tools from the theory of quasi-stationary distributions
(QSD) [MV12,KS+11] to characterize the population state before extinction and the rate
of extinction.
In the simplest models, the PhD student will seek for results as explicit as possible. In
more complex models like structured population models or the TSS, the main part of the
analysis will be numerical.
Principales activités
Tasks
The first task of the PhD student will be to get familiar with the biological and statistical
methods to account for observation or sampling bias in ecology, and more specifically in
paleontology. The PhD student will also learn the basic tools and methods from the theory
of large deviations and of quasi-stationary distributions.
The second task of the PhD student will be to study the general question of the project
on very simple models of evolution with explicit population dynamics. The first model
will deal with a species with only two phenotypes, the initial one being disadvantaged in
terms of survival. The main question will be to characterize the dynamics of the model
conditionally on being observed under the two analytical frameworks 1) (large deviations)
and 2) (QSD) described above.
The PhD student will progressively increase the complexity of the models, from models
with linear advantageous mutations, in the sense that each possible mutation can only
improve the survival properties of the population, to TSS models with extinction. When
the mathematical analysis of the model will not be possible, the PhD student will study nu-
merically the evolutionary dynamics conditionally on observation. The relevant numerical
method for such a study is unkown, as far as we know, so part of the work will consist in
developing appropriate numerical tools.
In parallel, biological questions will be addressed in the light of the obtained mathe-
matical results. For example, can the evolutionary observation bias studied above explain
the stasis periods often observed in paleontological records? We can expect that rare obser-
vations bias the evolutionary dynamics towards faster adaptation. One way to answer this
question is to study how the speed of evolution in the CEAD is modified when conditioned
on rare observations. Another question is the lilliput effect, i.e. the observed decrease
in animal body size in genera that have survived a major extinction. This phenomenon
could be explained by incorporating in the model known allometries between body size
and population size and see how, in a context of relatively fast extinction, rare observations
biases population size. During his exploration of the literature, the PhD student may find
other general observations in paleontology to which his mathematical findings will be
applied.
A final ambitious question could also be considered during the PhD: how is the phe-
nomenon of evolutionary branching influenced by conditioning on rare observations? This
is probably a difficult mathematical questions, that can be tackled only numerically, but this
could help understand why species radiations are empirically faster than observed today
paleontological records.
Bibliography
[CL07] Nicolas Champagnat. Amaury Lambert. Evolution of discrete populations and the canonical diffusion of adaptive dynamics. Ann. Appl. Probab. 17(1), 102-155, 2007.
[CM11] N. Champagnat and S. Méléard. Polymorphic evolution sequence and evolutionary branching. Probab. Theory Relat. Fields, 151:45-94, 2011.
[DL96] U. Dieckmann and R. Law. The dynamical theory of coevolution: a derivation from stochastic ecological processes. J. Math. Biol., 34(5-6):579-612, 1996.
[Fo16] Michael Foote; On the measurement of occupancy in ecology and paleontology. Paleobiology 42(4), 707-729, 2016. doi: https://doi.org/10.1017/pab.2016.24
[MBB13] Philip D. Mannion, Roger B.J. Benson, Richard J. Butler, Vertebrate palaeobiodiversity patterns and the impact of sampling bias, Palaeogeography, Palaeoclimatology, Palaeoecology 372, 1-4, 2013.
[MV12] Sylvie Méléard, Denis Villemonais. Quasi-stationary distributions and population processes, Probability Surveys 9, 340-410, 2012.
[MGM+96] J.A.J. Metz, S.A.H. Geritz, G. Meszéena, F.J.A. Jacobs and J.S. van Heerwaarden. Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In
Stochastic and spatial structures of dynamical systems (Amsterdam, 1995), Konink. Nederl.
Akad. Wetensch. Verh. Afd. Natuurk. Eerste Reeks, 45, pp. 183-231. North-Holland,
Amsterdam, 1996.
[KS+11] Klebaner, F. C., Sagitov, S., Vatutin, V. A., Haccou, P., Jagers, P. Stochasticity in the adaptive dynamics of evolution: the bare bones. Journal of biological dynamics 5(2), 147-162, 2011.
[SJL82] Philip W. Signor, III, Jere H. Lipps, 1982. Sampling bias, gradual extinction patterns and
catastrophes in the fossil record. Geological Implications of Impacts of Large Asteroids and
Comets on the Earth, Leon T. Silver, Peter H. Schultz
[SS97] Solow AR, Smith W. On fossil preservation and the stratigraphic ranges of taxa. Paleobiology 23(3), 271-277, 1997. doi:10.1017/S0094837300019680
[St10] Tanja Stadler, Sampling-through-time in birth-death trees, Journal of Theoretical Biology
267(3), 396-404, 2010.
[SL16] Starrfelt J, Liow LH. How many dinosaur species were there? Fossil bias and true richness estimated using a Poisson sampling model. Philos Trans R Soc Lond B Biol Sci. 371(1691):20150219, 2016. doi: 10.1098/rstb.2015.0219.
Compétences
Skills
The candidate should have skills and knowledge in theoretical ecology, in particular
adaptive dynamics, and in stochastic modeling of population dynamics and the basic related
probabilitic tools. An experience, or a willingness to develop skills, in numerical analysis of
stochastic models will be highly appreciated. An interest in pluridisciplinary projects and
willingness to learn the scientific language of another discipline is also welcome.
Supervision
The PhD thesis will be co-supervised by Nicolas Champagnat and Sylvain Billiard
Contact
[email protected]; [email protected]
Avantages
- Subsidized meals
- Partial reimbursement of public transport costs
- Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
- Possibility of teleworking (after 6 months of employment) and flexible organization of working hours
- Professional equipment available (videoconferencing, loan of computer equipment, etc.)
- Social, cultural and sports events and activities
- Access to vocational training
- Social security coverage
Rémunération
2300 € gross/month
