Spine for interacting populations and sampling

We consider Markov jump processes describing structured populations with interactions via density dependance. We propose a Markov construction with a distinguished individual which allows to describe the random tree and random sample at a given time via a change of probability. This spine construction involves the extension of type space of individuals to include the state of the population. The jump rates outside the spine are also modified. We apply this approach to some issues concerning evolution of populations and competition. For single type populations, we derive the diagram phase of a growth fragmentation model with competition and the growth of the size of birth and death processes with multiple births. We also describe the ancestral lineages of a uniform sample in multitype populations.

[1]  J. Bertoin,et al.  A probabilistic approach to spectral analysis of growth-fragmentation equations , 2017, 1701.05061.

[2]  P. Kušter.  Asymptotic Growth of Controlled Galton-Watson Processes , 1985 .

[3]  D. Vere-Jones.  Markov Chains , 1972, Nature.

[4]  Viet Chi Tran,et al.  Dynamics of lineages in adaptation to a gradual environmental change , 2021 .

[5]  On the Asymptotic Behaviour of Discrete Time Stochastic Growth Processes , 1987 .

[6]  Olle Nerman,et al.  The asymptotic composition of supercritical, multi-type branching populations , 1996 .

[7]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[8]  Russell Lyons,et al.  A Simple Path to Biggins’ Martingale Convergence for Branching Random Walk , 1998, math/9803100.

[9]  A. Rouault,et al.  KPP equation and supercritical branching brownian motion in the subcritical speed area. Application to spatial trees , 1988 .

[10]  Geometric rate of growth in population-size-dependent branching processes , 1984 .

[11]  A. Kyprianou,et al.  Spines, skeletons and the strong law of large numbers for superdiffusions , 2013, 1309.6196.

[12]  S. Karlin,et al.  The differential equations of birth-and-death processes, and the Stieltjes moment problem , 1957 .

[13]  J. Engländer.  Spatial Branching in Random Environments and with Interaction , 2014 .

[14]  J. Fontbona,et al.  Ray–Knight representation of flows of branching processes with competition by pruning of Lévy trees , 2015, 1506.00046.

[15]  Supercritical multitype branching processes: the ancestral types of typical individuals , 2003, Advances in Applied Probability.

[16]  Matthew I. Roberts,et al.  The coalescent structure of continuous-time Galton–Watson trees , 2017, The Annals of Applied Probability.

[17]  Bertrand Cloez,et al.  Limit theorems for some branching measure-valued processes , 2011, Advances in Applied Probability.

[18]  J. Bertoin.  Compensated fragmentation processes and limits of dilated fragmentations , 2016 .

[19]  L. Addario-Berry,et al.  The front location in branching Brownian motion with decay of mass , 2017 .

[20]  J. Bertoin.  Markovian growth-fragmentation processes , 2017 .

[21]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[22]  Yueyun Hu,et al.  Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees , 2007, math/0702799.

[23]  Jean Bertoin,et al.  Random fragmentation and coagulation processes , 2006 .

[24]  Russell Lyons,et al.  A Conceptual Proof of the Kesten-Stigum Theorem for Multi-Type Branching Processes , 1997 .

[25]  Matthew I. Roberts.  A simple path to asymptotics for the frontier of a branching Brownian motion , 2011, 1106.4771.

[26]  Bertrand Cloez,et al.  A non‐conservative Harris ergodic theorem , 2019, Journal of the London Mathematical Society.

[27]  S. Varadhan,et al.  On a variational formula for the principal eigenvalue for operators with maximum principle. , 1975, Proceedings of the National Academy of Sciences of the United States of America.

[28]  A. Kyprianou,et al.  Branching Brownian motion in strip: survival near criticality , 2012, 1212.1444.

[29]  K. Athreya.  Change of measures for Markov chains and the LlogL theorem for branching processes , 2000 .

[30]  A. Wakolbinger,et al.  “Trees under attack”: a Ray–Knight representation of Feller’s branching diffusion with logistic growth , 2013, 1305.1202.

[31]  A. Marguet.  Uniform sampling in a structured branching population , 2016, Bernoulli.

[32]  S. Mischler,et al.  Spectral analysis of semigroups and growth-fragmentation equations , 2013, 1310.7773.

[33]  M. Manhart,et al.  Markov Processes , 2018, Introduction to Stochastic Processes and Simulation.

[34]  S. Méléard,et al.  Stochastic Models for Structured Populations , 2015 .

[35]  Andreas E. Kyprianou,et al.  Fixed Points of the Smoothing Transform: the Boundary Case , 2005 .

[36]  Poisson representations of branching Markov and measure-valued branching processes. , 2011, 1104.1496.

[37]  A. Kyprianou,et al.  Strong law of large numbers for branching diffusions , 2007, 0709.0272.

[38]  Andreas Greven,et al.  A variational approach to branching random walk in random environment , 1993 .

[39]  A. Etheridge,et al.  Survival and extinction in a locally regulated population , 2004 .

[40]  T. Kurtz.  Approximation of Population Processes , 1987 .

[41]  Russell Lyons,et al.  Conceptual proofs of L log L criteria for mean behavior of branching processes , 1995 .