Abstract
We consider a stochastic individual-based population model with competition, trait-structure affecting reproduction and survival, and changing environment. The changes of traits are described by jump processes, and the dynamics can be approximated in large population by a non-linear PDE with a non-local mutation operator. Using the fact that this PDE admits a non-trivial stationary solution, we can approximate the non-linear stochastic population process by a linear birth-death process where the interactions are frozen, as long as the population remains close to this equilibrium. This allows us to derive, when the population is large, the equation satisfied by the ancestral lineage of an individual uniformly sampled at a fixed time T, which is the path constituted of the traits of the ancestors of this individual in past times . This process is a time inhomogeneous Markov process, but we show that the time reversal of this process possesses a very simple structure (e.g. time-homogeneous and independent of T). This extends recent results where the authors studied a similar model with a Laplacian operator but where the methods essentially relied on the Gaussian nature of the mutations.
Funding Statement
This work has been supported by the Chair “Modélisation Mathématique et Biodiversité” of Veolia Environnement-École Polytechnique-Museum National d’Histoire Naturelle-Fondation X and by the European Union (ERC, SINGER, 101054787). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. V.C.T. also acknowledges support from Labex Bézout (ANR-10-LABX-58).
Citation
Benoît Henry. Sylvie Méléard. Viet Chi Tran. "Time reversal of spinal processes for linear and non-linear branching processes near stationarity." Electron. J. Probab. 28 1 - 27, 2023. https://doi.org/10.1214/23-EJP911
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