A compact containment result for nonlinear historical superprocess approximations for population models with trait-dependence

Abstract

E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. AbstractWe consider an approximating sequence of interacting population models with branching, mutation and competition. Each individual is characterized by its trait and the traits of its ancestors. Birth-and death-events happen at exponential times. Traits are hereditarily transmitted unless mutation occurs. The present model is an extension of the model used in [9], where for large populations with small individual biomasses and under additional assumptions, the diffusive limit is shown to converge to a nonlinear historical superprocess. The main goal of the present article is to verify a compact containment condition in the more general setup of Polish trait-spaces and general mutation kernels that allow for a dependence on the parent's trait. As a by-product, a result on the paths of individuals is obtained. An application to evolving genealogies on marked metric measure spaces is mentioned where genealogical distance, counted in terms of the number of births without mutation, can be regarded as a trait. Because of the use of exponential times in the modeling of birth-and deathevents the analysis of the modulus of continuity of the trait-history of a particle plays a major role in obtaining appropriate bounds.

Dynamics of lineages in adaptation to a gradual environmental change

“…Let us compute explicitly I, the expectation appearing in the right hand side of (50). Recall that our probability space is endowed with the probability measure P. Let (F B t ) t≥0 be the filtration of the Brownian motion B and define the new probability Q by…”

“…The proof of Theorem 1.1 is now sketched. Our approach mixes here two points of view based on the stochastic individual-based model: on the one hand, the spinal approach as developed for branching diffusion in [3,41,42,60,61] and on the other hand, the historical processes, as introduced in Dawson and Perkins [21,66,67] and Dynkin [28], and then developed in Méléard and Tran [62] (with a correction, see [50,74]).…”

Evolving phylogenies of trait-dependent branching with mutation and competition, Part I: Existence

“…The latter model has been extended in [Kli14] by considering general mutation operators on Polish trait spaces.…”

“…In this particular setting the claim is covered by [Kli14, Theorem 3.4] (applied to r(t, y) := β(π K (y t )) with π K : K × R + → K denoting the projection map on to the trait, b(t, y) := Cβ(π K (y t )), D(t, y) ≡ 0, U (t, y, y ′ ) ≡ 0 and α N ((κ, a), d(κ ′ , a ′ )) := α N (κ, dκ ′ )δ a+ 1 N 1{κ =κ ′ } (da ′ ). Here the left hand sides refer to the set-up used in [Kli14] and the right hand sides to our set-up. ).…”

“…In the present article, the need for such an analysis arises due to the non-ultrametric setup, where instead of defining genetic distance to be twice the time to the most recent common ancestor (MRCA) (cf., for example, [GPW13, (2.27)] in the context of tree-valued Moran dynamics), genetic distance is taken to be the sum of the number of mutations of two individuals back in time to their MRCA. The control of the modulus of continuity along the path of an individual now allows us to relate genetic distance to time by considering age as part of the type-space in [Kli14].…”