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First-order Asymptotics of the Mutant Sub-populations for a General Finite Trait Space (Theorem 2.1)by@mutation

First-order Asymptotics of the Mutant Sub-populations for a General Finite Trait Space (Theorem 2.1)

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In this paper, we aim to understand the evolution of the genetic composition of cancer cell populations.
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This paper is available on arxiv under CC 4.0 license.

Authors:

(1) Vianney Brouard, ENS de Lyon, UMPA, CNRS UMR 5669, 46 All´ee d’Italie, 69364 Lyon Cedex 07, France; E-mail: [email protected].

4 First-order asymptotics of the mutant sub-populations for a general finite trait space (Theorem 2.1)





Notice that with such a construction it immediately follows the monotone coupling





In the next definition, we are going to introduce an equivalence relation on Γ(V ). Two paths are said to be equivalent if they are the same up to cycles (in particular cycles formed by backward mutations are taken into account). More precisely, that there exists a minimal path, from which the two previous paths are using all the edges, but potentially also some other edges forming cycles. The aim of this equivalence relation is to say that among one class of equivalence, only the path with the minimal length may contribute for the asymptotic of the mutant sub-population sizes



Now we have all the preliminary results and definitions to prove Theorem 2.1.



Proof of Theorem 2.1. We show Equation (11). The proof of Equation (12) is similar and is left to the reader.






And the last term of the r.h.s converges to 0 because