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Long-time behaviour of an advection-selection equation

Abstract

We study the long-time behaviour of the advection-selection equation ∂tn(t, x) + ∇ • (f (x)n(t, x)) = (r(x) − ρ(t)) n(t, x), ρ(t) = \int_{R^d}{n(t, x)dx} t ≥ 0, x ∈ R^d, with an initial condition n(0, •) = n^0. In the field of adaptive dynamics, this equation typically describes the evolution of a phenotype-structured population over time. In this case, n(t, x) represents the density of the population characterised by a phenotypic trait x, the advection term '∇ • (f (x)n(t, x))' a cell differentiation phenomenon driving the individuals toward specific regions, and the selection term '(r(x) − ρ(t)) n(t, x)' the growth of the population, which is of logistic type through the total population size ρ(t) = \int_{R^d}{n(t, x)}dx. In the one-dimensional case x ∈ R, we prove that the solution to this equation can either converge to a weighted Dirac mass or to a function in L^1. Depending on the parameters n^0 , f and r, we determine which of these two regimes of convergence occurs, and we specify the weight and the point where the Dirac mass is supported, or the expression of the L^1-function which is reached.
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Dates and versions

hal-03927313 , version 1 (06-01-2023)

Identifiers

  • HAL Id : hal-03927313 , version 1

Cite

Jules Guilberteau, Camille Pouchol, Nastassia Pouradier Duteil. Long-time behaviour of an advection-selection equation. 2023. ⟨hal-03927313⟩
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