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Concentration phenomenon in some non-local equation

Bonnefon, Olivier; Coville, Jérôme; Legendre, Guillaume (2017), Concentration phenomenon in some non-local equation, Discrete and Continuous Dynamical Systems. Series B, 22, 3, p. 763 - 781. 10.3934/dcdsb.2017037

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Type
Article accepté pour publication ou publié
Date
2017
Journal name
Discrete and Continuous Dynamical Systems. Series B
Volume
22
Number
3
Publisher
Dept. of Mathematics
Pages
763 - 781
Publication identifier
10.3934/dcdsb.2017037
Metadata
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Author(s)
Bonnefon, Olivier
Coville, Jérôme cc
Legendre, Guillaume
Abstract (EN)
We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the integro-differential equation∂tu(t,x)=∫Ωm(x,y)(u(t,y)−u(t,x))dy+(a(x)−∫Ωk(x,y)u(t,y)dy)u(t,x),supplemented by the initial condition u(0,⋅)=u0 in Ω . Such a problem is used in population dynamics models to capture the evolution of a clonal population structured with respect to a phenotypic trait. In this context, the function u represents the density of individuals characterized by the trait, the domain of trait values Ω is a bounded subset of RN, the kernels k and m respectively account for the competition between individuals and the mutations occurring in every generation, and the function a represents a growth rate. When the competition is independent of the trait , we construct a positive stationary solution which belongs to the space of Radon measures on Ω. Moreover, in the case where this measure is regular and bounded, we prove its uniqueness and show that, for any non-negative initial datum in L1(Ω)∩L∞(Ω), the solution of the Cauchy problem converges to this limit measure in L2(Ω). We also exhibit an example for which the measure is singular and non-unique, and investigate numerically the long time behaviour of the solution in such a situation. The numerical simulations seem to reveal a dependence of the limit measure with respect to the initial datum.
Subjects / Keywords
Non uniqueness; Measure solution; Nonlocal Fisher equation; Concentration phenomenon; Non-local equation; demo-genetics; asymptotic behaviour

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