2013/08/29 by Jérôme Coville, Coville, Jerome · 3 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #Analysis of PDEs (math.AP) #Evolution and Genetic Dynamics #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models
paper · pdf · doi:10.48550/arxiv.1308.6471
openalex publication_date 2013/08/29 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
In this paper we are interested in the long time behaviour of the positive solutions of the mutation selection model with Neumann Boundary condition: (∂ u(x,t))/(dt)=u[r(x)-∫_ØK(x,y)|u|p(y) dy]+∇⋅(A(x)∇ u(x)), in \R+×Ø where Ø⊂ \RN is a bounded smooth domain, k(.,.) ∈ C( Ø× C(Ø), \R), p≥ 1 and A(x) is a smooth elliptic matrix. In a blind competition situation, i.e K(x,y)=k(y), we show the existence of a unique positive steady state which is positively globally stable. That is, the positive steady state attracts all the possible trajectories initiated from any non negative initial datum. When K is a general positive kernel, we also present a necessary and sufficient condition for the existence of a positive steady states. We prove also some stability result on the dynamic of the equation when the competition kernel K is of the form K(x,y)=k0(y)+\eps k1(x,y). That is, we prove that for sufficiently small \eps there exists a unique steady state, which in addition is positively asymptotically stable. The proofs of the global stability of the steady state essentially rely on non-linear relative entropy identities and an orthogonal decomposition. These identities combined with the decomposition provide us some a priori estimates and differential inequalities essential to characterise the asymptotic behaviour of the solutions.