2019/03/08 by Carl Mueller, Mueller, Carl, Leonid Mytnik +3 · 1 citation
Medicine · Mathematics · #Mathematical and Theoretical Epidemiology and Ecology Models #Stochastic processes and statistical mechanics #Mathematical Biology Tumor Growth
paper · pdf · doi:10.48550/arxiv.1903.03645
We study the asymptotic speed of a random front for solutions ut(x) to\nstochastic reaction-diffusion equations of the form n
partialtu=
farc12
partialx2u+f(u)+
sigma
sqrtu(1-u)
dotW(t,x),~t
ge\n0,~x
in
Rm, arising in population genetics. Here, f is a continuous\nfunction with f(0)=f(1)=0, and such that~|f(u)|\≤ K|u(1-u)|^\γ\nwith~\γ\≥ 1/2, and \W(t,x) is a space-time Gaussian white noise.\nWe assume that the initial condition u0(x) satisfies 0\≤ u0(x)\≤ 1 for\nall x\∈ Rm, u0(x)=1 for~x<L0 and u0(x)=0 for~x>R0. We show that\nwhen \σ>0, for each t>0 there exist~R(ut)<+\∞\nand~L(ut)<-\∞ such that ut(x)=0 for x>R(ut) and ut(x)=1\nfor~x<L(ut) even if f is not Lipschitz. We also show that for all\n\σ>0 there exists a finite deterministic speed~V(\σ)\∈ Rm so\nthat~R(ut)/t\→ V(\σ) as t\→+\∞, almost surely. This is in\ndramatic contrast with the deterministic case \σ=0 for nonlinearities of\nthe type f(u)=um(1-u) with 0<m<1 when solutions converge to 1 uniformly\non Rm as t\→+\∞. Finally, we prove that when \γ>1/2 there\nexists cf\∈ Rm, so that~\σ2V(\σ)\→ cf as~\σ\→+\∞\nand give a characterization of cf. The last result complements a lower bound\nobtained by Conlon and Doering citecd05 for the special case of\nf(u)=u(1-u) where a duality argument is available.\n