2,080 research outputs found
Fisher-KPP dynamics in diffusive Rosenzweig-MacArthur and Holling-Tanner models
We prove the existence of traveling fronts in diffusive Rosenzweig-MacArthur
and Holling-Tanner population models and investigate their relation with fronts
in a scalar Fisher-KPP equation. More precisely, we prove the existence of
fronts in a Rosenzweig-MacArthur predator-prey model in two situations: when
the prey diffuses at the rate much smaller than that of the predator and when
both the predator and the prey diffuse very slowly. Both situations are
captured as singular perturbations of the associated limiting systems. In the
first situation we demonstrate clear relations of the fronts with the fronts in
a scalar Fisher-KPP equation. Indeed, we show that the underlying dynamical
system in a singular limit is reduced to a scalar Fisher-KPP equation and the
fronts supported by the full system are small perturbations of the Fisher-KPP
fronts. We obtain a similar result for a diffusive Holling-Tanner population
model. In the second situation for the Rosenzweig-MacArthur model we prove the
existence of the fronts but without observing a direct relation with Fisher-KPP
equation. The analysis suggests that, in a variety of reaction-diffusion
systems that rise in population modeling, parameter regimes may be found when
the dynamics of the system is inherited from the scalar Fisher-KPP equation
(Un-)bounded transition fronts for the parabolic Anderson model and the randomized F-KPP equation
We investigate the uniform boundedness of the fronts of the solutions to the
randomized Fisher-KPP equation and to its linearization, the parabolic Anderson
model. It has been known that for the standard (i.e. deterministic) Fisher-KPP
equation, as well as for the special case of a randomized Fisher-KPP equation
with so-called ignition type nonlinearity, one has a uniformly bounded (in
time) transition front. Here, we show that this property of having a uniformly
bounded transition front fails to hold for the general randomized Fisher-KPP
equation. Nevertheless, we establish that this property does hold true for the
parabolic Anderson model
The Bramson delay in the non-local Fisher-KPP equation
We consider the non-local Fisher-KPP equation modeling a population with
individuals competing with each other for resources with a strength related to
their distance, and obtain the asymptotics for the position of the invasion
front starting from a localized population. Depending on the behavior of the
competition kernel at infinity, the location of the front is either , as in the local case, or for some
explicit . Our main tools here are a local-in-time Harnack
inequality and an analysis of the linearized problem with a suitable moving
Dirichlet boundary condition. Our analysis also yields, for any
, examples of Fisher-KPP type non-linearities such
that the front for the local Fisher-KPP equation with reaction term
is at
The Bramson delay in the non-local Fisher-KPP equation
We consider the non-local Fisher-KPP equation modeling a population with
individuals competing with each other for resources with a strength related to
their distance, and obtain the asymptotics for the position of the invasion
front starting from a localized population. Depending on the behavior of the
competition kernel at infinity, the location of the front is either , as in the local case, or for some
explicit . Our main tools here are alocal-in-time Harnack
inequality and an analysis of the linearized problem with a suitable moving
Dirichlet boundary condition. Our analysis also yields, for any
, examples of Fisher-KPP type non-linearities such
that the front for the localFisher-KPP equation with reaction term
is at
Geometric scaling as traveling waves
We show the relevance of the nonlinear Fisher and Kolmogorov-Petrovsky-
Piscounov (KPP) equation to the problem of high energy evolution of the QCD
amplitudes. We explain how the traveling wave solutions of this equation are
related to geometric scaling, a phenomenon observed in deep-inelastic
scattering experiments. Geometric scaling is for the first time shown to result
from an exact solution of nonlinear QCD evolution equations. Using general
results on the KPP equation, we compute the velocity of the wave front, which
gives the full high energy dependence of the saturation scale.Comment: 4 pages, 1 figure. v2: references adde
Spectral stability of the critical front in the extended Fisher-KPP equation
We revisit the existence and stability of the critical front in the extended Fisher-KPP equation, refining earlier results of Rottschäfer and Wayne [28] which establish stability of fronts without identifying a precise decay rate. We verify that the front is marginally spectrally stable: while the essential spectrum touches the imaginary axis at the origin, there are no unstable eigenvalues and no eigenvalue (or resonance) embedded in the essential spectrum at the origin. Together with the recent work of Avery and Scheel [3], this implies nonlinear stability of the critical front with sharp decay rate, as previously obtained in the classical Fisher-KPP equation. The main challenges are to regularize the singular perturbation in the extended Fisher-KPP equation and to track eigenvalues near the essential spectrum, and we overcome these difficulties with functional analytic methods
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