110 research outputs found
On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds
We consider the Fisher-KPP equation with a non-local interaction term. We
establish a condition on the interaction that allows for existence of
non-constant periodic solutions, and prove uniform upper bounds for the
solutions of the Cauchy problem, as well as upper and lower bounds on the
spreading rate of the solutions with compactly supported initial data
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
Super-linear spreading in local and non-local cane toads equations
In this paper, we show super-linear propagation in a nonlocal
reaction-diffusion-mutation equation modeling the invasion of cane toads in
Australia that has attracted attention recently from the mathematical point of
view. The population of toads is structured by a phenotypical trait that
governs the spatial diffusion. In this paper, we are concerned with the case
when the diffusivity can take unbounded values, and we prove that the
population spreads as . We also get the sharp rate of spreading in a
related local model
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