137 research outputs found
Propagation in a kinetic reaction-transport equation: travelling waves and accelerating fronts
In this paper, we study the existence and stability of travelling wave
solutions of a kinetic reaction-transport equation. The model describes
particles moving according to a velocity-jump process, and proliferating thanks
to a reaction term of monostable type. The boundedness of the velocity set
appears to be a necessary and sufficient condition for the existence of
positive travelling waves. The minimal speed of propagation of waves is
obtained from an explicit dispersion relation. We construct the waves using a
technique of sub- and supersolutions and prove their \eb{weak} stability in a
weighted space. In case of an unbounded velocity set, we prove a
superlinear spreading. It appears that the rate of spreading depends on the
decay at infinity of the velocity distribution. In the case of a Gaussian
distribution, we prove that the front spreads as
Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration
Invasion fronts in ecology are well studied but very few mathematical results
concern the case with variable motility (possibly due to mutations). Based on
an apparently simple reaction-diffusion equation, we explain the observed
phenomena of front acceleration (when the motility is unbounded) as well as
other quantitative results, such as the selection of the most motile
individuals (when the motility is bounded). The key argument for the
construction and analysis of traveling fronts is the derivation of the
dispersion relation linking the speed of the wave and the spatial decay. When
the motility is unbounded we show that the position of the front scales as
. When the mutation rate is low we show that the canonical equation
for the dynamics of the fittest trait should be stated as a PDE in our context.
It turns out to be a type of Burgers equation with source term.Comment: 7 page
Non-Markovian Random Walks and Non-Linear Reactions: Subdiffusion and Propagating Fronts
We propose a reaction-transport model for CTRW with non-linear reactions and
non-exponential waiting time distributions. We derive non-linear evolution
equation for mesoscopic density of particles. We apply this equation to the
problem of fronts propagation into unstable state of reaction-transport systems
with anomalous diffusion. We have found an explicit expression for the speed of
propagating front in the case of subdiffusion transport.Comment: 7 page
Hyperbolic traveling waves driven by growth
We perform the analysis of a hyperbolic model which is the analog of the
Fisher-KPP equation. This model accounts for particles that move at maximal
speed (\epsilon\textgreater{}0), and proliferate according to
a reaction term of monostable type. We study the existence and stability of
traveling fronts. We exhibit a transition depending on the parameter
: for small the behaviour is essentially the same as for
the diffusive Fisher-KPP equation. However, for large the traveling
front with minimal speed is discontinuous and travels at the maximal speed
. The traveling fronts with minimal speed are linearly stable in
weighted spaces. We also prove local nonlinear stability of the traveling
front with minimal speed when is smaller than the transition
parameter.Comment: 24 page
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
A non-linear degenerate equation for direct aggregation and traveling wave dynamics
The gregarious behavior of individuals of populations is an important factor in avoiding predators or for reproduction. Here, by using a random biased walk approach, we build a model which, after a transformation, takes the general form [u_{t}=[D(u)u_{x}]_{x}+g(u)] . The model involves a density-dependent non-linear diffusion coefficient [D] whose sign changes as the population density [u] increases. For negative values of [D] aggregation occurs, while dispersion occurs for positive values of [D] . We deal with a family of degenerate negative diffusion equations with logistic-like growth rate [g] . We study the one-dimensional traveling wave dynamics for these equations and illustrate our results with a couple of examples. A discussion of the ill-posedness of the partial differential equation problem is included
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