Modulated traveling fronts for a nonlocal Fisher-KPP equation: a dynamical systems approach

@article{Faye2014ModulatedTF,
  title={Modulated traveling fronts for a nonlocal Fisher-KPP equation: a dynamical systems approach},
  author={Gr{\'e}gory Faye and Matt Holzer},
  journal={arXiv: Analysis of PDEs},
  year={2014}
}
View PDF on arXiv
41 Citations
Highly Influential Citations
5
Background Citations
18
Methods Citations
3

Figures from this paper

41 Citations

Citation Type

Citation Type

Modulating traveling fronts for the Swift-Hohenberg equation in the case of an additional conservation law
Pattern formation in the doubly-nonlocal Fisher-KPP equation
We study the existence, bifurcations, and stability of stationary solutions for the doubly-nonlocal Fisher-KPP equation. We prove using Lyapunov-Schmidt reduction that under suitable conditions on…
Modulating traveling fronts in a dispersive Swift-Hohenberg equation coupled to an additional conservation law
  • Bastian Hilder
  • Mathematics
    Journal of Mathematical Analysis and Applications
  • 2022
PERIODIC TRAVELING WAVES AND ASYMPTOTIC SPREADING OF A MONOSTABLE REACTION-DIFFUSION EQUATIONS WITH NONLOCAL EFFECTS
This article concerns the dynamical behavior for a reaction-diffusion equation with integral term. First, by using bifurcation analysis and center manifold theorem, the existence of periodic…
Nonlinear convective stability of a critical pulled front undergoing a Turing bifurcation at its back: a case study
We investigate a specific reaction-diffusion system that admits a monostable pulled front propagating at constant critical speed. When a small parameter changes sign, the stable equilibrium behind…
SPATIAL DYNAMICS OF A NONLOCAL BISTABLE REACTION DIFFUSION EQUATION
This article concerns a nonlocal bistable reaction-diffusion equation with an integral term. By using Leray-Schauder degree theory, the shift functions and Harnack inequality, we prove the existence…
Double Hopf bifurcation in nonlocal reaction-diffusion systems with spatial average kernel
In this paper, we consider a general reaction-diffusion system with nonlocal effects and Neumann boundary conditions, where a spatial average kernel is chosen to be the nonlocal kernel. By virtue of…
Existence of traveling wave solutions to a nonlocal scalar equation with sign-changing kernel
Sharp discontinuous traveling waves in a hyperbolic Keller–Segel equation
In this work we describe a hyperbolic model with cell-cell repulsion with a dynamics in the population of cells. More precisely, we consider a population of cells producing a field (which we call…
EXISTENCE OF TRAVELING WAVE SOLUTIONS TO A NONLOCAL SCALAR EQUATION WITH SIGN-CHANGING KERNEL
In this paper, we study the existence of traveling wave solutions connecting two constant states to a nonlocal scalar equation with sign-changing kernel. A typical example of such kernel in the…
...
...

References

SHOWING 1-10 OF 35 REFERENCES
On bounded positive stationary solutions for a nonlocal Fisher–KPP equation
Wave-like Solutions for Nonlocal Reaction-diffusion Equations: a Toy Model
Traveling waves for the nonlocal Fisher Equation can exhibit much more complex behaviours than for the usual Fisher equation. A striking numerical observation is that a traveling wave with minimal…
Travelling front solutions of a nonlocal Fisher equation
TLDR
This work considers a scalar reaction-diffusion equation containing a nonlocal term (an integral convolution in space) of which Fisher’s equation is a particular case and shows that if the nonlocality is sufficiently weak in a certain sense then such travelling fronts exist.
The non-local Fisher-KPP equation: travelling waves and steady states
We consider the Fisher–KPP equation with a non-local saturation effect defined through an interaction kernel (x) and investigate the possible differences with the standard Fisher–KPP equation. Our…
Bifurcating fronts for the Taylor-Couette problem in infinite cylinders
We show the existence of bifurcating fronts for the weakly unstable Taylor—Couette problem in an infinite cylinder. These fronts connect a stationary bifurcating pattern, here the Taylor vortices,…
Rapid traveling waves in the nonlocal Fisher equation connect two unstable states
Spatial structures and generalized travelling waves for an integro-differential equation
Some models in population dynamics with intra-specific competition lead to integro-differential equations where the integral term corresponds to nonlocal consumption of resources [8][9]. The…
Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation
On the nonlocal Fisher-KPP equation: steady states, spreading speed and global bounds
We consider the Fisher–KPP (for Kolmogorov–Petrovsky–Piskunov) equation with a nonlocal interaction term. We establish a condition on the interaction that allows for existence of non-constant…
The Swift-Hohenberg equation with a nonlocal nonlinearity
...
...