Analysis of the periodically fragmented environment model: II—biological invasions and pulsating travelling fronts

Berestycki, Henri; Hamel, François; Roques, Lionel

doi:10.1016/j.matpur.2004.10.006

Cited by 210 publications

(307 citation statements)

References 18 publications

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“…However, if ζ − is not everywhere positive, the principal eigenvalue µ − may not be negative in general. In [7], nonlinearities f = f (x, s) (for x ∈ = R N ) satisfying…”

Section: (17)mentioning

confidence: 99%

“…Notice also that (1.13) implies (1.5) (see [20]), as well as (1.6). Under the condition µ − < 0, the existence of pulsating fronts in any direction e was proved in [7] for all speeds c ≥ c * (e), where c * (e) is still given by (1.11) (see also [25] for partial results in the one-dimensional case), and it was already known from [7] that no pulsating front exists with speed less than c * (e). However, the uniqueness of the front profiles in a given direction e and for a given speed c ≥ c * (e) was still an open problem, even in dimension 1.…”

Section: (17)mentioning

confidence: 99%

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Uniqueness and stability properties of monostable pulsating fronts

Hamel¹,

Roques²

2010

J. Eur. Math. Soc.

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106

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Abstract. We prove the uniqueness, up to shifts, of pulsating traveling fronts for reaction-diffusion equations in periodic media with Kolmogorov-Petrovskiȋ-Piskunov type nonlinearities. These results provide in particular a complete classification of all KPP pulsating fronts. Furthermore, in the more general case of monostable nonlinearities, we also derive several global stability properties and convergence to pulsating fronts for solutions of the Cauchy problem with front-like initial data. In particular, we prove the stability of KPP pulsating fronts with minimal speed, which is a new result even in the case when the medium is invariant in the direction of propagation.

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“…However, if ζ − is not everywhere positive, the principal eigenvalue µ − may not be negative in general. In [7], nonlinearities f = f (x, s) (for x ∈ = R N ) satisfying…”

Section: (17)mentioning

confidence: 99%

Section: (17)mentioning

confidence: 99%

Uniqueness and stability properties of monostable pulsating fronts

Hamel¹,

Roques²

2010

J. Eur. Math. Soc.

Self Cite

106

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“…We refer to (Aronson & Weinberger, 1957;Aronson & Weinberger, 1978;Berestycki, Hamel, & Nadirashvili, 2010;Kametaka, 1976;Liang & Zhao, 2007;Liang, Yi, & Zhao, 2006;Sattinger, 1976;Uchiyama, 1978;Weinberger, 1982;etc). for the study of (1.2) in the case that f (x, u) is independent of x and refer to (Berestycki, Hamel, & Nadirashvili, 2005;Berestycki, Hamel, & Roques, 2005;Freidlin & Gärtner, 1979;Hamel, 2008;Hudson & Zinner, 1995;Nadin, 2009;Nolen, Rudd, & Xin, 2005;Weinberger, 2002;etc). for the study of (1.2) in the case that f (x, u) is periodic in x; refer to (Coville & Dupaigne, 2005;Coville, Dávila, & Martínez, 2008;Li, Sun, & Wang, 2010;etc).…”

Section: U(t Y)dy − U(t X) + U(t X) Fmentioning

confidence: 99%

Existence of Positive Solutions of Fisher-KPP Equations in Locally Spatially Variational Habitat with Hybrid Dispersal

Kong¹

2017

JMR

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“…An interesting recent extension of this approach has been used to study pulsating fronts in periodically modulated nonlocal neural field equations (Coombes and Laing 2011). The theory of pulsating fronts has also been developed in a more general and rigorous setting (Gartner and Freidlin 1979;Xin 2000;Weinberger 2002;Berestycki et al 2005;El Smaily et al 2009). …”

Section: Pulsating Waves In the Presence Of Discretely Distributed Spmentioning

confidence: 99%

“…), which are fragmented by natural or artificial barriers (rivers, roads, cultivated fields etc.). In recent years, a number of analytical and numerical methods have been developed to study propagating invasive fronts in periodic and random media (Gartner and Freidlin 1979;Shigesada et al 1986;Shigesada and Kawasaki 1997;Xin 2000;Weinberger 2002;Kinezaki et al 2003;Cantrell and Cosner 2003;Berestycki et al 2005;El Smaily et al 2009). Heterogeneity is often incorporated by assuming that the diffusion coefficient and the growth rate of a population are periodically varying functions of space.…”

Section: Introductionmentioning

confidence: 99%

Propagation of CaMKII translocation waves in heterogeneous spiny dendrites

Bressloff

2012

J. Math. Biol.

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CaMKII (Ca 2+ -calmodulin-dependent protein kinase II) is a key regulator of glutamatergic synapses and plays an essential role in many forms of synaptic plasticity. It has recently been observed experimentally that stimulating a local region of dendrite not only induces the local translocation of CaMKII from the dendritic shaft to synaptic targets within spines, but also initiates a wave of CaMKII translocation that spreads distally through the dendrite with an average speed of order 1µm/s. We have previously developed a simple reaction-diffusion model of CaMKII translocation waves that can account for the observed wavespeed and predicts wave propagation failure if the density of spines is too high. A major simplification of our previous model was to treat the distribution of spines as spatially uniform. However, there are at least two sources of heterogeneity in the spine distribution that occur on two different spatial scales. First, spines are discrete entities that are joined to a dendritic branch via a thin spine neck of submicron radius, resulting in spatial variations in spine density at the micron level. The second source of heterogeneity occurs on a much longer length scale and reflects the experimental observation that there is a slow proximal to distal variation in the density of spines. In this paper, we analyze how both sources of heterogeneity modulate the speed of CaMKII translocation waves along a spiny dendrite. We adapt methods from the study of the spread of biological invasions in heterogeneous environments, including homogenization theory of pulsating fronts and Hamilton-Jacobi dynamics of sharp interfaces.

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Analysis of the periodically fragmented environment model: II—biological invasions and pulsating travelling fronts

Cited by 210 publications

References 18 publications

Uniqueness and stability properties of monostable pulsating fronts

Uniqueness and stability properties of monostable pulsating fronts

Existence of Positive Solutions of Fisher-KPP Equations in Locally Spatially Variational Habitat with Hybrid Dispersal

Propagation of CaMKII translocation waves in heterogeneous spiny dendrites

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