Generalized Transition Waves and Their Properties

Berestycki, Henri; Hamel, François

doi:10.1002/cpa.21389

Cited by 173 publications

(212 citation statements)

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“…This conceptualization of pushed and pulled solutions, whose mathematical definitions will be given in a future work, has the advantage of being intuitive and adaptable to more complex models that do not necessarily admit traveling wave solutions. For instance, we should now be able to determine the pushed-pulled nature of the solutions of (i) integro-differential equations including long-distance dispersal events and resulting in accelerating waves (39,40), (ii) reaction-diffusion equations with spatially heterogeneous coefficients that lead to pulsating or generalized transition waves (41,42), (iii) reaction-diffusion equations with forced speed, which have been used in ref. 43 to study the effects of a shifting climate on the dynamics of a biological species.…”

Section: Numerical Computationsmentioning

confidence: 99%

Allee effect promotes diversity in traveling waves of colonization

Roques

Garnier

Hamel

et al. 2012

Proc. Natl. Acad. Sci. U.S.A.

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168

236

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Most mathematical studies on expanding populations have focused on the rate of range expansion of a population. However, the genetic consequences of population expansion remain an understudied body of theory. Describing an expanding population as a traveling wave solution derived from a classical reaction-diffusion model, we analyze the spatio-temporal evolution of its genetic structure. We show that the presence of an Allee effect (i.e., a lower per capita growth rate at low densities) drastically modifies genetic diversity, both in the colonization front and behind it. With an Allee effect (i.e., pushed colonization waves), all of the genetic diversity of a population is conserved in the colonization front. In the absence of an Allee effect (i.e., pulled waves), only the furthest forward members of the initial population persist in the colonization front, indicating a strong erosion of the diversity in this population. These results counteract commonly held notions that the Allee effect generally has adverse consequences. Our study contributes new knowledge to the surfing phenomenon in continuous models without random genetic drift. It also provides insight into the dynamics of traveling wave solutions and leads to a new interpretation of the mathematical notions of pulled and pushed waves. R apid increases in the number of biological invasions by alien organisms (1) and the movement of species in response to their climatic niches shifting as a result of climate change have caused a growing number of empirical and observational studies to address the phenomenon of range expansion. Numerous mathematical approaches and simulations have been developed to analyze the processes of these expansions (2, 3). Most results focus on the rate of range expansion (4), and the genetic consequences of range expansion have received little attention from mathematicians and modelers (5). However, range expansions are known to have an important effect on genetic diversity (6, 7) and generally lead to a loss of genetic diversity along the expansion axis due to successive founder effects (8). Simulation studies have already investigated the role of the geometry of the invaded territory (9-11), the importance of long-distance dispersal and the shape of the dispersal kernel (12-14), the effects of local demography (15), or existence of a juvenile stage (13). Further research is needed to obtain mathematical results supporting these empirical and simulation studies, as such results could determine the causes of diversity loss and the factors capable of increasing or reducing it.In a simulation study using a stepping-stone model with a lattice structure, Edmonds et. al (16) analyzed the fate of a neutral mutation present in the leading edge of an expanding population. Although in most cases the mutation remains at a low frequency in its original position, in some cases the mutation increases in frequency and propagates among the leading edge. This phenomenon is described as "surfing" (15). Surfing is caused by the strong genetic drift ...

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Section: Numerical Computationsmentioning

confidence: 99%

Allee effect promotes diversity in traveling waves of colonization

Roques

Garnier

Hamel

et al. 2012

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

168

236

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“…The latter can be done because (2.28) continues to hold if we replace ε 2 by min{ε 2 , c γ /4} and R 2 by R 2 + 4c −1 γ ln + u 0 ∞ . First, we claim that lim sup 6) where the rate of these decays depends on the same parameters as T ε in (C) does, except of ε (by "rate" we mean a functionT :…”

Section: Proofs Of Theorems 27 and 29mentioning

confidence: 97%

“…We note that the condition (2.29) for our transition solutions also has a counterpart in [6]. There an invasion of u − by u + is defined to be a transition front connecting u ± for which …”

Section: Entire Solutions With Bounded Widthsmentioning

confidence: 97%

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Propagation of Reactions in Inhomogeneous Media

Zlatoš

2016

Comm Pure Appl Math

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Abstract. Consider reaction-diffusion equation u t = ∆u + f (x, u) with x ∈ R d and general inhomogeneous ignition reaction f ≥ 0 vanishing at u = 0, 1. Typical solutions 0 ≤ u ≤ 1 transition from 0 to 1 as time progresses, and we study them in the region where this transition occurs. Under fairly general qualitative hypotheses on f we show that in dimensions d ≤ 3, the Hausdorff distance of the super-level sets {u ≥ ε} and {u ≥ 1 − ε} remains uniformly bounded in time for each ε ∈ (0, 1). Thus, u remains uniformly in time close to the characteristic function of {u ≥ 1 2 } in the sense of Hausdorff distance of super-level sets. We also show that {u ≥ 1 2 } expands with average speed (over any long enough time interval) between the two spreading speeds corresponding to any x-independent lower and upper bounds on f . On the other hand, these results turn out to be false in dimensions d ≥ 4, at least without further quantitative hypotheses on f . The proof for d ≤ 3 is based on showing that as the solution propagates, small values of u cannot escape far ahead of values close to 1. The proof for d ≥ 4 involves construction of a counter-example for which this fails.Such results were before known for d = 1 but are new for general non-periodic media in dimensions d ≥ 2 (some are also new for homogeneous and periodic media). They extend in a somewhat weaker sense to monostable, bistable, and mixed reaction types, as well as to transitions between general equilibria u − < u + of the PDE, and to solutions not necessarily satisfying u − ≤ u ≤ u + .

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“…A generalization of the notions of planar and pulsating travelling waves to heterogeneous equations like (E) has been given by Berestycki and Hamel in [4,5]. Heuristically, this definition means that the solution x → u(t, x) connects 0 to 1 for all t, and that the widths of the spatial interfaces I ε (t) = {x ∈ R, ε < u(t, x) < 1 − ε} are bounded with respect to t ∈ R for all ε ∈ (0, 1/2).…”

Section: Spatial Transition Waves For General Heterogeneous Equationsmentioning

confidence: 99%

Critical travelling waves for general heterogeneous one-dimensional reaction–diffusion equations

Nadin

2015

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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Generalized Transition Waves and Their Properties

Cited by 173 publications

References 52 publications

Allee effect promotes diversity in traveling waves of colonization

Allee effect promotes diversity in traveling waves of colonization

Propagation of Reactions in Inhomogeneous Media

Critical travelling waves for general heterogeneous one-dimensional reaction–diffusion equations

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