Traveling waves in the nonlocal KPP-Fisher equation: Different roles of the right and the left interactions

Hası́k, Karel; Kopfová, Jana; Nábělková, Petra; Trofimchuk, Sergeĭ

doi:10.1016/j.jde.2015.12.035

Cited by 22 publications

(34 citation statements)

References 34 publications

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“…In addition, in Subsection 2.2, we state a somewhat weaker version of Theorem 1.4, Theorem 2.5. This result does not require any sub-tangency restriction from g. In Subsection 2.3, we are also illustrating our findings on an explicit example allowing a rather complete analytical and numerical analysis (this type of 'toy models' was proposed in [26], see also [10,17]). In particular, the computations done in Subsection 2.3 suggest that c = c(τ ) is decreasing function of τ and that each monotone wavefront is unique (up to a translation).…”

Section: Introduction and Main Resultssupporting

confidence: 67%

Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities

Trofimchuk

Volpert

2019

Nonlinearity

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We study the existence of monotone wavefronts for a general family of bistable reaction-diffusion equations with delayed reaction term g. Differently from previous works, we do not assume the monotonicity of g(u, v) with respect to the delayed variable v that does not allow to apply the comparison techniques. Thus our proof is based on a variant of the Hale-Lin functional-analytic approach to heteroclinic solutions of functional differential equations where Lyapunov-Schmidt reduction is done in appropriate weighted spaces of C 2 -smooth functions. This method requires a detailed analysis of associated linear differential Fredholm operators and their formal adjoints. For two different types of v−unimodal functions g(u, v), we prove the existence of a maximal continuous family of bistable monotone wavefronts.. Depending on the type of unimodality (equivalently, on the sign of the wave speed), two different scenarios can be observed for the bistable waves: 1) independently on the size of delay, each bistable wavefront is monotone; 2) wavefronts are monotone for moderate values of delays and can oscillate for large delays.AMS classification scheme numbers: 34K12, 35K57, 92D25

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Section: Introduction and Main Resultssupporting

confidence: 67%

Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities

Trofimchuk

Volpert

2019

Nonlinearity

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“…Broadly speaking, the cited works show that the wave uniqueness can be established when either the evolution equation or the waves are monotone, or when the Lipschitz constant of nonlinear reaction term is dominated by its derivative at the unstable equilibrium (the Diekmann-Kaper condition). On the other hand, recent studies [8,17,25] reveal that the uniqueness property can fail to hold even for monotone waves of some non-local monostable equations. To get a clearer picture of the situation, consider the following KPP-Fisher delayed equation (see [2,10,14,15,17,18,19,20,32] for more detail and references concerning this model):…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In addition, if h ≥ 0.57 then all non-constant positive wave solutions to (1) are slowly oscillating in the space variable, see [10,18]. In this situation, neither the comparison techniques nor the Berestycki-Nirenberg sliding solutions method, nor the Diekmann-Kaper approach can be used to prove the uniqueness of all traveling waves to (1) (the property conjectured in [17]). Thus only partial waves' uniqueness results for the Hutchinson diffusive equation (1) were available so far.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…On the left [respectively, right] side of (17) we have a strictly convex upward [respectively, downward] function, so that equation (17) can have at most two real roots counting multiplicity. In fact, since p > q and positive function 0 −h e zs dµ + (s) is non-increasing in z, we deduce the existence of ǫ * > 0 such that (17) has exactly two simple real roots if and only if ǫ ∈ (0, ǫ * ) and has a double real root if and only if ǫ = ǫ * .…”

Section: Appendixmentioning

confidence: 99%

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A simple approach to the wave uniqueness problem

Solar

Trofimchuk

2019

Journal of Differential Equations

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We propose a new approach for proving uniqueness of semi-wavefronts in generally nonmonotone monostable reaction-diffusion equations with distributed delay. This allows to solve an open problem concerning the uniqueness of non-monotone (hence, slowly oscillating) semi-wavefronts to the KPP-Fisher equation with delay. Similarly, a broad family of the Mackey-Glass type diffusive equations is shown to possess a unique (up to translation) semi-wavefront for each admissible speed.

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“…It is important to stress that the above uniqueness result is valid only within the sub-class of monotone wavefronts. As it was shown in [17], the uniqueness property does not hold within the larger class of all wavefronts even for the nonlocal 2 KPP-Fisher equation (i.e. when γ = 0).…”

Section: Introductionmentioning

confidence: 91%

Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

Trofimchuk¹,

Pinto²,

Trofimchuk³

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We are revisiting the topic of travelling fronts for the food-limited (FL) model with spatio-temporal nonlocal reaction. These solutions are crucial for understanding the whole model dynamics. Firstly, we prove the existence of monotone wavefronts. In difference with all previous results formulated in terms of 'sufficiently small parameters', our existence theorem indicates a reasonably broad and explicit range of the model key parameters allowing the existence of monotone waves. Secondly, numerical simulations realized on the base of our analysis show appearance of non-oscillating and non-monotone travelling fronts in the FL model. These waves were never observed before. Finally, invoking a new approach developed recently by Solar et al, we prove the uniqueness (for a fixed propagation speed, up to translation) of each monotone front. Keywords: food-limited model, monotone wave, nonlocal interaction, existence, uniqueness 2010 MSC: 34K60, 35K57, 92D25 R K(s, y, τ )u(t − s, x − y)dy, and K : R + × R × R + → R + is a function satisfyingI c,τ (λ) := R+×R K(s, y, τ )e −λ(cs+y) dsdy ∈ R, I c,τ (0) = 1, lim λ→−λ0(c) + I c,τ (λ) = +∞,

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Traveling waves in the nonlocal KPP-Fisher equation: Different roles of the right and the left interactions

Cited by 22 publications

References 34 publications

Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities

Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities

A simple approach to the wave uniqueness problem

Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

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