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

Trofimchuk, Sergeĭ; Volpert, Vitaly

doi:10.1088/1361-6544/ab0e23

Cited by 9 publications

(11 citation statements)

References 39 publications

(180 reference statements)

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“…As several previous works show (e.g. see [9,13,20]), such a kind of nonlinear birth functions g allows to detect all essential geometric features of traveling waves that appear in the unimodal models. In Section 5, we show that for each k ∈ (1, 3) all traveling fronts to equation (1) considered with g given on Figure 1 can be determined in an explicit way.…”

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confidence: 89%

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On pushed wavefronts of monostable equation with unimodal delayed reaction

Hası́k¹,

Kopfová²,

Nábělková³

et al. 2021

DCDS

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We study the Mackey-Glass type monostable delayed reactiondiffusion equation with a unimodal birth function g(u). This model, designed to describe evolution of single species populations, is considered here in the presence of the weak Allee effect (g(u 0 ) > g (0)u 0 for some u 0 > 0). We focus our attention on the existence of slow monotonic traveling fronts to the equation: under given assumptions, this problem seems to be rather difficult since the usual positivity and monotonicity arguments are not effective. First, we solve the front existence problem for small delays, h ∈ [0, hp], where hp, given by an explicit formula, is optimal in a certain sense. Then we take a representative piece-wise linear unimodal birth function which makes possible explicit computation of traveling fronts. In this case, we find out that a) increase of delay can destroy asymptotically stable pushed fronts; b) the set of all admissible wavefront speeds has usual structure of a semi-infinite interval [c * , +∞); c) for each h ≥ 0, the pushed wavefront is unique (if it exists); d) pushed wave can oscillate slowly around the positive equilibrium for sufficiently large delays.

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confidence: 89%

“…We will analyze the situation when χ κ has exactly three real zeros, one positive and two negative (counting multiplicity), µ 3 ≤ µ 2 < 0 < µ 1 . In such a case, every complex zero µ j of χ κ is simple [20,Lemma A.2] and has its real part µ j < µ 2 [8,Lemma 1.1]. Importantly, the latter estimate can be improved.…”

mentioning

confidence: 99%

On pushed wavefronts of monostable equation with unimodal delayed reaction

Hası́k¹,

Kopfová²,

Nábělková³

et al. 2021

DCDS

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“…Existence of waves described by this equation was proved in [41,42]. Since conventional monotonicity conditions and the maximum principle are not applicable in this case, the proof of the wave existence requires sophisticated mathematical techniques.…”

Section: Propagation Of Wavesmentioning

confidence: 99%

Nonlocal Reaction–Diffusion Model of Viral Evolution: Emergence of Virus Strains

et al. 2020

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This work is devoted to the investigation of virus quasi-species evolution and diversification due to mutations, competition for host cells, and cross-reactive immune responses. The model consists of a nonlocal reaction–diffusion equation for the virus density depending on the genotype considered to be a continuous variable and on time. This equation contains two integral terms corresponding to the nonlocal effects of virus interaction with host cells and with immune cells. In the model, a virus strain is represented by a localized solution concentrated around some given genotype. Emergence of new strains corresponds to a periodic wave propagating in the space of genotypes. The conditions of appearance of such waves and their dynamics are described.

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“…The constant c is the wave speed and the solution φ(z) travels in either the left or the right direction without changing its shape. Substituting solution (48) into model (41) we get to the wave equation…”

Section: Theorem 3 the Reduced Model (30) Has A General Solution Of mentioning

confidence: 99%

“…Identifying the sufficient conditions for the global stability of wave solutions is the aim of our future studies. The existence of monotone wavefronts [3,29,48] and the global asymptotic stability of traveling waves [26,36,41,42] have been investigated for a number of delayed population models. Nonetheless, it remains an open problem to prove the global stability of wavefronts corresponding to the general model (26).…”

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confidence: 99%

Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model

Bani-Yaghoub¹,

Ou²,

Yao³

2020

Discrete &Amp; Continuous Dynamical Systems - S

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The present work investigates the effects of maturation and dispersal delays on dynamics of single species populations. Both delays have been incorporated in a single species nonlocal hyperbolic-parabolic population model, which admits traveling and stationary wave solutions. We reduce the model into various forms and obtain the corresponding analytical solutions. Analysis of the reduced models indicates that the dispersal delay can result in loss of monotonicity, where the solutions oscillate as they converge to a positive equilibrium. The stability analysis of the general model reveals that the maturation time delay admits a Hopf bifurcation threshold, which is expressed as a function of the dispersal delay. The numerical simulations of the general model suggest that the global stability of the stationary wave solutions is lost when the dispersal delay is increased from zero. In conclusion, population models with maturation and dispersal delays can give new insights into the complex dynamics of single species.

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

Cited by 9 publications

References 39 publications

On pushed wavefronts of monostable equation with unimodal delayed reaction

On pushed wavefronts of monostable equation with unimodal delayed reaction

Nonlocal Reaction–Diffusion Model of Viral Evolution: Emergence of Virus Strains

Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model

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