arXiv subjects

Sergei Trofimchuk

Publications and source records attributed to Sergei Trofimchuk.

At least 19 recordsLinked to original sources

arXiv2025

Global stability of Wright-type equations with negative Schwarzian

Simplicity of the $37/24$-global stability criterion announced by E.M. Wright in 1955 and rigorously proved by B. Bánhelyi et al in 2014 for the delayed logistic equation raised the question of its possible extension for other population models. In our study, we answer this question by extending the $37/24$-stability condition for the Wright-type equations with decreasing smooth nonlinearity $f$ which has a negative Schwarzian and satisfies the standard negative feedback and boundedness assumptions. The proof contains the construction and careful analysis of qualitative properties of certain bounding relations. To validate our conclusions, these relations are evaluated at finite sets of points; for this purpose, we systematically use interval analysis.

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arXiv2024

On the global dynamics of a forest model with monotone positive feedback and memory

We continue to study (see arXiv:2401.08618, https://doi.org/10.48550/arXiv.2401.08618) a renewal equation $ϕ(t)=\frak Fϕ_t$ proposed in [C. Barril et al., J. Math. Biology, https://doi.org/10.1007/s00285-024-02084-x] to model trees growth. This time we are considering the case when the per capita reproduction rate $β(x)$ is a non-monotone (unimodal) function of tree's height $x$. Note that the height of some species of trees can impact negatively seed viability, in a kind of autogamy depression. Similarly to previous works, it is also assumed that the growth rate $g(x)$ of an individual of height $x$ is a strictly decreasing function. Here we analyse the connection between dynamics of the associated one-dimensional map $F(b)= {\frak F}b,$ $b \in {\mathbb R}_+$, and the delayed (hence infinite-dimensional) model $ϕ(t)=\frak Fϕ_t$. Our key observation is that this model is of monotone positive feedback type since $F$ is strictly increasing on ${\mathbb R}_+$ independently on the monotonicity properties of $β$.

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arXiv2024

Global dynamics of a size-structured forest model

We study a size-structured model proposed in [1] C. Barril, À. Calsina, O. Diekmann, J. Z. Farkas, On competition through growth reduction, e-print arXiv:2303.02981, to describe the dynamics of trees growth in the forest. Our approach to the associated renewal equation is rather different from the methods in [1] and is based on ideas developed in [2] F. Herrera, S. Trofimchuk, Dynamics of one-dimensional maps and Gurtin-MacCamy's population model. Part I: asymptotically constant solutions, Ukrainian Math. J., (in Memory of O. Sharkovsky), 75 (2023), 1635-1651, https://doi.org/10.3842/umzh.v75i12.7678. Assuming relatively weak restrictions on the reproduction, death and growth rates $β, μ, g$, we establish the permanence properties of the semiflow $\frak F^t$ generated by the renewal equation and prove that it possesses a compact global attractor of points $\mathcal A$. Next we show that the opposite types of monotonicity of $β, g$ assure that $\frak F^t$ is also monotone and that in this case $\mathcal A$ coincides with a unique asymptotically stable equilibrium attracting neighbourhoods of compact sets with non-zero initial data.

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arXiv2023

Bistable wavefronts in the delayed Belousov-Zhabotinsky reaction

We study the Murray adaptation of the Noyes-Field five-step model of the Belousov-Zhabotinsky (BZ) reaction in the case when a tuning parameter $r$, which determines the level of the bromide ion far ahead of the propagating wave, is bigger than 1 and when the delay in generation of the bromous acid is taken into account. The existence of wavefronts in the delayed BZ system was previously established only in the monostable situation with $r \in (0,1]$, the physically relevant bistable situation where $r >1$ (in real experiments $r$ varies between 5 and 50) was left open. We complete the study by showing that the BZ system with $r >1$ admits monotone traveling fronts. Note that one of the stable equilibria of the BZ model is not isolated. This circumstance does not allow the direct application of the topological or analytical methods previously elaborated for the analysis of the existence of bistable waves.

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arXiv2022

Two reasons for the appearance of pushed wavefronts in the Belousov-Zhabotinsky system with spatiotemporal interaction

We prove the existence of the minimal speed of propagation $c_*(r,b,K) \in [2\sqrt{1-r},2]$ for wavefronts in the Belousov-Zhabotinsky system with a spatiotemporal interaction defined by the convolution with (possibly, "fat-tailed") kernel $K$. The model is assumed to be monostable non-degenerate, i.e. $r\in (0,1)$. The slowest wavefront is termed pushed or non-linearly determined if its velocity $c_*(r,b,K) > 2\sqrt{1-r}$. We show that $c_*(r,b,K)$ is close to 2 if i) positive system's parameter $b$ is sufficiently large or ii) if $K$ is spatially asymmetric to one side (e.g. to the left: in such a case, the influence of the right side concentration of the bromide ion on the dynamics is more significant than the influence of the left side). Consequently, this reveals two reasons for the appearance of pushed wavefronts in the Belousov-Zhabotinsky reaction.

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arXiv2021

Wavefront's stability with asymptotic phase in the delayed monostable equations

We extend the class of initial conditions for scalar delayed reaction-diffusion equations $u_t (t,x)=u_{xx}(t,x)+f(u(t, x), u(t-h, x))$ which evolve in solutions converging to monostable traveling waves. Our approach allows to compute, in the moving reference frame, the phase distortion $α$ of the limiting travelling wave with respect to the position of solution at the initial moment $t=0$. In general, $α\not=0$ for the Mackey-Glass type diffusive equation. Nevertheless, $α=0$ for the KPP-Fisher delayed equation: the related theorem also improves existing stability conditions for this model.

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arXiv2021

The peak-and-end rule and differential equations with maxima: a view on the unpredictability of happiness

In the 1990s, after a series of experiments, the behavioral psychologist and economist Daniel Kahneman and his colleagues formulated the following Peak-End evaluation rule: "The remembered utility of pleasant or unpleasant episodes is accurately predicted by averaging the Peak (most intense value) of instant utility (or disutility) recorded during an episode and the instant utility recorded near the end of the experience", (D. Kahneman et al., 1997, QJE, p. 381). Hence, the simplest mathematical model for time evolution of the experienced utility function $u=u(t)$ can be given by the scalar differential equation $u'(t)=a u(t) + b \max \{u(s) : s\in [t-h,t]\}+f(t) \ (*),$ where $f$ represents exogenous stimuli, $h$ is the maximal duration of the experience, and $a, b \in \mathbb R$ are some averaging weights. In this work, we study equation $(*)$ and show that, for a range of parameters $a, b, h$ and a periodic sine-like term $f$, the dynamics of $(*)$ can be completely described in terms of an associated one-dimensional dynamical system generated by a piece-wise continuous map from a finite interval into itself. We illustrate our approach with two examples. In particular, we show that the hedonic utility $u(t)$ (`happiness') can exhibit chaotic behavior.

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arXiv2020

On pushed wavefronts of monostable equation with unimodal delayed reaction

We study the Mackey-Glass type monostable delayed reaction-diffusion 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, $h \in [0,h_p]$, where $h_p$ (given by an explicit formula) is optimal in a certain sense. Then we take a representative piece-wise linear unimodal birth function making 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 $[c_*, +\infty)$; c) for each $h\geq 0$, 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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arXiv2019

Nonlinearly determined wavefronts of the Nicholson's diffusive equation: when small delays are not harmless

By proving the existence of non-monotone and non-oscillating wavefronts for the Nicholson's blowflies diffusive equation (the NDE), we answer an open question raised in [16]. Surprisingly, these wavefronts can be observed only for sufficiently small delays. Similarly to the pushed fronts, obtained waves are not linearly determined. In contrast, a broader family of eventually monotone wavefronts for the NDE is indeed determined by properties of the spectra of the linearized equations. Our proofs use essentially several specific characteristics of the blowflies birth function (its unimodal form and the negativity of its Schwarz derivative, among others). One of the key auxiliary results of the paper shows that the Mallet-Paret--Cao--Arino theory of super-exponential solutions for scalar equations can be extended for some classes of second order delay differential equations. For the new type of non-monotone waves to the NDE, our numerical simulations also confirm their stability properties established by Mei et al.

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arXiv2019

On the geometric diversity of wavefronts for the scalar Kolmogorov ecological equation

In this work, we answer three fundamental questions concerning monostable travelling fronts for the scalar Kolmogorov ecological equation with diffusion and spatiotemporal interaction: these are the questions about their existence, uniqueness and geometric shape. In the particular case of the food-limited model, we give a rigorous proof of the existence of a peculiar, yet substantive non-linearly determined class of non-monotone and non-oscillating wavefronts. As regards to the scalar models coming from applications, this kind of wave solutions is analyzed here for the first time.

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arXiv2019

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

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.

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arXiv2019

Nonstandard quasi-monotonicity: an application to the wave existence in a neutral KPP-Fisher equation

We revisit Wu and Zou non-standard quasi-monotonicity approach for proving existence of monotone wavefronts in monostable reaction-diffusion equations with delays. This allows to solve the problem of existence of monotone wavefronts in a neutral KPP-Fisher equation. In addition, using some new ideas proposed recently by Solar et al., we establish the uniqueness (up to a translation) of these monotone wavefronts.

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arXiv2018

A simple approach to the wave uniqueness problem

We propose a new approach for proving uniqueness of semi-wavefronts in generally non-monotone 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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arXiv2018

Global continuation of monotone waves for a unimodal bistable reaction-diffusion equation with delay

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.

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arXiv2017

A cyclic system with delay and its characteristic equation

A nonlinear cyclic system with delay and the overall negative feedback is considered. The characteristic equation of the linearized system is studied in detail. Sufficient conditions for the oscillation of all solutions and for the existence of monotone solutions are derived in terms of roots of the characteristic equation.

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arXiv2017

Travelling waves for a bistable reaction-diffusion equation with delay

The paper is devoted to a reaction-diffusion equation with delay arising in modelling the immune response. We prove the existence of travelling waves in the bistable case using the Leray-Schauder method. In difference with the previous works, we do not assume here quasi-monotonicity of the delayed reaction term.

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arXiv2016

Almost periodic evolution systems with impulse action at state-dependent moments

We study the existence of almost periodic solutions for semi-linear abstract parabolic evolution equations with impulse action at state-dependent moments. In particular, we present conditions excluding the beating phenomenon in these systems. The main result is illustrated with an example of impulsive diffusive logistic equation.

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arXiv2015

Monotone waves for non-monotone and non-local monostable reaction-diffusion equations

We propose a criterion for the existence of monotone wavefronts in non-monotone and non-local monostable diffusive equations of the Mackey-Glass type. This extends recent results by Gomez et al proved for the particular case of equations with local delayed reaction. In addition, we demonstrate the uniqueness (up to a translation) of obtained monotone wavefront within the class of all monotone wavefronts (such a kind of conditional uniqueness was recently established for the non-local KPP-Fisher equation by Fang and Zhao). Moreover, we show that if delayed reaction is local then this uniqueness actually holds within the class of all wavefronts and therefore the minimal fronts under consideration (either pulled or pushed) should be monotone. Similarly to the case of the KPP-Fisher equations, our approach is based on the construction of an appropriate fundamental solution for associated boundary value problem for linear integral-differential equation.

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