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Elena Trofimchuk

Publications and source records attributed to Elena Trofimchuk.

7 recordsLinked to original sources
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.

math.CA↗
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.

math.AP↗
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.

math.CA↗
arXiv2013

Traveling waves for a model of the Belousov-Zhabotinsky reaction

Following J.D. Murray, we consider a system of two differential equations that models traveling fronts in the Noyes-Field theory of the Belousov-Zhabotinsky (BZ) chemical reaction. We are also interested in the situation when the system incorporates a delay $h\geq 0$. As we show, the BZ system has a dual character: it is monostable when its key parameter $r \in (0,1]$ and it is bistable when $r >1$. For $h=0, r\not=1$, and for each admissible wave speed, we prove the uniqueness of monotone wavefronts. Next, a concept of regular super-solutions is introduced as a main tool for generating new comparison solutions for the BZ system. This allows to improve all previously known upper estimations for the minimal speed of propagation in the BZ system, independently whether it is monostable, bistable, delayed or not. Special attention is given to the critical case $r=1$ which to some extent resembles to the Zeldovich equation.

math.CA↗
arXiv2011

Pushed traveling fronts in monostable equations with monotone delayed reaction

We study the existence and uniqueness of wavefronts to the scalar reaction-diffusion equations $u_{t}(t,x) = Δu(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone delayed reaction term $g: \R_+ \to \R_+$ and $h >0$. We are mostly interested in the situation when the graph of $g$ is not dominated by its tangent line at zero, i.e. when the condition $g(x) \leq g'(0)x,$ $x \geq 0$, is not satisfied. It is well known that, in such a case, a special type of rapidly decreasing wavefronts (pushed fronts) can appear in non-delayed equations (i.e. with $h=0$). One of our main goals here is to establish a similar result for $h>0$. We prove the existence of the minimal speed of propagation, the uniqueness of wavefronts (up to a translation) and describe their asymptotics at $-\infty$. We also present a new uniqueness result for a class of nonlocal lattice equations.

math.AP↗
arXiv2006

Admissible wavefront speeds for a single species reaction-diffusion equation with delay

We consider equation $u_t(t,x) = Δu(t,x)- u(t,x) + g(u(t-h,x)) (*) $, when $g:\R_+\to \R_+$ has exactly two fixed points: $x_1= 0$ and $x_2=κ>0$. Assuming that $g$ is unimodal and has negative Schwarzian, we indicate explicitly a closed interval $\mathcal{C} = \mathcal{C}(h,g'(0),g'(κ)) = [c_*,c^*]$ such that $(*)$ has at least one (possibly, nonmonotone) travelling front propagating at velocity $c$ for every $c \in \mathcal{C}$. Here $c_*>0$ is finite and $c^* \in \R_+ \cup \{+\infty\}$. Every time when $\mathcal{C}$ is not empty, the minimal bound $c_*$ is sharp so that there are not wavefronts moving with speed $c < c_*$. In contrast to reported results, the interval $\mathcal{C}$ can be compact, and we conjecture that some of equations $(*)$ can indeed have an upper bound for propagation speeds of travelling fronts. As particular cases, Eq. $(*)$ includes the diffusive Nicholson's blowflies equation and the Mackey-Glass equation with nonmonotone nonlinearity.

math.DS↗