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.Comment: 20 pages, submitte

Mackey-Glass type delay differential equations near the boundary of absolute stability

For equations $x'(t) = -x(t) + \zeta f(x(t-h)), x \in \R, f'(0)= -1, \zeta > 0,$ with $C^3$-nonlinearity $f$ which has negative Schwarzian derivative and satisfies $xf(x) < 0$ for $x\not=0$, we prove convergence of all solutions to zero when both $\zeta -1 >0$ and $h(\zeta-1)^{1/8}$ are less than some constant (independent on $h,\zeta$). This result gives additional insight to the conjecture about the equivalence between local and global asymptotical stabilities in the Mackey-Glass type delay differential equations.Comment: 16 pages, 1 figure, accepted for publication in the Journal of Mathematical Analysis and Application

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.Comment: 28 pages, 6 figures, submitte

On the geometry of wave solutions of a delayed reaction-diffusion equation

The aim of this paper is to study the existence and the geometry of positive bounded wave solutions to a non-local delayed reaction-diffusion equation of the monostable type.Comment: 25 pages, several important modifications are made. Some references added to the previous versio

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.Comment: 23 pages, to appear in the Journal of Differential Equation

Pushed traveling fronts in monostable equations with monotone delayed reaction

We study the existence and uniqueness of wavefronts to the scalar reaction-diffusion equations $u_{t}(t,x) = \Delta u(t,x) - u(t,x) + g(u(t-h,x)),$ 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.Comment: 17 pages, submitte

Slowly oscillating wave solutions of a single species reaction-diffusion equation with delay

We study positive bounded wave solutions u(t,x)=ϕ(ν⋅x+ct), ϕ(−∞)=0, of equation ut(t,x)=δu(t,x)−u(t,x)+g(u(t−h,x)), x∈Rm(*). It is supposed that Eq. (∗) has exactly two non-negative equilibria: u1≡0 and u2≡κ>9. The birth function g∈C(R+,R+) satisfies a few mild conditions: it is unimodal and differentiable at 0,κ. Some results also require the positive feedback of g:[g(maxg),maxg]→R+ with respect to κ. If additionally ϕ(+∞)=κ, the above wave solution u(t,x) is called a travelling front. We prove that every wave ϕ(ν⋅x+ct) is eventually monotone or slowly oscillating about κ. Furthermore, we indicate c∗∈R+∪+∞ such that (∗) does not have any travelling front (neither monotone nor non-monotone) propagating at velocity c>c∗. Our results are based on a detailed geometric description of the wave profile ϕ. In particular, the monotonicity of its leading edge is established. We also discuss the uniqueness problem indicating a subclass G of ’asymmetric’ tent maps such that given g∈G, there exists exactly one travelling front for each fixed admissible speed