Admissible wavefront speeds for a single species reaction-diffusion equation with delay
Elena Trofimchuk Sergei Trofimchuk
We consider equation $u_t(t,x) = \Delta u(t,x)- u(t,x) + g(u(t-h,x))$(*) , when $g:\R_+\to \R_+$ has exactly two fixed points: $x_1= 0$ and $x_2=\kappa>0$. Assuming that $g$ is unimodal and has negative Schwarzian, we indicate explicitly a closed interval $\mathcal{C} = \mathcal{C}(h,g'(0),g'(\kappa)) =$ [ c*, c * ] such that (*) has at least one (possibly, non-monotone) 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 non-monotone nonlinearity.
keywords: Time-delayed reaction-diffusion equation heteroclinic solutions single species population models. non-monotone positive travelling fronts
Global stability in a regulated logistic growth model
E. Trofimchuk Sergei Trofimchuk
We investigate global stability of the regulated logistic growth model (RLG) $n'(t)=rn(t)(1-n(t-h)/K-cu(t))$, $u'(t)=-au(t)+bn(t-h)$. It was proposed by Gopalsamy and Weng [1, 2] and studied recently in [4, 5, 6, 9]. Compared with the previous results, our stability condition is of diff erent kind and has the asymptotical form. Namely, we prove that for the fixed parameters $K$ and $\mu=bcK/a$ (which determine the levels of steady states in the delayed logistic equation $n'(t)=rn(t)(1-n(t-h)/K)$ and in RLG) and for every $hr < \sqrt{2}$ the regulated logistic growth model is globally stable if we take the dissipation parameter a sufficiently large. On the other hand, studying the local stability of the positive steady state, we observe the improvement of stability for the small values of a: in this case, the inequality $rh<\pi (1+\mu)/2$ guarantees such a stability.
keywords: global stability regulated logistic model. delay differential equations Schwarz derivative
Periodic solutions and their stability of a differential-difference equation
Anatoli F. Ivanov Sergei Trofimchuk
Existence, stability, and shape of periodic solutions are derived for the differential-difference equation $\varepsilon\dot x(t)+x(t)=f(x([t-1])), 0<\varepsilon\<\<1,$ where $[\cdot]$ is the integer part function. The equation can be viewed as a special discretization (discrete version) of the singularly perturbed differential delay equation $\varepsilon\dot x(t)+x(t)=f(x(t-1))$. The principal analysis is based on reduction to the two-dimensional map $F: (u,v)\to (v, f(u)+ [v-f(u)]e^{-1/\varepsilon}),$ many relevant properties of which follow from those of the one-dimensional map $f$.
keywords: Reduction to discrete maps Interval maps Periodic solutions and their stability Singular perturbations Differential delay and difference equations
Slowly oscillating wavefronts of the KPP-Fisher delayed equation
Karel Hasik Sergei Trofimchuk
This paper concerns the semi-wavefronts (i.e. bounded solutions $u=\phi(x\cdot\nu +ct) >0,$ $ |\nu|=1, $ satisfying $\phi(-\infty)=0$) to the delayed KPP-Fisher equation $u_t(t,x) = \Delta u(t,x) + u(t,x)(1-u(t-\tau,x)), \ u \geq 0,\ x \in \mathbb{R}^m.$ First, we show that the profile $\phi$ of each semi-wavefront should be either monotone or eventually sine-like slowly oscillating around the positive equilibrium. Then a solution to the problem of existence of semi-wavefronts is provided. Next, we prove that the semi-wavefronts are in fact wavefronts (i.e. additionally $\phi(+\infty)=1$) if $c \geq 2$ and $\tau \leq 1$; our proof uses dynamical properties of an auxiliary one-dimensional map with the negative Schwarzian. However, we also show that, for $c \geq 2$ and $\tau \geq 1.87$, each semi-wavefront profile $\phi(t)$ should develop non-decaying oscillations around $1$ as $t \to +\infty$.
keywords: Wright's equation. Upper and lower solutions monotone traveling waves slowly oscillating fronts
Wright type delay differential equations with negative Schwarzian
Eduardo Liz Manuel Pinto Gonzalo Robledo Sergei Trofimchuk Victor Tkachenko
We prove that the well-known $3/2$ stability condition established for the Wright equation (WE) still holds if the nonlinearity $p(\exp(-x)-1)$ in WE is replaced by a decreasing or unimodal smooth function $f$ with $f'(0)<0$ satisfying the standard negative feedback and below boundedness conditions and having everywhere negative Schwarz derivative.
keywords: Schwarz derivative global stability delay differential equations. Wright conjecture 3/2 stability condition
On a generalized Yorke condition for scalar delayed population models
Teresa Faria Eduardo Liz José J. Oliveira Sergei Trofimchuk
For a scalar delayed differential equation $\dot x(t)=f(t,x_t)$, we give sufficient conditions for the global attractivity of its zero solution. Some technical assumptions are imposed to insure boundedness of solutions and attractivity of non-oscillatory solutions. For controlling the behaviour of oscillatory solutions, we require a very general condition of Yorke type, together with a 3/2-condition. The results are particularly interesting when applied to scalar differential equations with delays which have served as models in populations dynamics, and can be written in the general form $\dot x(t)=(1+x(t))F(t,x_t)$. Applications to several models are presented, improving known results in the literature.
keywords: Yorke condition Delayed population model global attractivity 3/2- condition.
Pushed traveling fronts in monostable equations with monotone delayed reaction
Elena Trofimchuk Manuel Pinto Sergei Trofimchuk
We study the wavefront solutions of the scalar reaction-diffusion equations $u_{t}(t,x) = \Delta u(t,x) - u(t,x) + g(u(t-h,x)),$ with monotone reaction term $g: \mathbb{R}_{+} → \mathbb{R}_+$ 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 $h>0$. To this end, we describe the asymptotics of all wavefronts (including critical and non-critical fronts) at $-\infty$. We also prove the uniqueness of wavefronts (up to a translation). In addition, a new uniqueness result for a class of nonlocal lattice equations is presented.
keywords: monotone traveling waves minimal speed. pushed fronts Upper and lower solutions asymptotic integration
Yorke and Wright 3/2-stability theorems from a unified point of view
Eduardo Liz Victor Tkachenko Sergei Trofimchuk
We consider a family of scalar delay differential equations $x'(t) = f(t, x_t)$, with a nonlinearity $f$ satisfying a negative feedback condition combined with a boundedness condition. We present a global stability criterion for this family, which in particular unifies the celebrated 3/2-conditions given for the Yorke and the Wright type equations. We illustrate our results with some applications.
keywords: delay differential equations. 3\2 stability condition global stability
On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction
Elena Trofimchuk Manuel Pinto Sergei Trofimchuk
In this paper, we answer the question about the existence of the minimal speed of front propagation in a delayed version of the Murray model of the Belousov-Zhabotinsky (BZ) chemical reaction. It is assumed that the key parameter $r$ of this model satisfies $0< r \leq 1$ that makes it formally monostable. By proving that the set of all admissible speeds of propagation has the form $[c_*,+\infty)$, we show here that the BZ system with $r \in (0,1]$ is actually of the monostable type (in general, $c_*$ is not linearly determined). We also establish the monotonicity of wavefronts and present the principal terms of their asymptotic expansions at infinity (in the critical case $r=1$ inclusive).
keywords: Belousov Zhabotinsky super-solution monostable. minimal speed

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