Admissible wavefront speeds for a single species reaction-diffusion equation with delay
Elena Trofimchuk Sergei Trofimchuk
Discrete & Continuous Dynamical Systems - A 2008, 20(2): 407-423 doi: 10.3934/dcds.2008.20.407
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
Discrete & Continuous Dynamical Systems - B 2005, 5(2): 461-468 doi: 10.3934/dcdsb.2005.5.461
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
Conference Publications 2009, 2009(Special): 385-393 doi: 10.3934/proc.2009.2009.385
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
Discrete & Continuous Dynamical Systems - A 2014, 34(9): 3511-3533 doi: 10.3934/dcds.2014.34.3511
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
Discrete & Continuous Dynamical Systems - A 2003, 9(2): 309-321 doi: 10.3934/dcds.2003.9.309
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
Discrete & Continuous Dynamical Systems - A 2005, 12(3): 481-500 doi: 10.3934/dcds.2005.12.481
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
Discrete & Continuous Dynamical Systems - A 2013, 33(5): 2169-2187 doi: 10.3934/dcds.2013.33.2169
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
Conference Publications 2003, 2003(Special): 580-589 doi: 10.3934/proc.2003.2003.580
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
Discrete & Continuous Dynamical Systems - B 2014, 19(6): 1769-1781 doi: 10.3934/dcdsb.2014.19.1769
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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