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### Open Access Journals

DCDS

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.
DCDS-B

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 different 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.

PROC

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 $f$.

DCDS

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 $t \to +\infty$.

DCDS

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.

DCDS

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.

DCDS

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.

PROC

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.

DCDS-B

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).

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