We prove the existence of a continuous family of positive and generally nonmonotone travelling fronts for delayed reaction-diffusion equations u t (t, x) = u(t, x) − u(t, x) + g(u(t − h, x)) ( * ), when g ∈ C 2 (R + , R + ) has exactly two fixed points: x 1 = 0 and x 2 = K > 0. Recently, nonmonotonic waves were observed in numerical simulations by various authors. Here, for a wide range of parameters, we explain why such waves appear naturally as the delay h increases. For the case of g with negative Schwarzian, our conditions are rather optimal; we observe that the well known Mackey-Glass-type equations with diffusion fall within this subclass of ( * ). As an example, we consider the diffusive Nicholson's blowflies equation.

We study positive bounded wave solutions u(t, x) = φ(ν · x + ct), φ(−∞) = 0, of equation u t (t, x) = u(t, x) − u(t, x) + g(u(t − h, x)), x ∈ R m ( * ). This equation is assumed to have two non-negative equilibria: u 1 ≡ 0 and u 2 ≡ κ > 0. The birth function g ∈ C(R + , R + ) is unimodal and differentiable at 0 and κ. Some results also require the feedback condition (g(s) − κ)(s − κ) < 0, with s ∈ [g(max g), max g] \ {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 Eq. ( * ) 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 positive travelling front for each fixed admissible speed.

In this paper, we establish efficient existence criteria for monotone traveling fronts u = ϕ(ν · x + ct), ϕ(−∞) = 0, ϕ(+∞) = κ of the monostable (and, in general, nonquasi‐monotone) delayed reaction–diffusion equations ut(t, x) − Δu(t, x) = f(u(t, x), u(t − h, x)). The function f is of class C1,γ and it is assumed to satisfy f(0, 0) = f(κ, κ) = 0 together with other monostability restrictions. Our theory covers several important cases including Mackey–Glass‐type diffusive equations and Kolmogorov‐Petrovskii‐Piskunov‐Fisher‐type equations. The proofs are based on a variant of the Hale–Lin functional‐analytic approach to heteroclinic solutions where Lyapunov–Schmidt reduction is realized in a ‘mobile’ weighted space of C2‐smooth functions. This method requires a detailed analysis of a family of associated linear differential Fredholm operators: at this stage, the discrete Lyapunov functionals by Mallet‐Paret and Sell are essential to the method.

Trofimchuk, S (reprint author), Univ Talca, Inst Matemat & Fis, Casilla 747, Talca, Chile.In the early 2000's, Gourley (2000), Wu et al. (2001), Ashwin et al. (2002) initiated the study of the positive wavefronts in the delayed Kolmogorov-Petrovskii-Piskunov-Fisher equation u(t)(t, x) = Delta u(t, x) + u (t, x)(1 - u(t - h, x)), u >= 0, x is an element of R(m). (*) Since then, this model has become one of the most popular objects in the studies of traveling waves for the monostable delayed reaction-diffusion equations. In this paper, we give a complete solution to the problem of existence and uniqueness of monotone waves in Eq. (*). We show that each monotone traveling wave can be found via an iteration procedure. The proposed approach is based on the use of special monotone integral operators (which are different from the usual Wu-Zou operator) and appropriate upper and lower solutions associated to them. The analysis of the asymptotic expansions of the eventual traveling fronts at infinity is another key ingredient of our approach. (c) 2010 Elsevier Inc. All rights reserved

We consider scalar delay differential equations x ′ (t) = −δx(t) + f (t, xt) ( * ) with nonlinear f satisfying a sort of negative feedback condition combined with a boundedness condition. The well known Mackey-Glass type equations, equations satisfying the Yorke condition, and equations with maxima all fall within our considerations. Here, we establish a criterion for the global asymptotical stability of a unique steady state to ( * ). As an example, we study Nicholson's blowflies equation, where our computations support the Smith's conjecture about the equivalence between global and local asymptotical stabilities in this population model.

This paper is concerned with a scalar nonlinear convolution equation which appears naturally in the theory of traveling waves for monostable evolution models. First, we prove that each bounded positive solution of the convolution equation should either be asymptotically separated from zero or it should converge (exponentially) to zero. This dichotomy principle is then used to establish a general theorem guaranteeing the uniform persistence and existence of semi-wavefront solutions to the convolution equation. Finally, we apply our abstract results to several well-studied classes of evolution equations with asymmetric non-local and non-monotone response. We show that, contrary to the symmetric case, these equations can possess at the same time the stationary, the expansion and the extinction waves.Comment: 15 pages, submitte

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Abstract. For a discrete dynamical system given by a compact Hausdorff space X and a continuous selfmap f of X the connection between minimality, invertibility and openness of f is investigated. It is shown that any minimal map is feebly open, i.e., sends open sets to sets with nonempty interiors (and if it is open then it is a homeomorphism). Further, it is shown that if f is minimal and A ⊆ X then both f (A) and f −1 (A) share with A those topological properties which describe how large a set is. Using these results it is proved that any minimal map in a compact metric space is almost one-to-one and, moreover, when restricted to a suitable invariant residual set it becomes a minimal homeomorphism. Finally, two kinds of examples of noninvertible minimal maps on the torus are given-these are obtained either as a factor or as an extension of an appropriate minimal homeomorphism of the torus.