In this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction–diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li–Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee–Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutions.

In this paper we will deal with Lipschitz continuous perturbations of Morse-Smale semigroups with only equilibrium points as critical elements. We study the behavior of the structure of equilibrium points and their connections when subjected to non-differentiable perturbations. To this end we define more general notions of hyperbolicity and transversality, which do not require differentiability.

Although Leishmania infantum is well-known as the aethiological agent of visceral leishmaniasis (VL), in some Central American countries it may cause atypical non-ulcerated cutaneous leishmaniasis (NUCL). However, the mechanisms favoring its establishment in the skin are still unknown. Lipophosphoglycan (LPG) is the major Leishmania multivirulence factor involved in parasite-host interaction. In the case of viscerotropic L. infantum, it causes an immunosuppression during the interaction with macrophages. Here, we investigated the biochemical and functional roles of LPGs from four dermotropic L. infantum strains from Honduras during in vitro interaction with murine macrophages. LPGs were extracted, purified and their repeat units analysed. They did not have side chains consisting of Gal(β1,4)Man(α1)-PO 4 common to all LPGs. Peritoneal macrophages from BALB/c and C57BL/6 were exposed to LPG for nitric oxide (NO) and cytokine (TNF-α and, IL-6) production. LPGs from dermotropic strains from Honduras triggered higher NO and cytokine levels compared to those from viscerotropic strains. In conclusion, LPGs from dermotropic strains are devoid of side-chains and exhibit high pro-inflammatory activity.