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We model indirect transmission, via contact with viruses, of avian influenza in migratory and nonmigratory birds, taking into account age structure. Migration is modeled via a reaction-advection equation on a closed loop parametrized by arc length (the migration flyway) that starts and ends at the location where birds breed in summer. Our modeling keeps the birds together as a flock, the position of which is implicitly determined and known for all future time. Births occur when the flock passes the breeding location and are modeled using ideas of impulsive differential equations. For a migratory species the model derivation starts from age-structured reaction-advection equations with location-dependent parameters that describe local conditions. In the derivation of delay equations for the time-dependent variables representing numbers of juvenile and adult birds, these location-dependent parameters are evaluated at the flock's position, so that seasonal effects are captured indirectly but through rigorous modeling whereby we keep track of the flock's exact position and local conditions there. Sufficient conditions are obtained for the local stability of the disease-free equilibrium (for a nonmigratory species) and for the disease-free periodic solution (for a migratory species). [ABSTRACT FROM AUTHOR]

The delayed feedback control (DFC) is a control method for stabilizing unstable periodic orbits in nonlinear autonomous differential equations. We give an important relationship between the characteristic multipliers of the linear variational equation around an unstable periodic solution of the equation and those of its delayed feedback equation. The key of our proof is a result about the spectrum of a matrix which is a difference of commutative matrices. The relationship, moreover, allows us to design control gains of the DFC such that the unstable periodic solution is stabilized. In other words, the validity of the DFC is proved mathematically. As an application for the Rossler equation, we determine the best range of k such that the unstable periodic orbit is stabilized by taking a feedback gain K = kE. [ABSTRACT FROM AUTHOR]

This paper focuses on consensus problems for a class of linear systems with distributed delay that are encountered in modeling traffic flow dynamics. In the application problems the distributed delay, whose kernel is a ?-distribution with a gap, represents the human drivers' behavior in the average. The aim of the paper is to give a characterization of the regions in the corresponding delay parameter space, where a consensus is reached for all initial conditions. The structure and properties of the system are fully exploited, which leads to explicit and computationally tractable expressions. As a by-product a stability theory for distributed delay systems with a ?-distribution kernel is developed. Also explicit expressions for the consensus function(al) of time-delay systems with constant and distributed delays that solve a consensus problem are provided. Several illustrative examples complete the presentation. [ABSTRACT FROM AUTHOR]

The problem of the asymptotic stability independent of delays for a class of linear systems including multiple delays is addressed. Both cases where the delays are allowed to vary independently of each other and where they are restricted to a one-dimensional subspace of the delay-parameter space are considered. It the latter case it turns out that the resulting dependency between the delays (rationally independent, rationally dependent, commensurate) plays an important role. The stability conditions are expressed in terms of the spectral properties of some appropriate complex matrices. As a consequence of the stability study, a complete characterization of the delay interference phenomenon is given. Furthermore, a connection is established with the stability theory for continuous-time delay-difference equations, subjected to delay perturbations. Various illustrative examples complete the paper. [ABSTRACT FROM AUTHOR]

We consider the spatio-temporal patterns of disease spread involving structured populations. We start with a general model framework in population biology and spatial ecology where the individual's spatial movement behaviors depend on its maturation status, and we show how delayed reaction diffusion equations with nonlocal interactions arise naturally. We then consider the impact of this delayed nonlocal interaction on the disease spread by revisiting the spatial spread of rabies in continental Europe during the period between 1945 and 1985. We show how the distinction of territorial patterns between juvenile and adult foxes, the main carriers of the rabies under consideration, yields a class of partial differential equations involving delayed and nonlocal terms that are implicitly defined by a hyperbolic-parabolic equation, and we show how incorporating this distinction into the model leads to a formula describing the relation of the minimal wave speed and the maturation time of foxes. We show how the homotopy argument developed by Chow, Lin, and Mallet-Paret can be applied to obtain the existence of a heteroclinic orbit between a disease-free equilibrium and an endemic state for the spatially averaged system of delay differential equations, and we illustrate how the technique developed by Faria, Huang, and Wu can be used to establish the existence of a family of traveling wavefronts in the neighborhood of the heteroclinic orbit for the corresponding spatial model. [ABSTRACT FROM AUTHOR]

We consider networks of coupled scalar maps, with weighted connections which may include a time delay, and study the stability of equilibria with respect to the delays and connection structure. We prove that the largest eigenvalue of the graph Laplacian determines the effect of the connection topology on stability. The stability region in the parameter plane shrinks with increasing values of the largest eigenvalue, or of the time delay of the same parity. In particular, all bipartite graphs have an identical stability region, regardless of the delay or graph size, which is also the smallest stability region among those of all graphs. Furthermore, for certain parameter ranges, unstable (and possibly chaotic) maps can be stabilized via diffusive coupling with an odd time delay, provided that the network does not have a nontrivial and connected bipartite component. On the other hand, stabilization is not possible for even values of the delay or for bipartite networks. [ABSTRACT FROM AUTHOR]