There is a growing interest in the study of evolutionary dynamics on populations with some non-homogeneous structure. In this paper we follow the model of Lieberman et al. (Lieberman et al. 2005 Nature 433, 312–316) of evolutionary dynamics on a graph. We investigate the case of non-directed equally weighted graphs and find solutions for the fixation probability of a single mutant in two classes of simple graphs. We further demonstrate that finding similar solutions on graphs outside these classes is far more complex. Finally, we investigate our chosen classes numerically and discuss a number of features of the graphs; for example, we find the fixation probabilities for different initial starting positions and observe that average fixation probabilities are always increased for advantageous mutants as compared with those of unstructured populations

In this paper a random link activation–deletion (RLAD) model is proposed that gives rise to a stochastically evolving network. This dynamic network is then coupled to a simple susceptible-infectious-suceptible (SIS) dynamics on the network, and the resulting spectrum of model behaviour is explored via simulation and a novel pairwise model for dynamic networks. First, the dynamic network model is systematically analysed by considering link-type independent and dependent network dynamics coupled with globally constrained link creation. This is done rigorously with some analytical results and we highlight where such analysis can be performed and how these simpler models provide a benchmark to test and validate full simulations. The pairwise model is used to study the interplay between SIS-type dynamics on the network and link-type- dependent activation–deletion. Assumptions of the pairwise model are identified and their implications interpreted in a way that complements our current understanding. Furthermore, we also discuss how the strong assumptions of the closure relations can lead to disagreement between the simulation and pairwise model. Unlike on a static network, the resulting spectrum of behaviour is more complex with the prevalence of infections exhibiting not only a single steady state, but also bistability and oscillations.

The changes of energies and lifetimes are calculated for an atom fixed near a flat dielectric mirror, or at the midpoint of the gap between two such mirrors. The theory is formulated only for the non‐retarded regime governed wholly by Coulomb forces, where all separations are much smaller than the dominant absorption wavelengths of the atom. It proceeds by constructing and then quantizing a simple model for the dielectric that admits dissipation and covers the resonant case important for experiments, where a downward atomic transition frequency can coincide with the (discrete) frequency of surface polaritons on a hypothetically non‐dissipative half‐space. The results are found first for zero temperature and then for T > 0. For completeness, an appendix extends the model to an unbounded medium.

We propose a mathematical model for opinion formation in a society that is built of two groups, one group of `ordinary¿ people and one group of `strong opinion leaders¿. Our approach is based on an opinion formation model introduced in Toscani (Toscani 2006 Commun. Math. Sci.4, 481¿496) and borrows ideas from the kinetic theory of mixtures of rarefied gases. Starting from microscopic interactions among individuals, we arrive at a macroscopic description of the opinion formation process that is characterized by a system of Fokker¿Planck-type equations. We discuss the steady states of this system, extend it to incorporate emergence and decline of opinion leaders and present numerical results.

Real-time dynamic substructuring is a new component testing method for simulating the dynamics of complex engineering systems. The physical component is tested within a computer-generated `virtual¿ environment using real-time control techniques. Delays in communication which occur between the component and the virtual environment can potentially destabilize the simulation. In this paper, the mechanism for this instability is examined using a beam-oscillator systemas a case study.We will show how the stability and the amplitude response of the system change with the time delay. Numerical simulations of the reduced system as well as a full-delayed beam equation are performed. A series of experimental tests is carried out on a beam-oscillator system. Comparison of the theoretical, numerical and experimental results is presented and these agree remarkably well.