Numerical equilibrium analysis for structured consumer resource models

In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the (two-parameter) plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability (and their derivatives) as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for “Daphnia consuming algae” models in C-code. The results obtained by way of this implementation are shown in the form of graphs

Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations

In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models

On Local Bifurcations in Neural Field Models with Transmission Delays

Neural field models with transmission delay may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results

Variational principle for the Wheeler-Feynman electrodynamics

We adapt the formally-defined Fokker action into a variational principle for the electromagnetic two-body problem. We introduce properly defined boundary conditions to construct a Poincare-invariant-action-functional of a finite orbital segment into the reals. The boundary conditions for the variational principle are an endpoint along each trajectory plus the respective segment of trajectory for the other particle inside the lightcone of each endpoint. We show that the conditions for an extremum of our functional are the mixed-type-neutral-equations with implicit state-dependent-delay of the electromagnetic-two-body problem. We put the functional on a natural Banach space and show that the functional is Frechet-differentiable. We develop a method to calculate the second variation for C2 orbital perturbations in general and in particular about circular orbits of large enough radii. We prove that our functional has a local minimum at circular orbits of large enough radii, at variance with the limiting Kepler action that has a minimum at circular orbits of arbitrary radii. Our results suggest a bifurcation at some radius below which the circular orbits become saddle-point extrema. We give a precise definition for the distributional-like integrals of the Fokker action and discuss a generalization to a Sobolev space of trajectories where the equations of motion are satisfied almost everywhere. Last, we discuss the existence of solutions for the state-dependent delay equations with slightly perturbated arcs of circle as the boundary conditions and the possibility of nontrivial solenoidal orbits

Limits on Neutrino Oscillations from the CHOOZ Experiment

We present new results based on the entire CHOOZ data sample. We find (at 90% confidence level) no evidence for neutrino oscillations in the anti_nue disappearance mode, for the parameter region given by approximately Delta m**2 > 7 x 10**-4 eV^2 for maximum mixing, and sin**2(2 theta) = 0.10 for large Delta m**2. Lower sensitivity results, based only on the comparison of the positron spectra from the two different-distance nuclear reactors, are also presented; these are independent of the absolute normalization of the anti_nue flux, the cross section, the number of target protons and the detector efficiencies.Comment: 19 pages, 11 figures, Latex fil

Integration of continuous-time dynamics in a spiking neural network simulator

Contemporary modeling approaches to the dynamics of neural networks consider two main classes of models: biologically grounded spiking neurons and functionally inspired rate-based units. The unified simulation framework presented here supports the combination of the two for multi-scale modeling approaches, the quantitative validation of mean-field approaches by spiking network simulations, and an increase in reliability by usage of the same simulation code and the same network model specifications for both model classes. While most efficient spiking simulations rely on the communication of discrete events, rate models require time-continuous interactions between neurons. Exploiting the conceptual similarity to the inclusion of gap junctions in spiking network simulations, we arrive at a reference implementation of instantaneous and delayed interactions between rate-based models in a spiking network simulator. The separation of rate dynamics from the general connection and communication infrastructure ensures flexibility of the framework. We further demonstrate the broad applicability of the framework by considering various examples from the literature ranging from random networks to neural field models. The study provides the prerequisite for interactions between rate-based and spiking models in a joint simulation