On the net reproduction rate of continuous structured populations with distributed states at birth

We consider a nonlinear structured population model with a distributed recruitment term. The question of the existence of non-trivial steady states can be treated (at least!) in three different ways. One approach is to study spectral properties of a parametrized family of unbounded operators. The alternative approach, on which we focus here, is based on the reformulation of the problem as an integral equation. In this context we introduce a density dependent net reproduction rate and discuss its relationship to a biologically meaningful quantity. Finally, we briefly discuss a third approach, which is based on the finite rank approximation of the recruitment operator.Comment: To appear in Computers and Mathematics with Application

Test particle propagation in magnetostatic turbulence. 2: The local approximation method

An approximation method for statistical mechanics is presented and applied to a class of problems which contains a test particle propagation problem. All of the available basic equations used in statistical mechanics are cast in the form of a single equation which is integrodifferential in time and which is then used as the starting point for the construction of the local approximation method. Simplification of the integrodifferential equation is achieved through approximation to the Laplace transform of its kernel. The approximation is valid near the origin in the Laplace space and is based on the assumption of small Laplace variable. No other small parameter is necessary for the construction of this approximation method. The n'th level of approximation is constructed formally, and the first five levels of approximation are calculated explicitly. It is shown that each level of approximation is governed by an inhomogeneous partial differential equation in time with time independent operator coefficients. The order in time of these partial differential equations is found to increase as n does. At n = 0 the most local first order partial differential equation which governs the Markovian limit is regained

Pseudospectral discretization of nonlinear delay equations: new prospects for numerical bifurcation analysis

We apply the pseudospectral discretization approach to nonlinear delay models described by delay differential equations, renewal equations or systems of coupled renewal equations and delay differential equations. The aim is to derive ordinary differential equations and to investigate the stability and bifurcation of equilibria of the original model by available software packages for continuation and bifurcation for ordinary differential equations. Theoretical and numerical results confirm the effectiveness and the versatility of the approach, opening a new perspective for the bifurcation analysis of delay equations, in particular coupled renewal and delay differential equations

Manifestations of projection-induced memory: General theory and the tilted single file.

Over the years the field of non-Markovian stochastic processes and anomalous diffusion evolved from a specialized topic to mainstream theory, which transgressed the realms of physics to chemistry, biology and ecology. Numerous phenomenological approaches emerged, which can more or less successfully reproduce or account for experimental observations in condensed matter, biological and/or single-particle systems. However, as far as their predictions are concerned these approaches are not unique, often build on conceptually orthogonal ideas, and are typically employed on an ad hoc basis. It therefore seems timely and desirable to establish a systematic, mathematically unifying and clean approach starting from more fine-grained principles. Here we analyze projection-induced ergodic non-Markovian dynamics, both reversible as well as irreversible, using spectral theory. We investigate dynamical correlations between histories of projected and latent observables that give rise to memory in projected dynamics, and rigorously establish conditions under which projected dynamics is Markovian or renewal. A systematic metric is proposed for quantifying the degree of non-Markovianity. As a simple, illustrative but non-trivial example we study single file diffusion in a tilted box, which, for the first time, we solve exactly using the coordinate Bethe ansatz. Our results provide a solid foundation for a deeper and more systematic analysis of projection-induced non-Markovian dynamics and anomalous diffusion

Solitary Waves Under the Competition of Linear and Nonlinear Periodic Potentials

In this paper, we study the competition of linear and nonlinear lattices and its effects on the stability and dynamics of bright solitary waves. We consider both lattices in a perturbative framework, whereby the technique of Hamiltonian perturbation theory can be used to obtain information about the existence of solutions, and the same approach, as well as eigenvalue count considerations, can be used to obtained detailed conditions about their linear stability. We find that the analytical results are in very good agreement with our numerical findings and can also be used to predict features of the dynamical evolution of such solutions.Comment: 13 pages, 4 figure