Patterns and rules for sensitivity and elasticity in population projection matrices

Sensitivity and elasticity analysis of population projection matrices (PPMs) are established tools in the analysis of structured populations, allowing comparison of the contributions made by different demographic rates to population growth. In some commonly used structures of PPM, however, there are mathematically inevitable patterns in the relative sensitivity and elasticity of certain demographic rates. We take a simulation approach to investigate these mathematical constraints for a range of PPM models. Our results challenge some previously proposed constraints on sensitivity and elasticity. We also identify constraints beyond those which have already been proven mathematically, and promote them as candidates for future mathematical proof. A general theme among these rules is that changes to the demographic rates of older or larger individuals have less impact on population growth than do equivalent changes among younger or smaller individuals. However, the validity of these rules in each case depends on the choice between sensitivity and elasticity, the growth rate of the population and the PPM structure used. If the structured population conforms perfectly to the assumptions of the PPM used to model it, the rules we describe represent biological reality, allowing us to prioritise management strategies in the absence of detailed demographic data. Conversely, if the model is a poor fit to the population (specifically; if demographic rates within stages are heterogeneous) such analyses could lead to inappropriate management prescriptions. Our results emphasise the importance of choosing a structured population model which fits the demographics of the population

Robustness of infinite dimensional systems

The results contained within this thesis concern an abstract framework for a robustness analysis of exponential stability of infinite dimensional systems. The abstract analysis relies on the strong relationship between exponential stability and L2-stability which exists for many classes of linear systems. In Chapter 1a "stability radius", for systems governed by semigroups, is developed, for a class of "structured" perturbations of its generator. The abstract theory is illustrated by examples of perturbations of the boundary data for homogeneous boundary value problems and also perturbations arising due to neglected delay terms in differential delay equations. In Chapter 2a related problem of a non standard linear quadratic problem is studied, which leads to a stability analysis for certain nonlinear systems. In Chapter 3 an abstract L2-stability theory is developed and then applied to integrodifferential equations and time-varying systems, to investigate the robustness of exponential stability of such systems

Learning of spatio–temporal codes in a coupled oscillator system

©2009 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.In this paper, we consider a learning strategy that allows one to transmit information between two coupled phase oscillator systems (called teaching and learning systems) via frequency adaptation. The dynamics of these systems can be modelled with reference to a number of partially synchronized cluster states and transitions between them. Forcing the teaching system by steady but spatially nonhomogeneous inputs produces cyclic sequences of transitions between the cluster states, that is, information about inputs is encoded via a “winnerless competition” process into spatio–temporal codes. The large variety of codes can be learned by the learning system that adapts its frequencies to those of the teaching system. We visualize the dynamics using “weighted order parameters (WOPs)” that are analogous to “local field potentials” in neural systems. Since spatio–temporal coding is a mechanism that appears in olfactory systems, the developed learning rules may help to extract information from these neural ensembles

Dynamics on networks of cluster states for globally coupled phase oscillators

Copyright © by Society for Industrial and Applied Mathematics. Unauthorized reproduction of this article is prohibited. Its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.Systems of globally coupled phase oscillators can have robust attractors that are heteroclinic networks. We investigate such a heteroclinic network between partially synchronized states where the phases cluster into three groups. For the coupling considered there exist 30 different three-cluster states in the case of five oscillators. We study the structure of the heteroclinic network and demonstrate that it is possible to navigate around the network by applying small impulsive inputs to the oscillator phases. This paper shows that such navigation may be done reliably even in the presence of noise and frequency detuning, as long as the input amplitude dominates the noise strength and the detuning magnitude, and the time between the applied pulses is in a suitable range. Furthermore, we show that, by exploiting the heteroclinic dynamics, frequency detuning can be encoded as a spatiotemporal code. By changing a coupling parameter we can stabilize the three-cluster states and replace the heteroclinic network by a network of excitable three-cluster states. The resulting “excitable network” has the same structure as the heteroclinic network and navigation around the excitable network is also possible by applying large impulsive inputs. We also discuss features that have implications for related models of neural activity

Adaptive high-gain fast sampling P-control

In this paper we show that well-known high-gain universal adaptive P-controllers can be implemented digitally, via adaptive sampling, provided that the length of the sampling interval increases sufficiently fast, as the proportional gain increases. Both stabilization and lambda-tracking of arbitrary bounded and essentially smooth reference signals are considered

Adaptive high-gain λ-[lambda]-tracking with variable sampling rate

It is well known that proportional output feedback control can stabilize any relative-degree one, minimum-phase system if the sign of the feedback is correct and the proportional gain is high enough. Moreover, there exists simple adaptation laws for tuning the proportional gain (the so-called high-gain adaptive controllers) which are not based on system identification or plant parameter estimation algorithms or injection of probing signals. If tracking of signals is desired, then these simple controllers are also applicable without invoking an internal model if the tracking error is not necessarily supposed to converge to zero but towards a ball around zero of arbitrarily small but prespecified radius lambda>0. In this note we consider a sampled version of the high-gain adaptive lamda-tracking controller. The motivation for sampling arises from the possibility that the output of a system may not be available continuously, but only at discrete time instants. The problem is that the stiffness of the system increases as the proportional gain is increased. Our result shows that adaptive sampling tracking is possible if the product hk of the decreasing sampling rate h and the increasing proportional gain k decreases at a rate proportional to 1/log k

Simple adaptive stabilization of high-gain stabilizable systems

It is shown that the simple adaptive feedback strategy u(t)=1ln k(t) cos1/ln(t).k(t)=y(y)² is a universal adaptive stabilizer for the class of single-input, single-output, finite-dimensional, linear systems which are stabilizable by either negative or positive high-gain feedback

Adaptive sampling control of high-gain stabilizable systems

It is well known that proportional output feedback control can stabilize any relative-degree one, minimum-phase system if the sign of the feedback is correct and the proportional gain is high enough. Moreover, there exist simple adaptation laws for tuning the proportional gain (so-called high-gain adaptive controllers) which do not need to know the system and do not attempt to identify system parameters. In this paper the authors consider sampled versions of the highgain adaptive controller. The motivation for sampling arises from the possibility that the output of a system may not be available continuously, but only at sampled times. The main point of interest is the need to develop techniques for adapting the sampling rate, since the stiffness of the system increases as the proportional gain is increased. Our main result shows that adaptive sampling stabilization is possible if the product hk of the decreasing sampling interval h and the increasing proportional gain k decreases at a rate proportional to 1= log k