Using Numerical Bifurcation Analysis to Study Pattern Formation in Mussel Beds

Soft-bottomed mussel beds provide an important example of ecosystem-scale self-organisation. Field data from some intertidal regions shows banded patterns of mussels, running parallel to the shore. This paper demonstrates the use of numerical bifurcation methods to investigate in detail the predictions made by mathematical models concerning these patterns. The paper focusses on the “sediment accumulation model” proposed by Liu et al (Proc. R. Soc. Lond. B 14 (2012), 20120157). The author calculates the parameter region in which patterns exist, and the sub-region in which these patterns are stable as solutions of the original model. He then shows how his results can be used to explain numerical observations of history-dependent wavelength selection as parameters are varied slowly

The stability of model ecosystems

Ecologists would like to understand how complexity persists in nature. In this thesis I have taken two fundamentally different routes to study ecosystem stability of model ecosystems: classical community ecology and classical population ecology. In community ecology models, we can study the mathematical mechanisms of stability in general, large model ecosystems. In population ecology models, fewer species are studied but greater detail of species interactions can be incorporated. Within these alternative contexts, this thesis contributes to two consuming issues concerning the stability of ecological systems: the ecosystem stability-complexity debate; and the causes of cyclic population dynamics. One of the major unresolved issues in community ecology is the relationship between ecosystem stability and complexity. In 1958 Charles Elton made the conjecture that the stability of an ecological system was coupled to its complexity and this could be a “wise principle of co-existence between man and nature” with which ecologists could argue the case for the conservation of nature for all species, including man. The earliest and simplest model systems were randomly constructed and exhibited a negative association between stability and complexity. This finding sparked the stability-complexity debate and initiated the search for organising principles that enhanced stability in real ecosystems. One of the universal laws of ecology is that ecosystems contain many rare and few common species. In this thesis, I present analytical arguments and numerical results to show that the stability of an ecosystem can increase with complexity when the abundance distribution is characterized by a skew towards many rare species. This work adds to the growing number of conditions under which the negative stability - complexity relationship can been inverted in theoretical studies. While there is growing evidence that the stability-complexity debate is progressing towards a resolution, community ecology has become increasingly subject to major criticism. A long-standing criticism is the reliance on local stability analysis. There is growing recognition that a global property called permanence is a more satisfactory definition of ecosystem stability because it tests only whether species can coexist. Here I identify and explain a positive correlation between the probability of local stability and permanence, which suggests local stability is a better measure of species coexistence than previously thought. While this offers some relief, remaining issues cause the stability-complexity debate to evade clear resolution and leave community ecology in a poor position to argue for the conservation of natural diversity for the benefit of all species. In classical population ecology, a major unresolved issue is the cause of non-equilibrium population dynamics. In this thesis, I use models to study the drivers of cyclic dynamics in Scottish populations of mountain hares (Lepus timidus), for the first time in this system. Field studies currently favour the hypothesis that parasitism by a nematode Trichostrongylus retortaeformis drives the hare cycles, and theory predicts that the interaction should induce cycling. Initially I used a simple, strategic host-parasite model parameterised using available empirical data to test the superficial concordance between theory and observation. I find that parasitism could not account for hare cycles. This verdict leaves three options: either the parameterisation was inadequate, there were missing important biological details or simply that parasites do not drive host cycles. Regarding the first option, reliable information for some hare-parasite model parameters was lacking. Using a rejection-sampling approach motivated by Bayesian methods, I identify the most likely parameter set to predict observed dynamics. The results imply that the current formulation of the hare-parasite model can only generate realistic dynamics when parasite effects are significantly larger than current empirical estimates, and I conclude it is likely that the model contains an inadequate level of detail. The simple strategic model was mathematically elegant and allowed mathematical concepts to be employed in analysis, but the model was biologically naïve. The second model is the antipode of the first, an individual based model (IBM) steeped in biological reality that can only be studied by simulation. Whilst most highly detailed tactical models are developed as a predictive tool, I instead structurally perturb the IBM to study the ecological processes that may drive population cycles in mountain hares. The model allows delayed responses to life history by linking maternal body size and parasite infection to the future survival and fecundity of offspring. By systematically removing model structure I show that these delayed life history effects are weakly destabilising and allow parameters to lie closer to empirical estimates to generate observed hare population cycles. In a third model I structurally modify the simple strategic host-parasite model to make it spatially explicit by including diffusion of mountain hares and corresponding advection of parasites (transportation with host). From initial simulations I show that the spatially extended host-parasite equations are able to generate periodic travelling waves (PTWs) of hare and parasite abundance. This is a newly documented behaviour in these widely used host-parasite equations. While PTWs are a new potential scenario under which cyclic hare dynamics could be explained, further mathematical development is required to determine whether adding space can generate realistic dynamics with parameters that lie closer to empirical estimates. In the general thesis discussion I deliberate on whether a hare-parasite model has been identified which can be considered the right balance between abstraction and relevant detail for this system

Pattern formation in the wake of external mechanisms

University of Minnesota Ph.D. dissertation. June 2016. Major: Mathematics. Advisor: Arnd Scheel. 1 computer file (PDF); xiii, 189 pages.Pattern formation in nature has intrigued humans for centuries, if not millennia. In the past few decades researchers have become interested in harnessing these processes to engineer and manufacture self-organized and self-regulated devices at various length scales. Since many natural pattern forming processes nucleate or grow from a homogeneous unstable state, they typically create defects, caused by thermal and other inherent sources of noise, which can hamper effectiveness in applications. One successful experimental method for controlling the pattern forming process is to use an external mechanism which moves through a system, transforming it from a stable state to an unstable state from which the pattern forming dynamics can take hold. In this thesis, we rigorously study partial differential equations which model how such triggering mechanisms can select and control patterns. We first use dynamical systems techniques to study the case where a spatial trigger perturbs a pattern forming freely invading front in a scalar partial differential equation. We study such perturbations for the two generic types of scalar invasion fronts, known as pulled and pushed fronts, which roughly correspond to fronts which invade either through a linear or nonlinear mechanism. Our results give the existence of perturbed fronts and provide expansions in the speed of the triggering mechanism for the wavenumber perturbation of the pattern formed. With the hope of moving towards the more complicated geometries which can arise in two spatial dimensions, where many dynamical systems methods cannot be readily applied, we also develop a functional analytic method for the study of Hopf bifurcation in the presence of continuous spectrum. Our method, while still giving computable information about the bifurcating solution, is more direct than previously proposed methods. We develop this method in the context of a triggered Cahn-Hilliard equation, in one spatial dimension, which has been used to study many triggered pattern forming systems. Furthermore, we use these abstract results to characterize an explicit example and also use our method to give a simplified proof of the bifurcation of oscillatory shock solutions in viscous conservation laws

Abstracts on Radio Direction Finding (1899 - 1995)

The files on this record represent the various databases that originally composed the CD-ROM issue of "Abstracts on Radio Direction Finding" database, which is now part of the Dudley Knox Library's Abstracts and Selected Full Text Documents on Radio Direction Finding (1899 - 1995) Collection. (See Calhoun record https://calhoun.nps.edu/handle/10945/57364 for further information on this collection and the bibliography). Due to issues of technological obsolescence preventing current and future audiences from accessing the bibliography, DKL exported and converted into the three files on this record the various databases contained in the CD-ROM. The contents of these files are: 1) RDFA_CompleteBibliography_xls.zip [RDFA_CompleteBibliography.xls: Metadata for the complete bibliography, in Excel 97-2003 Workbook format; RDFA_Glossary.xls: Glossary of terms, in Excel 97-2003 Workbookformat; RDFA_Biographies.xls: Biographies of leading figures, in Excel 97-2003 Workbook format]; 2) RDFA_CompleteBibliography_csv.zip [RDFA_CompleteBibliography.TXT: Metadata for the complete bibliography, in CSV format; RDFA_Glossary.TXT: Glossary of terms, in CSV format; RDFA_Biographies.TXT: Biographies of leading figures, in CSV format]; 3) RDFA_CompleteBibliography.pdf: A human readable display of the bibliographic data, as a means of double-checking any possible deviations due to conversion

Structure-preserving numerical methods for differential problems

It is the purpose of this talk to analyze the behaviour of multi-value numerical methods acting as structure-preserving integrators for the numerical solution of ordinary and partial differential equations (PDEs), with special emphasys to Hamiltonian problems, reaction-diffusion problems and stochastic differential equations (SDEs). The methodology we aim to follow is unifying, i.e. we propose problem-oriented numerical solvers, able to accurately and efficiently reproduce typical properties and behaviors of the above mentioned problems. Thus, according to this clear perspective, the methods we consider are adapted to the problem, in contrast with general purpose solvers that, on the contrary, do not take into account specific features of the problems. A descriptions of the issues regarding each of the above mentioned problems now follows, with the aim to describe how this unifying structure-preserving approach is adapted in concrete to each of the mentioned operators. As regards Hamiltonian problems, we analyze the nearly conservative behavior of multi-value numerical methods. Such methods, even though they cannot be symplectic, may act as structure-preserving integrators if the fulfill the properties of G-symplecticity \cite{bh2}, symmetry and bounded parasitic components which come into the numerical solution due to the multi-value nature of the solver. In particular, we provide a rigorous long-term error analysis regarding energy conservation for Hamiltonian problems \cite{bh1}, obtained by means of backward error analysis arguments, leading to sharp estimates for the parasitic solution components and for the error in the Hamiltonian. We also discuss the way these features characterize partitioned multi-value methods, specific for solving separable Hamiltonian problems, by pointing out how they can lead to overall explicit numerical schemes, also in comparison with existing symplectic partitioned Runge-Kutta methods \cite{but}. As regards PDEs, we present novel finite difference schemes for problems with periodic or oscillatory solutions of interest in the mathematical modeling of oscillatory biological systems \cite{sherratt94,sherratt09}. We mainly focus our attention on problem-oriented numerical schemes as in \cite{ref2,ref1}, based on adapted finite difference formulae arising from a twofold level of adaptation to the problem: along space, by approximating the spatial derivatives appearing in the operator by means of finite differences based on non-polynomial fitting techniques; along time, by integrating the semi-discretized problems via special purpose numerical solvers. The coefficients of the resulting methods will depend on the unknown values of parameters related to the problem (e.g. the values of the frequencies of the oscillations): suitable techniques leading to estimates of the parameters will be discussed. Moreover, further issues on the possibility to improve the efficiency of the solvers by assessing adapted IMEX numerical schemes will also be briefly described. As regards SDEs, the perspective is that of analyzing the potential of stochastic linear multistep methods to act as structure-preserving integrators, with special emphasys to numerically retaining dissipativity properties possessed by the problem \cite{bd}. Exponential mean square contractivity is analyzed, through results revealing some conditional nonlinear stability properties leading to accurate bounds for the stepsize. Numerical experiments on a selection of nonlinear problems are presented. The presented results deal with a series of joint works in collaboration with Evelyn Buckwar (Johannes Kepler University of Linz), John C. Butcher (University of Auckland), Ernst Hairer (University of Geneva) and Beatrice Paternoster (University of Salerno). \begin{thebibliography}{4} \bibitem{bd} E. Buckwar, R. D'Ambrosio, {\em Mean square contractivity of stochastic linear multistep methods}, in preparation. \bibitem{but} J.C. Butcher, R. D'Ambrosio, {\em Partitioned general linear methods for separable Hamiltonian problems}, in preparation. \bibitem{bh1} R. D'Ambrosio, E. Hairer, {\em Long-term stability of multi-value methods for ordinary differential equations}, J. Sci. Comput. 60(3), 627--640 (2014). \bibitem{bh2} R. D'Ambrosio, E. Hairer, C.J. Zbinden, {\em G-symplecticity implies conjugate-symplecticity of the underlying one-step method}, BIT 53, 867-872 (2013). \bibitem{ref2} R. D'Ambrosio, B. Paternoster, {\em Numerical solution of a diffusion problem by exponentially fitted finite difference methods}, Springer Plus 3, 425--431 (2014). \bibitem{ref1} R. D'Ambrosio, B. Paternoster, {\em Numerical solution of reaction-diffusion systems of $\lambda$-$\omega$ type by trigonometrically fitted methods}, submitted. \bibitem{sherratt94} J.A. Sherratt, {\em On the evolution of periodic plane waves in reaction-diffusion systems of $\lambda$-$\omega$ type}, SIAM J. Appl. Math. 54(5), 1374--1385 (1994). \bibitem{sherratt09} M.J. Smith, J.D.M. Rademacher, J.A. Sherratt, {\em Absolute stability of wavetrains can explain spatiotemporal dynamics in reaction-diffusion systems of lambda-omega type}, SIAM J. Appl. Dyn. Systems 8, 1136--1159 (2009). \end{thebibliography