Inspired by the experiments with the three strains of E. coli bacteria as well as the three morphs of Uta stansburiana lizards, a model of cyclic dominance was proposed to investigate the mechanisms facilitating the maintenance of biodiversity in spatially structured populations. Subsequent studies enriched the original model with various biologically motivated extension repeating the proposed mathematical analysis and computer simulations. The research presented in this thesis unifies and generalises these models by combining the birth, selection-removal, selection-replacement and mutation processes as well as two forms of mobility into a generic metapopulation model. Instead of the standard mathematical treatment, more controlled analysis with inverse system size and multiscale asymptotic expansions is presented to derive an approximation of the system dynamics in terms of a well-known pattern forming equation. The novel analysis, capable of increased accuracy, is evaluated with improved numerical experiments performed with bespoke software developed for simulating the stochastic and deterministic descriptions of the generic metapopulation model. The emergence of spiral waves facilitating the long term biodiversity is confirmed in the computer simulations as predicted by the theory. The derived conditions on the stability of spiral patterns for different values of the biological parameters are studied resulting in discoveries of interesting phenomena such as spiral annihilation or instabilities caused by nonlinear diffusive terms.
We consider a two-dimensional model of three species in rock-paper-scissors competition and study the self-organisation of the population into fascinating spiraling patterns. Within our individual-based metapopulation formulation, the population composition changes due to cyclic dominance (dominance-removal and dominance-replacement), mutations, and pair-exchange of neighboring individuals. Here, we study the influence of mobility on the emerging patterns and investigate when the pair-exchange rate is responsible for spiral waves to become elusive in stochastic lattice simulations. In particular, we show that the spiral waves predicted by the system’s deterministic partial equations are found in lattice simulations only within a finite range of the mobility rate. We also report that in the absence of mutations and dominance-replacement, the resulting spiraling patterns are subject to convective instability and far-field breakup at low mobility rate. Possible applications of these resolution and far-field breakup phenomena are discussed.
Rock is wrapped by paper, paper is cut by scissors and scissors are crushed by rock. This simple game is popular among children and adults to decide on trivial disputes that have no obvious winner, but cyclic dominance is also at the heart of predator-prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms and competition in microbial populations. Cyclical interactions also emerge spontaneously in evolutionary games entailing volunteering, reward, punishment, and in fact are common when the competing strategies are three or more, regardless of the particularities of the game. Here, we review recent advances on the rock–paper–scissors (RPS) and related evolutionary games, focusing, in particular, on pattern formation, the impact of mobility and the spontaneous emergence of cyclic dominance. We also review mean-field and zero-dimensional RPS models and the application of the complex Ginzburg-Landau equation, and we highlight the importance and usefulness of statistical physics for the successful study of large-scale ecological systems. Directions for future research, related, for example, to dynamical effects of coevolutionary rules and invasion reversals owing to multi-point interactions, are also outlined.
The spatiotemporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatiotemporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising "rock-paper-scissors" interactions via dominance removal and replacement, reproduction, mutations, pair exchange, and hopping of individuals. By combining analytical and numerical methods, we obtain the model's phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the model's Hopf bifurcation. Our results allow us to clarify when spatial "rock-paper-scissors" competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility.
Species diversity in ecosystems is often accompanied by the self-organisation of the population into fascinating spatio-temporal patterns. Here, we consider a two-dimensional three-species population model and study the spiralling patterns arising from the combined effects of generic cyclic dominance, mutation, pair-exchange and hopping of the individuals. The dynamics is characterised by nonlinear mobility and a Hopf bifurcation around which the system's phase diagram is inferred from the underlying complex Ginzburg-Landau equation derived using a perturbative multiscale expansion. While the dynamics is generally characterised by spiralling patterns, we show that spiral waves are stable in only one of the four phases. Furthermore, we characterise a phase where nonlinearity leads to the annihilation of spirals and to the spatially uniform dominance of each species in turn. Away from the Hopf bifurcation, when the coexistence fixed point is unstable, the spiralling patterns are also affected by nonlinear diffusion.
An extended abstract and a poster presented at European Conference on Complex Systems 2013 (ECCS2013), 16th-20th of September 2013, World Trade Center, Barcelona, Spain.
The spiralling patterns arising from the combined effects of cyclic dominance, mutation, pair-exchange and hopping of individuals, are studied in a two-dimensional three-species population model. The system's phase diagram around a Hopf bifurcation is inferred from the underlying complex Ginzburg-Landau equation derived with a perturbative multiscale expansion. While the dynamics is generally characterised by spiralling patterns, they are shown to be stable in only one of the four phases. Furthermore, a spiral annihilation phase is discovered, in which spatially uniform dominance of each species occurs in turn.
A talk given at Mathematical Models in Ecology and Evolution 2013 conference (MMEE2013), 12th-15th of August 2013, University of York, York, United Kingdom.
A talk given at Applied Mathematics Postgraduate Seminar in 2013, University of Leeds, Leeds, United Kingdom.
A talk given at Applied Mathematics Postgraduate Seminar in 2011, University of Leeds, Leeds, United Kingdom.
Evolutionary dynamics in finite populations is known to fixate eventually in the absence of mutation. We here show that a similar phenomenon can be found in stochastic game dynamical batch learning, and investigate fixation in learning processes in a simple 2x2 game, for two-player games with cyclic interaction, and in the context of the best-shot network game. The analogues of finite populations in evolution are here finite batches of observations between strategy updates. We study when and how such fixation can occur, and present results on the average time-to-fixation from numerical simulations. Simple cases are also amenable to analytical approaches and we provide estimates of the behaviour of so-called escape times as a function of the batch size. The differences and similarities with escape and fixation in evolutionary dynamics are discussed.
The notion of risk awareness is implemented for the first time in the standard replicator equations. The system is analysed for a general 2x2 game and a non-zero-sum rock-paper-scissors in a well-mixed regime with the power spectra of fluctuations computed by the van Kampen expansion. At critical values of the risk awareness, bifurcations occur and new stable fixed points appear. These have a dramatic effect on the population dynamics and the fixation probabilities, reversing the expected behaviour. The model is subsequently applied in two population reinforced learning. Additional fixed points are also present in this setup and their appearance introduces chaos in the dynamical system. Algorithms developed for a general n-dimensional game, capable of deterministic and stochastic simulations with and without risk, are also presented.
Third harmonic cavities have been designed and fabricated by FNAL to be used at the FLASH/XFEL facility at DESY to minimise the energy spread along the bunches. Modes in these cavities are analysed and the sensitivity to frequency errors are assessed. A circuit model is employed to model the monopole bands. The monopole circuit model is enhanced to include successive cell coupling, in addition to the usual nearest neighbour coupling. A mode matching code is used to facilitate rapid simulations, incorporating fabrication errors. Curves surfaces are approximated by a series of abrupt transitions and the validity of this approach is examined.
A single photon counting PMT and a Nd:YVO4 laser were used to investigate diffraction of light through two thick slits. Self-interference was produced by reducing the intensity of laser beam to a single photon level. Interference pattern was observed at photon flux of 51 kHz with probability of two photons interacting equal to 8.5x10-5 implying a wave nature of light quanta.
Undergraduate and postgraduate univerisity notes.
A-Level Electronics Project.
* 21st Century C: C Tips From The New School - Ben Klemens
* Algorithms In C, Parts 1-4 - Robert Sedgewick
* Code Reading, The Open Source Perspective - Diomidis Spinellis
* C In A Nutshell - Peter Prinz, Tony Crawford
* C Programming: A Modern Approach - K. N. King
* Computing For Scientists: Principles Of Programming With Fortran 90 And C++ - R. J. Barlow and A. R. Barnett
* Effective Perl Programming - Joseph N. Hall and Randal L. Schwartz
* Expert C Programming: Deep Secrets - Peter van der Linden
* Hacking: The Art Of Exploitation - Jon Erickson
* Hacking Vim: A Cookbook To Get The Most Out Of The Latest Vim Editor - Kim Schulz
* Java, An Object Oriented Language - Michael Smith
* Java Backpack Reference Guide - Peter J DePasquale
* Learning The Vi And Vim Editors - Arnold Robbins, Elbert Hannah, Linda Lamb
* Linux System Programming - Robert Love
* More Effective C++ - Scott Meyers
* Numerical Methods In Finance With C++ - Maciej J. Capinski, Tomasz Zastawniak
* Parallel Programming In C With MPI And OpenMP - Michael J. Quinn
* Practical C++ Programming - Steve Oualline
* Practice Of Programming, The - Brian W. Kernighan, Rob Pike
* Pro Git - Scott Chacon
* Programming Interviews Exposed - John Mongan, Noah Suojanen
* Unix Programming Environment, The - Brian W. Kernighan, Rob Pike
* Writing Scientific Software: A Guide To Good Style - Suely Oliveira, David E. Stewart