Absorbing-state phase transitions with extremal dynamics

Extremal dynamics represents a path to self-organized criticality in which the order parameter is tuned to a value of zero. The order parameter is associated with a phase transition to an absorbing state. Given a process that exhibits a phase transition to an absorbing state, we define an ``extremal absorbing" process, providing the link to the associated extremal (nonabsorbing) process. Stationary properties of the latter correspond to those at the absorbing-state phase transition in the former. Studying the absorbing version of an extremal dynamics model allows to determine certain critical exponents that are not otherwise accessible. In the case of the Bak-Sneppen (BS) model, the absorbing version is closely related to the "$f$-avalanche" introduced by Paczuski, Maslov and Bak [Phys. Rev. E {\bf 53}, 414 (1996)], or, in spreading simulations to the "BS branching process" also studied by these authors. The corresponding nonextremal process belongs to the directed percolation universality class. We revisit the absorbing BS model, obtaining refined estimates for the threshold and critical exponents in one dimension. We also study an extremal version of the usual contact process, using mean-field theory and simulation. The extremal condition slows the spread of activity and modifies the critical behavior radically, defining an ``extremal directed percolation" universality class of absorbing-state phase transitions. Asymmetric updating is a relevant perturbation for this class, even though it is irrelevant for the corresponding nonextremal class.Comment: 24 pages, 11 figure

Reaction Diffusion Models in One Dimension with Disorder

We study a large class of 1D reaction diffusion models with quenched disorder using a real space renormalization group method (RSRG) which yields exact results at large time. Particles (e.g. of several species) undergo diffusion with random local bias (Sinai model) and react upon meeting. We obtain the large time decay of the density of each specie, their associated universal amplitudes, and the spatial distribution of particles. We also derive the spectrum of exponents which characterize the convergence towards the asymptotic states. For reactions with several asymptotic states, we analyze the dynamical phase diagram and obtain the critical exponents at the transitions. We also study persistence properties for single particles and for patterns. We compute the decay exponents for the probability of no crossing of a given point by, respectively, the single particle trajectories ($\theta$) or the thermally averaged packets ($\bar{\theta}$). The generalized persistence exponents associated to n crossings are also obtained. Specifying to the process $A+A \to \emptyset$ or A with probabilities $(r,1-r)$, we compute exactly the exponents $\delta(r)$ and $\psi(r)$ characterizing the survival up to time t of a domain without any merging or with mergings respectively, and $\delta_A(r)$ and $\psi_A(r)$ characterizing the survival up to time t of a particle A without any coalescence or with coalescences respectively. $\bar{\theta}, \psi, \delta$ obey hypergeometric equations and are numerically surprisingly close to pure system exponents (though associated to a completely different diffusion length). Additional disorder in the reaction rates, as well as some open questions, are also discussed.Comment: 54 pages, Late

[Book of abstracts]

USPCAPESCNPqFAPESPICMC Summer Meeting on Differential Equations (2016 São Carlos

Thresholds for epidemiological outbreaks

Tese de doutoramento em Ciências (área de conhecimento em Matemática)The characterization of the critical thresholds in epidemic models is probably the most important feature of the mathematical epidemiology research due to the drastic change of the disease spread on the critical threshold. Hence, the study of the critical thresholds and the epidemic behaviour near these thresholds, especially in the SIS and the SIRI epidemiological models, is present among all the chapters of this thesis. In chapter 2, we introduce the stochastic SIS model and study the dynamical evolution of the mean value, the variance and the higher moments of the infected individuals. To establish the dynamical equations for all the moments we develop recursive formulas and we observe that the dynamic of the m first moments of infecteds depend on the m + 1 moment. Using the moment closure method we close the dynamical equations for the m first moments of infecteds and we developed for every m a recursive formula to compute the equilibria manifold of these equations. In chapter 3, we consider equilibria manifold obtained from the dynamical equations for the m first moments of infecteds on the SIS model and we study when the stable equilibria can be a good approximation of the quasi-stationary mean value of infecteds. We discover that the steady states give a good approximation of the quasi-stationary states of the SIS model not only for large populations of individuals but also for small ones and not only for large infection rate values but also for infection rate values close to its critical values. In chapter 4, we consider the spatial stochastic SIS model formulated with creation and annihilation operators. We study the perturbative series expansion of the gap between the dominant and subdominant eigenvalues of the evolution operator of the model and we compute explicitly the first terms of the series expansion of the gap. In chapter 5, we present the reinfection epidemic SIRI model and study the dynamical equations for the state variables. We compute the phase transition diagram in the mean field approximation and observe the so called reinfection threshold. Moreover, we compute the phase transition lines analytically in pair approximation improving the mean field results.A caracterização de limiares críticos em modelos epidemiológicos é provavelmente o aspecto mais importante da investigação matemática em epidemiologia, devido à mudança drástica da propagação epidémica no limiar crítico. Assim, o estudo dos limiares críticos e do comportamento epidémico junto destes limiares críticos, especialmente no modelo SIS e no modelo SIRI, está presente em todos os capítulos desta tese. No capítulo 2, introduzimos o modelo estocástico SIS e estudamos a evolução dinâmica do valor médio, da variância e dos momentos de ordem mais elevada da quantidade de indivíduos infectados. Para estabelecer as equações dinâmicas para os momentos de todas as ordens desenvolvemos fórmulas recursivas e observamos que a dinâmica dos m primeiros momentos depende do momento de ordem m + 1. Utilizando o método “moment closure” fechamos as equações dinâmicas para os m primeiros momentos da quantidade de indivíduos infectados e desenvolvemos para cada m uma fórmula recursiva que permite obter os equilíbrios resultantes da dinâmica dos momentos. No capítulo 3, consideramos os equilíbrios obtidos a partir das equações de variação dos m primeiros momentos da quantidade de indivíduos infectados, no modelo SIS, e estudamos quando é que os equilíbrios estáveis constituem boas aproximações do valor médio quase-estacionário desta quantidade. Descobrimos que estes equilíbrios fornecem uma boa aproximação dos estados quase-estacionários do modelo SIS, não só para populações de grande dimensão mas também de pequena dimensão e não só para valores elevados da taxa de infecção mas também para valores da taxa de infecção próximos do valor crítico. No capítulo 4, consideramos o modelo estocástico SIS com estrutura espacial, formulado a partir dos operadores de criação e de aniquilação. Estudamos a expansão em série da diferença entre o valor próprio dominante e subdominante do operador evolução deste modelo e calculamos explicitamente os primeiros termos desta expansão. No capítulo 5, apresentamos o modelo epidemiológico de reinfecção SIRI e estudamos as equações dinâmicas para as variáveis de estado. Calculamos o diagrama de transição de fase para a aproximação de campo médio e observamos o chamado limiar crítico de reinfecção. Além disso, calculamos as linhas de transição de fase analiticamente para a aproximação par, melhorando os resultados obtidos na aproximação de campo médio.Fundação para a Ciência e a Tecnologia (FCT) - SFRH/BD/37433/2007

Velocity Modification of Power Spectrum from Absorbing Medium

Quantitative description of the statistics of intensity fluctuations within spectral line data cubes introduced in our earlier work is extended to the absorbing media. A possibility of extracting 3D velocity and density statistics from both integrated line intensity as well as from the individual channel maps is analyzed. We find that absorption enables the velocity effects to be seen even if the spectral line is integrated over frequencies. This regime that is frequently employed in observations is characterized by a non-trivial relation between the spectral index of velocities and the spectral index of intensity fluctuations. For instance when density is dominated by fluctuations at large scales, i.e. when correlations scale as r^{-\gamma}, \gamma<0, the intensity fluctuations exhibit a universal spectrum of fluctuations ~K^{-3} over a range of scales. When small scale fluctuations of density contain most of the energy, i.e. when correlations scale as r^{-\gamma}, \gamma>0, the resulting spectrum of the integrated lines depends on the scaling of the underlying density and scales as K^{-3+\gamma}. We show that if we take the spectral line slices that are sufficiently thin we recover our earlier results for thin slice data without absorption. As the result we extend the Velocity Channel Analysis (VCA) technique to optically thick lines enabling studies of turbulence in molecular clouds. In addition, the developed mathematical machinery enables a quantitative approach to solving other problems that involved statistical description of turbulence within emitting and absorbing gas.Comment: 51 page, 3 figures. Accepted to Astrophysical Journa

Individual and environmental drivers of resource use in an endangered vulture: Integrating movement, spatial and social ecology

The study of animal movement makes possible to understand how the spatial context influences individuals and populations, something especially important for species with high movement abilities able to wander routinely over large areas and face diverse humanmediated threats at different spatial scales. Vultures are long-lived scavenging vertebrates with delayed maturity and low fecundity, often exhibiting complex social behaviours. European and most of the Old World vulture populations have been historically dependent on the spatial distribution of human-provided resources. Currently, they are endangered worldwide suffering regional dramatic declines of up to 90% mainly provoked by direct and indirect persecution, infrastructure development and decrease in food resources. Effective vulture conservation requires spatially-explicit understanding of how vultures cope with resource availability and human induced transformation. In this thesis, we used spatial data collected between 2013 and 2018 from 49 GPStagged Canarian Egyptian vultures (Neophron percnopterus majorensis). The bulk of this endemic subspecies is concentrated in Fuerteventura Island where it depends on extensive goat farms, currently changing towards semi-intensive and intensive regimes. In addition, it suffers from high mortality in power lines. Movement information is combined with individual characteristics, life-history stages, and detailed data on the spatial distribution of territories, feeding resources, and human facilities. Particularly, we examined i) drivers of the use of power lines, ii) foraging tactics according to spatiotemporal variation in food predictability and individual characteristics, iii) drivers of use of livestock farms, and iv) farmers’ perception about the target species in relation to the observed spatial behaviour. Vultures’ behaviour was influenced by the distribution, predictability and amount of feeding resources, as well as by the spatial distribution of conspecifics and sources of human disturbance. At the same time, individual attributes such as sex, age, territorial status or social rank interacted with environmental cues to shape movements. Overall, vultures somewhat avoided humans by selecting farms and electric pylons far from roads or urban areas. Importantly, the predictable food resources provided by the main feeding station determined an intensive use of electric pylons and farms in its proximity. Use of both resources were however complex according to individual traits and life-history stages. Territorial status and social position in the dominance hierarchy shaped sex-specific patterns of feeding preferences, with dominant breeding females, but not males, using predictable food more intensively and choosing to settle in nearby territories. The distribution of territories affected in turn the selection of pylons and farms by both territorial and non-breeding individuals, highlighting the close interdependency and feedback between the spatial structuring of the population and the distribution of resources. Non-territorial individuals avoided resources located close to occupied breeding territories meanwhile territorial ones preferred those close to their nest, these spatial constrains being less obvious during the non-breeding season. After disentangling how those factors influence the intensity of use electric pylons, we combined predictions combined with actual mortality showing that prioritizing mitigation measures on relatively few pylons could drastically reduce accidents. Finally, farmers perceived Egyptian vultures as the most beneficial avian scavenger in the island, but owners of larger farms, which were more visited but frequently more mechanized than smaller ones, were less aware of vulture presence in their exploitations. This suggests a potential influence of modernization in livestock practices on disconnecting people from ecosystem services provided by wildlife. Moreover, the consistency between awareness and GPS data increased when vultures were more present in farms or their surroundings, indicating that scarce and endangered species are more susceptible of suffering misperception. Our findings are important from an applied point of view in a context of rapid changes in traditional livestock practices and power development, offering sound scientific knowledge that allows to make informed management decisions. The complex network of factors and patterns should be considered in the management of electric infrastructures, predictable and semi-predictable resources, or social actions for vulture conservation. General solutions ignoring population structure and the spatial distribution of territories, feeding sources or human footprint should be avoided since those could lead to unbalanced impacts between population fractions that could compromise the effectiveness of management and conservation actions

Large deviations and dynamical phase transitions in stochastic chemical networks

Chemical reaction networks offer a natural nonlinear generalisation of linear Markov jump processes on a finite state-space. In this paper, we analyse the dynamical large deviations of such models, starting from their microscopic version, the chemical master equation. By taking a large-volume limit, we show that those systems can be described by a path integral formalism over a Lagrangian functional of concentrations and chemical fluxes. This Lagrangian is dual to a Hamiltonian, whose trajectories correspond to the most likely evolution of the system given its boundary conditions. The same can be done for a system biased on time-averaged concentrations and currents, yielding a biased Hamiltonian whose trajectories are optimal paths conditioned on those observables. The appropriate boundary conditions turn out to be mixed, so that, in the long time limit, those trajectories converge to well-defined attractors. We are then able to identify the largest value that the Hamiltonian takes over those attractors with the scaled cumulant generating function of our observables, providing a non-linear equivalent to the well-known Donsker-Varadhan formula for jump processes. On that basis, we prove that chemical reaction networks that are deterministically multistable generically undergo first-order dynamical phase transitions in the vicinity of zero bias. We illustrate that fact through a simple bistable model called the Schl\"ogl model, as well as multistable and unstable generalisations of it, and we make a few surprising observations regarding the stability of deterministic fixed points, and the breaking of ergodicity in the large-volume limit

Integrated analysis of the movement and ecology of wild dingoes in the arid zone

The behaviour and use of space by mammalian predators influences the structure and function of ecosystems. Mammalian predators are often highly mobile, and in resource limited environments they need to move through spatially distinct habitats in search of prey, water, refuge, and conspecifics. This increases connectivity between ecosystems and can have important ecological implications for food web stability. Underlying a predator’s decision to move is the need to balance requirements for energy acquisition and expenditure. Advances in remote monitoring technology are increasing our ability to gain detailed, quantitative insights into the movement ecology and ecophysiology of wild animals. However, methodological and analytical complexities have impeded the integration of different biologging tools. In this thesis I explored the biology of Australia’s largest terrestrial predator, the dingo Canis dingo, and adopted an integrative approach to studying the ecology of wild individuals. My research focused on the genetics, behaviour, resource selection, and energetics of a population of dingoes in remote central Australia. The population exhibited high genetic purity (mean purity > 90%) and clear phenotypic variation in coat colour. Genetic analysis of 83 individuals revealed high levels of relatedness, and both promiscuous and monogamous mating strategies. Morphological analysis of prey remains in dingo scats collected from our study site showed European rabbits Oryctolagus cuniculus dominated their diet, which was mirrored in a meta-analysis of the diet of dingoes across the 5.4 million km2 arid zone. Whenever food availability boomed during resource pulses, hyperabundant small and medium sized mammals became their primary prey. Dingoes showed considerable dietary similarities throughout the arid zone, despite some sites being separated by a distance equivalent to that between Spain and Russia. Using captive dingoes fitted with accelerometers, we developed a classification model that predicted 14 behaviours from accelerometry data collected at the very low frequency of 1 Hz. The high accuracy (mean = 87%) of these predictions, even at a low sampling frequency, suggests that reliable, fine-scale behavioural observations of wild animals can be made over a longer period than was previously thought possible. To investigate resource use at both the individual and population level, we obtained > 150,000 GPS locations from 18 dingoes over a two year period. At the population level, dingoes selected strongly for watercourses and avoided salt lakes, which is not surprising considering the survival of canids in desert ecosystems is contingent upon access to free water and refuges. Interestingly, there was extreme individual variation in space use by dingoes, as well as seasonal differences in activity patterns where females shifted their behaviour from crepuscular to diurnal during the pup whelping and rearing seasons. We recorded accelerometry data from seven wild dingoes, and were able to apply our classification model to predict fine time-scale behaviours. We then used these behaviours to create activity-specific time-energy budgets by incorporating energetic data previously reported in the literature. This is one of the first attempts at integrating location, accelerometry, and energetic data, and allowed the comprehensive assessment of the daily costs of living in a wild canid. Our results revealed that ambient temperature (Ta) drives the activity and energetics of dingoes in the arid zone, with substantially lower activity when external temperatures were high, equating to lower daily energy expenditure in summer than in winter. Moreover, the negative relationship between dingo activity and Ta during the day implies that high heat gain from solar radiation is a factor that limits diurnal activity in an arid environment.Thesis (Ph.D.) -- University of Adelaide, School of Biological Sciences, 201

Phenomenology of local scale invariance: from conformal invariance to dynamical scaling

Statistical systems displaying a strongly anisotropic or dynamical scaling behaviour are characterized by an anisotropy exponent theta or a dynamical exponent z. For a given value of theta, we construct local scale transformations which can be viewed as scale transformations with a space-time-dependent dilatation factor. Two distinct types of local scale transformations are found. The first type may describe strongly anisotropic scaling of static systems with a given value of theta, whereas the second type may describe dynamical scaling with a dynamical exponent z. Local scale transformations act as a dynamical symmetry group of certain non-local free-field theories. Known special cases of local scale invariance are conformal invariance for theta=1 and Schrodinger invariance for theta=2. The hypothesis of local scale invariance implies that two-point functions of quasiprimary operators satisfy certain linear fractional differential equations, which are constructed from commuting fractional derivatives. The explicit solution of these yields exact expressions for two-point correlators at equilibrium and for two-point response functions out of equilibrium. Aparticularly simple and general form is found for the two-time autoresponse function. These predictions are explicitly confirmed at the uniaxial Lifshitz points in the ANNNI and ANNNS models and in the aging behaviour of simple ferromagnets such as the kinetic Glauber-Ising model and the kinetic spherical model with a non-conserved order parameter undergoing either phase-ordering kinetics or non-equilibrium critical dynamics.Comment: Latex, 73 pages, with 9 figures. Minor corrections, final for

The Impact of Climate Change on a Tropical Carnivore: From Individual to Species

Climate change is impacting species globally. Predicting which species will be impacted, where, when, and by how much, is vital to conserve biodiversity in a warming world. In this thesis, I evaluate the likely impacts of climate change on an endangered species, the African wild dog, Lycaon pictus, for which direct impacts of high ambient temperature on behaviour and recruitment have previously been identified. // Wild dogs hunt mainly in daylight, and I show they are unlikely to be able to adapt to a warming climate by hunting at night. I found nocturnal hunting was constrained by the availability of moonlight, and by the need to guard pups in the den, restricting the use of cooler night-time hours. I also show high ambient temperatures are associated with increased adult mortality, appearing to increase mortality due to human causes and disease, which is linked to human pressures through transmission from domestic dogs. // Having quantified the impacts of ambient temperature on key vital rates, I develop an Individual-Based Model to project the likely effects of climate change on population growth. I show that population projections for this species are sensitive to the emissions scenario and population size, with population collapse predicted for smaller populations under the worst-case scenario. // Finally, I use my Individual-Based Model to make spatially explicit predictions of population changes throughout the species’ remaining range. My model predicts that populations in cooler coastal regions will suffer the smallest population declines, along with populations located in East Africa. Predicted threat status of the species was dependant on the emissions scenario. // My study shows how behavioural and demographic data can be used to inform conservation planning in a changing climate. My findings also inform efforts to incorporate climate change impacts into assessments of species’ threat status by the IUCN Red List