Stochastic Skellam model

We consider the dynamics of a biological population described by the Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) equation in the case where the spatial domain consists of alternating favorable and adverse patches whose sizes are distributed randomly. For the one-dimensional case we define a stochastic analogue of the classical critical patch size. We address the issue of persistence of a population and we show that the minimum fraction of the length of favorable segments to the total length is always smaller in the stochastic case than in a periodic arrangement. In this sense, spatial stochasticity favors viability of a population.     

Phase transitions in cellular automata models of spatial susceptible--infected--resistant--susceptible epidemics

Spatially explicit models have become widely used in today's mathematical ecology and epidemiology to study the persistence of populations. For simplicity, population dynamics is often analysed by using ordinary differential equations (ODEs) or partial differential equations (PDEs) in the one-dimensional (1D) space. An important question is to predict species extinction or persistence rate by mean of computer simulation based on the spatial model. Recently, it has been reported that stable turbulent and regular waves are persistent based on the spatial susceptible--infected--resistant--susceptible (SIRS) model by using the cellular automata (CA) method in the two-dimensional (2D) space [Proc. Natl. Acad. Sci. USA 101, 18246 (2004)]. In this paper, we address other important issues relevant to phase transitions of epidemic persistence. We are interested in assessing the significance of the risk of extinction in 1D space. Our results show that the 2D space can considerably increase the possibility of persistence of spread of epidemics when the degree distribution of the individuals is uniform, i.e. the pattern of 2D spatial persistence corresponding to extinction in a 1D system with the same parameters. The trade-offs of extinction and persistence between the infection period and infection rate are observed in the 1D case. Moreover, near the trade-off (phase transition) line, an independent estimation of the dynamic exponent can be performed, and it is in excellent agreement with the result obtained by using the conjectured relationship of directed percolation. We find that the introduction of a short-range diffusion and a long-range diffusion among the neighbourhoods can enhance the persistence and global disease spread in the space.     

Percolation theory for the distribution and abundance of species

We develop and test new models that unify the mathematical relationships among the abundance of a species, the spatial dispersion of the species, the number of patches occupied by the species, the edge length of the occupied patches, and the scale on which the distribution of species is mapped. The models predict that species distributions will exhibit percolation critical thresholds, i.e., critical population abundances at which the fragmented patches (as measured by the number of patches and edge length) start to coalesce to form large patches.     

Space-variant filtering with holographic multifacet elements

We present a spatial filtering method for space-variant operations. The method is based on the combination of a holographic lenslet array and an appropriate filter array. The lenslets produce local Fourier transforms of the object which are filtered in parallel and independently by the spatial filters. The most obvious space-variant setup based on lenslet arrays would consist of many narrow 4f-setups, arranged side by side. The total object would be partitioned into many small object patches, as many as there are lens elements in the lenslet array. However this setup shows the undesired effect of local inversion. Therefore we choose a setup with only one lenslet array and an imaging lens which avoids the local inversion.     

Proximity networks and epidemics

Disease spread in most biological populations requires the proximity of agents. In populations where the individuals have spatial mobility, the contact graph is generated by the “collision dynamics” of the agents, and thus the evolution of epidemics couples directly to the spatial dynamics of the population. We first briefly review the properties and the methodology of an agent-based simulation (EPISIMS) to model disease spread in realistic urban dynamic contact networks. Using the data generated by this simulation, we introduce the notion of dynamic proximity networks which takes into account the relevant time-scales for disease spread: contact duration, infectivity period, and rate of contact creation. This approach promises to be a good candidate for a unified treatment of epidemic types that are driven by agent collision dynamics. In particular, using a simple model, we show that it can account for the observed qualitative differences between the degree distributions of contact graphs of diseases with short infectivity period (such as air-transmitted diseases) or long infectivity periods (such as HIV).     

Global Migration Can Lead to Stronger Spatial Selection than Local Migration

The outcome of evolutionary processes depends on population structure. It is well known that mobility plays an important role in affecting evolutionary dynamics in group structured populations. But it is largely unknown whether global or local migration leads to stronger spatial selection and would therefore favor to a larger extent the evolution of cooperation. To address this issue, we quantify the impacts of these two migration patterns on the evolutionary competition of two strategies in a finite island model. Global migration means that individuals can migrate from any one island to any other island. Local migration means that individuals can only migrate between islands that are nearest neighbors; we study a simple geometry where islands are arranged on a one-dimensional, regular cycle. We derive general results for weak selection and large population size. Our key parameters are: the number of islands, the migration rate and the mutation rate. Surprisingly, our comparative analysis reveals that global migration can lead to stronger spatial selection than local migration for a wide range of parameter conditions. Our work provides useful insights into understanding how different mobility patterns affect evolutionary processes.     

Epidemics in small world networks

For many infectious diseases, a small-world network on an underlying regular lattice is a suitable simplified model for the contact structure of the host population. It is well known that the contact network, described in this setting by a single parameter, the small-world parameter p, plays an important role both in the short term and in the long term dynamics of epidemic spread. We have studied the effect of the network structure on models of immune for life diseases and found that in addition to the reduction of the effective transmission rate, through the screening of infectives, spatial correlations may strongly enhance the stochastic fluctuations. As a consequence, time series of unforced Susceptible-Exposed-Infected-Recovered (SEIR) models provide patterns of recurrent epidemics with realistic amplitudes, suggesting that these models together with complex networks of contacts are the key ingredients to describe the prevaccination dynamical patterns of diseases such as measles and pertussis. We have also studied the role of the host contact strucuture in pathogen antigenic variation, through its effect on the final outcome of an invasion by a viral strain of a population where a very similar virus is endemic. Similar viral strains are modelled by the same infection and reinfection parameters, and by a given degree of cross immunity that represents the antigenic distance between the competing strains. We have found, somewhat surprisingly, that clustering on the network decreases the potential to sustain pathogen diversity.     

Persistence in Extended Dynamical Systems

Persistence in spatially extended dynamical systems (e.g., coarsening and other nonequilibrium systems) is reviewed. We discuss, in particular, the spatial correlations in the persistent regions and their evolution in time. We also discuss the dependence of the persistence behavior on the dynamics of the system, and consider the specific example of different updating rules in the temporal evolution of the system. Finally, we discuss the universal behavior shown by persistence in various stochastic models belonging to the directed percolation universality class.     

Persistence of manifolds in nonequilibrium critical dynamics

We study the persistence probability P(t) that, starting from a random initial condition, the magnetization of a d'-dimensional manifold of a d-dimensional spin system at its critical point does not change sign up to time t. For d'>0 we find three distinct late-time decay forms for P(t): exponential, stretched exponential, and power law, depending on a single parameter zeta=(D-2+eta)/z, where D=d-d' and eta,z are standard critical exponents. In particular, we predict that for a line magnetization in the critical d=2 Ising model, P(t) decays as a power law while, for d=3, P(t) decays as a power of t for a plane magnetization but as a stretched exponential for a line magnetization. Numerical results are consistent with these predictions.     

Changes in the gradient percolation transition caused by an Allee effect

The establishment and spreading of biological populations depends crucially on population growth at low densities. The Allee effect is a problem in those populations where the per capita growth rate at low densities is reduced. We examine stochastic spatial models in which the reproduction rate changes across a gradient g so that the population undergoes a 2D-percolation transition. Without the Allee effect, the transition is continuous and the width w of the hull scales as in conventional (i.e., uncorrelated) gradient percolation, w ∝ g(-0.57). However, with a strong Allee effect the transition is first order and w ∝ g(-0.26).     

Spatial coherence resonance induced by coloured noise and parameter diversity in a neuronal network

Spatial coherence resonance in a two-dimensional neuronal network induced by additive Gaussian coloured noise and parameter diversity is studied. We focus on the ability of additive Gaussian coloured noise and parameter diversity to extract a particular spatial frequency (wave number) of excitatory waves in the excitable medium of this network. We show that there exists an intermediate noise level of the coloured noise and a particular value of diversity, where a characteristic spatial frequency of the system comes forth. Hereby, it is verified that spatial coherence resonance occurs in the studied model. Furthermore, we show that the optimal noise intensity for spatial coherence resonance decays exponentially with respect to the noise correlation time. Some explanations of the observed nonlinear phenomena are also presented.     

On the relationship between cyclic and hierarchical three-species predator-prey systems and the two-species Lotka-Volterra model

Stochastic spatial predator-prey competition models represent paradigmatic systems to understand the emergence of biodiversity and the stability of ecosystems. We aim to clarify the relationship and connections between interacting three-species models and the classic two-species Lotka-Volterra (LV) model that entails predator-prey coexistence with long-lived population oscillations. To this end, we utilize mean-field theory and Monte Carlo simulations on two-dimensional square lattices to explore the temporal evolution characteristics of two different interacting three-species predator-prey systems, namely: (1) a cyclic rock-paper-scissors (RPS) model with conserved total particle number but strongly asymmetric reaction rates that lets the system evolve towards one “corner” of configuration space; (2) a hierarchical “food chain” where an additional intermediate species is inserted between the predator and prey in the LV model. For the asymmetric cyclic model variant (1), we demonstrate that the evolutionary properties of both minority species in the (quasi-)steady state of this stochastic spatial three-species “corner” RPS model are well approximated by the two-species LV system, with its emerging characteristic features of localized population clustering, persistent oscillatory dynamics, correlated spatio-temporal patterns, and fitness enhancement through quenched spatial disorder in the predation rates. In contrast, we could not identify any regime where the hierarchical three-species model (2) would reduce to the two-species LV system. In the presence of pair exchange processes, the system remains essentially well-mixed, and we generally find the Monte Carlo simulation results for the spatially extended hierarchical model (2) to be consistent with the predictions from the corresponding mean-field rate equations. If spreading occurs only through nearest-neighbor hopping, small population clusters emerge; yet the requirement of an intermediate species cluster obviously disrupts spatio-temporal correlations between predator and prey, and correspondingly eliminates many of the intriguing fluctuation phenomena that characterize the stochastic spatial LV system.     

Relation between the spatial growth rate of a traveling wave and the temporal growth rate of a standing wave

The relation between the spatial growth rate of a traveling wave and the temporal growth rate of the corresponding standing wave is examined. It is shown that, when radiation propagates in a gain medium with a sufficiently narrow gain line and a high amplification coefficient in the line center, the frequency dependences of the spatial and temporal growth rates of the field amplitude can differ significantly. In particular, at a fixed population inversion, the unbounded narrowing of the gain line, which results in an unbounded increase in the spatial growth rate and the narrowing of its frequency profile, is accompanied by neither an unbounded increase in the maximum value of the temporal growth rate nor an unbounded narrowing of the frequency profile of this growth rate.     

Placement and dimension optimization of shunted piezoelectric patches for vibration reduction

Passive structural vibration reduction by means of shunted piezoelectric patches is addressed in this paper. We present a strategy to optimize, in terms of damping efficiency, the geometry of piezoelectric patches as well as their placement on the host elastic structure. This procedure is based on the maximization of the modal electro-mechanical coupling factor (MEMCF) of the mechanical vibration mode to which the shunt is tuned. To illustrate the method, a general analytical model of a laminated beam is proposed. Two particular configurations are investigated: (i) a beam with two collocated piezoelectric patches connected in series or in parallel to the shunt and (ii) a cantilever beam with one patch. After a modal expansion, original closed-form solutions of the MEMCF are exhibited, which enables to compute optimal values for the placement, length and thickness of the piezoelectric patches that maximize the MEMCF. A dimensionless model is used so that this study can be used to design any smart beam, whatever be its dimensions. More general results about the coupling mechanisms between the piezoelectric patches and the host structure are also raised. In particular, it is found that the patches thickness is an essential parameter and that several configurations are possible, depending on the considered vibration mode. Experiments are also proposed to validate the model.     

Dynamical response of multi-patch, flux-based models to the input of infected people: Epidemic response to initiated events

The time course of an epidemic can be modeled using the differential equations that describe the spread of disease and by dividing people into “patches” of different sizes with the migration of people between these patches. We used these multi-patch, flux-based models to determine how the time course of infected and susceptible populations depends on the disease parameters, the geometry of the migrations between the patches, and the addition of infected people into a patch. We found that there are significantly longer lived transients and additional “ancillary” epidemics when the reproductive rate R is closer to 1, as would be typical of SARS (Severe Acute Respiratory Syndrome) and bird flu, than when R is closer to 10, as would be typical of measles. In addition we show, both analytical and numerical, how the time delay between the injection of infected people into a patch and the corresponding initial epidemic that it produces depends on R.     

Generalised expressions for the association and dissociation rate constants of molecules with multiple binding sites

ABSTRACT

Many proteins exhibit multiple binding patches. A patch may harbour a key chemical modification site, but may also simply act as a trap for the binding to another site. Here we consider the scenario in which one molecule (enzyme) binds another molecule (substrate) which contains two sites. We present microscopic expressions for the rate at which the enzyme binds to a particular site on the substrate, both for the scenario in which the enzyme directly binds the site without first visiting the other site, and for the case in which it may visit the other site an arbitrary number of times before binding to the site of interest. We also present the expressions for the corresponding dissociation reactions. These expressions can be used to compute in a single rare-event simulation of the dissociation pathway not only both the intrinsic and effective dissociation rate constants but also both association rate constants.     

The effect of spatial structure in adaptive evolution

We study the dynamics of adaptation in a spatially structured population. The model assumes local competition for replication, where each organism interacts only with its nearest neighbors and is inspired by experimental methods that can be used to study the process of adaptive evolution in microbes. In such experiments microbial populations are grown on petri dishes and allowed to adapt by serial passage. We compare the rate of adaptation in a structured population where the structure is maintained intact to those where movement of individuals can occur. We observe that the rate of adaptive evolution is higher and the mean effect of fixed beneficial mutations is lower in intact structures than in structures with mixing.     

Reconstruction of source distributions from sound pressures measured over discontinuous regions: multipatch holography and interpolation

In the present work, a method of alternating orthogonal projections is described in the context of near-field acoustical holography; it allows missing (or "not measured") data to be recovered, thus relieving the strictness of measurement requirements related to the use of the discrete Fourier transform. The method described here provides the detailed foundation for the patch holography procedure that has previously been introduced to mitigate finite measurement aperture effects by allowing the sound field to be iteratively extended beyond the measurement aperture. It is also shown that the latter iterative algorithm can be used regardless of the spatial distribution of measured data: i.e., patches can be discontinuous. Numerical simulations performed by using a synthetic sound field created by a point-driven, simply supported plate were used to demonstrate the latter point. In particular, a multipatch holography procedure is described that allows a source distribution to be reconstructed from the hologram pressure measured over multiple, unconnected patches. It is finally shown that a related approach allows spatial resolution enhancement by interpolation between measured points.     

Sources and sinks: a stochastic model of evolution in heterogeneous environments

We study evolution driven by spatial heterogeneity in a stochastic model of source-sink ecologies. A sink is a habitat where mortality exceeds reproduction so that a local population persists only due to immigration from a source. Immigrants can, however, adapt to conditions in the sink by mutation. To characterize the adaptation rate, we derive expressions for the first arrival time of adapted mutants. The joint effects of migration, mutation, birth, and death result in two distinct parameter regimes. These results may pertain to the rapid evolution of drug-resistant pathogens and insects.