Allee dynamics: Growth, extinction and range expansion

Bias Temperature Instability (BTI) in transistors has become a key reliability bottleneck with sub-45nm CMOS technologies. The most common models to characterize BTI are the Reaction-Diffusion (RD) and Atomistic trap-based models. This paper presents comparative impact analysis of RD and Atomistic trap-based BTI models for the SRAM Sense Amplifier. The evaluation metric, the sensing delay is analyzed for both models for the different workloads and supply voltages for 45nm technology node. The results show that the sensing delay degradation is slightly higher in RD model than Atomistic trap-based model for different workloads. Nevertheless, we observe a similar trend for both models. For example the BTI impact degradation is 6:69% for RD model and 6:57% for Atomistic trap-based model when worst case workload is applied for a 108s life time.

Are Assumptions about the Model Type Necessary in Reaction-Diffusion Modeling? A FRAP Application

Reaction-diffusion (RD) models are widely used to study the spatio-temporal evolution of pattern formation during development. Nonlinear RD models are often analytically intractable, and require numerical solution methods. Interrogation of RD models for a large physiological range of parameters covers many orders of magnitude, establishing situations where solutions are stiff and solvers fail to provide accurate results to the time-dependent problem. The spatial dependence of these parameters, and the nonlinearity of the underlying dynamics, impose additional challenges. We developed an efficient approach for simulating stiff RD models of pattern formation and we used supercomputer clusters to carry out a large screen of spatially varying parameters. The proposed approach generated data for screening of RD systems within a reasonable amount of time (a few days), which scales down further if additional cluster nodes are available. The approaches outlined herein are applicable to any systems biology problem requiring numerical approximation of RD equations with spatially non-uniform properties and stiff nonlinear reactions.

Reaction-diffusion (RD) models are widely used to study the spatio-temporal evolution of pattern formation during development. Nonlinear RD models are usually analytically intractable, and require numerical solution methods. Interrogation of RD models for a large physiological range of parameters covers many orders of magnitude- establishing situations where solutions are stiff and solvers fail to provide accurate results to the time-dependent problem. The spatial dependence of these parameters, and the nonlinearity of the underlying dynamics, impose additional challenges. We developed an efficient approach of simulating stiff RD models of pattern formation and we used supercomputer clusters to carry out a large scale screen of spatially varying parameters for biological pattern formation. The approaches outlined herein are applicable to any systems biology problem requiring numerical approximation of RD equations with spatially non-uniform properties and stiff non-linear reactions.

At elevated temperatures, pMOS transistors show a considerable drift in fundamental device parameters such as the threshold voltage when a large negative bias is applied. This phenomenon, known as negative bias temperature instability, is regarded as one of the most important reliability concerns in highly scaled pMOS transistors. Modeling efforts date back to the reaction-diffusion (RD) model proposed by Jeppson and Svensson forty years ago which has been continuously refined since then. So far, the change in the interface state density predicted by the RD model is directly used to approximate the threshold voltage shift. Here we present a coupling of the RD model to the semiconductor equations which is required to go beyond that approximation and to study degradation during realistic device operating conditions. It is also shown that such a coupled treatment is required to accurately model the behavior during the measurement phase. In addition, the RD model is extended to improve the prediction both in the stress and the relaxation phase by accounting for trap-controlled transport of the released hydrogen species.

Cells owe their internal organization to self‐organized protein patterns, which originate and adapt to growth and external stimuli via a process that is as complex as it is little understood. Here, we study the emergence, stability, and state transitions of multistable Min protein oscillation patterns in live Escherichia coli bacteria during growth up to defined large dimensions. De novo formation of patterns from homogenous starting conditions is observed and studied both experimentally and in simulations. A new theoretical approach is developed for probing pattern stability under perturbations. Quantitative experiments and simulations show that, once established, Min oscillations tolerate a large degree of intracellular heterogeneity, allowing distinctly different patterns to persist in different cells with the same geometry. Min patterns maintain their axes for hours in experiments, despite imperfections, expansion, and changes in cell shape during continuous cell growth. Transitions between multistable Min patterns are found to be rare events induced by strong intracellular perturbations. The instances of multistability studied here are the combined outcome of boundary growth and strongly nonlinear kinetics, which are characteristic of the reaction–diffusion patterns that pervade biology at many scales.

The reaction-diffusion (RD) model of epidemic spreading generally assume that contagion occurs only at the nodes of network, i.e., the links are used only for migration/diffusion of agents. However, in reality, we observe that contagion occurs also among the travelers staying in the same car, train or plane etc., which consist of the links of network. To reflect the contagious effect of links, we here present a traveling-contagion model where contagion occurs not only at nodes but also at links. Considering that the population density in transportation is generally much larger than that in districts, we introduce different infection rates for the nodes and links, respectively, whose two extreme cases correspond to the type-I and type-II reactions in the RD model [V. Colizza, R. Pastor-Satorras, A. Vespignani, Nat. Phys. 3, 276 (2007)]. Through studying three typical diffusion processes, we reveal both numerically and theoretically that the contagion at links can accelerate significantly the epidemic spreading. This finding is helpful in designing the controlling strategies of epidemic spreading.

In order to maximize the utility of the optical scattering technology in the area of bacterial colony identification, it is necessary to have a thorough understanding of how bacteria species grow into different morphological aggregation and subsequently function as distinctive optical amplitude and phase modulators to alter the incoming Gaussian laser beam. In this paper, a 2-dimentional reaction-diffusion (RD) model with nutrient concentration, diffusion coefficient, and agar hardness as variables is investigated to explain the correlation between the various environmental parameters and the distinctive morphological aggregations formed by different bacteria species. More importantly, the morphological change of the bacterial colony against time is demonstrated by this model, which is able to characterize the spatio-temporal patterns formed by the bacteria colonies over their entire growth curve. The bacteria population density information obtained from the RD model is mathematically converted to the amplitude/phase modulation factor used in the scalar diffraction theory which predicts the light scattering patterns for bacterial colonies. The conclusions drawn from the RD model combined with the scalar diffraction theory are useful in guiding the design of the optical scattering instrument aiming at bacteria colony detection and classification.

In arid regions, ecosystems are fragile, and vegetation exhibits high sensitivity to changes in climatic conditions. Vegetation patterns-non-uniform macroscopic structures formed by vegetation through temporal and spatial self-organization-serve as critical indicators of an ecosystem's adaptive capacity, post-disturbance resilience, and early warning signals of ecosystem degradation. Investigating the formation mechanisms of vegetation patterns using reaction-diffusion (RD) models represents a vital approach to deciphering vegetation evolution dynamics, with significant implications for protecting arid ecosystems. However, heterogeneous steady-state solutions of RD systems, such as Turing patterns, often reside in multistable regions. This implies that minute variations in initial conditions may lead to markedly divergent outcomes. When initial vegetation distribution data are imprecise, predictions of vegetation evolution trends and steady-state distributions in a given spatial position using RD models become highly sensitive to initial errors-a case where "minor discrepancies in input yield vastly divergent results." This study applies the three-dimensional variational data assimilation method to a RD model coupling vegetation, soil moisture, and surface water dynamics in arid regions. The results demonstrate that incorporating a modest amount of observational data can substantially enhance the model's predictive accuracy for vegetation evolution trajectories.

Conventional H/H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> and poly H/H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> Reaction-Diffusion (RD) models are compared, and the poly version is explored as a more physically likely model for predicting interface trap (N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IT</sub> ) generation during Negative Bias Temperature Instability (NBTI) in p-MOSFETs. Stochastic implementations of the conventional H/H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> RD model and the poly H/H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> RD model are realized, and their equivalence to continuum implementations are investigated for large area devices. Impact of dimensionality (1D, 2D, 3D) and device size (W, L) are explored for stochastic implementation. Stochastic simulations for small area devices using the poly H/H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> RD model show long term 1/6 power law time exponent during stress. A comprehensive framework consisting of H/H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> RD model for N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">IT</sub> along with empirical models for hole trapping (N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HT</sub> ) and bulk trap generation (N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">OT</sub> ) is able to predict experimental data for a wide variety of large area devices for different experimental conditions. Variation of small area device degradation has been simulated and compared to experimental results.

In this work we present a rigorous investigation of the negative bias temperature instability (NBTI) recovery process during measurement intervals in comparison to the numerical solution of an extended reaction-diffusion (RD) model. In contrast to previous work, the RD model has been implemented in a multi-dimensional device simulator and is solved self-consistently together with the semiconductor device equations. This allows us to directly use many commonly approximated quantities such as the oxide electric field and the interface hole concentration in a self-consistent manner. In addition, the influence of the trapped charges can be more accurately considered by using a distributed Shockley-Read-Hall interface trap-charge model which has been coupled to the RD model. Thus, due to the self-consistent solution procedure, also the feedback of these charged interface-states on the Poisson equation is considered which influences the observed threshold voltage shift. Experimental data confirm the model which has been calibrated to a wide range of temperatures using a single set of parameters.

Every morphological, behavioral, or even developmental character expression of living beings is coded in its genotype and is expressed in its phenotype. Nevertheless, the interplay between phenotypic and ontogenetic plasticities, that is, the capability to manifest trait variations, is a current field of research that needs morphometric, numerical, or even mathematical modeling investigations. In the present work, we are searching for a phenotypic index able to identify the underlying correlation among phenotypic, ontogenetic, and geographic distribution of the evolutionary development of species of the same genus. By studying the case of Pseudoplatystoma fishes, we use their skin patterns as an auxiliary trait that can be reproduced by means of a reaction diffusion (RD) model. From this model, we infer the phenotypic index in terms of one of the parameters appearing in the mathematical equations. To achieve this objective, we perform extensive numerical simulations and analysis of the model equations and link the parameter variations with different environmental and physicochemical conditions in which the individuals develop, and which may be regulated by the ontogenetic plasticity of the species. Our numerical study indicates that the patterns predicted by a set of reaction diffusion equations are not uniquely determined by the value of the parameters of the equation, but also depend on how the process is initiated and on the spatial distribution of values of these parameters. These factors are therefore significant, since they show that an individual's growth dynamics and apparent secondary transport processes, like advection, can be determinant for the alignment of motifs in a skin pattern. Our results allow us to discern the correlation between phenotypic, ontogenetic, and geographic distribution of the different species of Pseudoplatystoma fishes, thus indicating that RD models represent a useful taxonomic tool able to quantify evolutionary indexes.

We study a simple reaction-diffusion population model [proposed by A. Windus and H. J. Jensen, J. Phys. A: Math. Theor. 40, 2287 (2007)] on scale-free networks. In the case of fully random diffusion, the network topology cannot affect the critical death rate, whereas the heterogeneous connectivity can cause smaller steady population density and critical population density. In the case of modified diffusion, we obtain a larger critical death rate and steady population density, at the meanwhile, lower critical population density, which is good for the survival of species. The results were obtained using a mean-field-like framework and were confirmed by computer simulations.

The onset of pulse propagation is studied in a reaction-diffusion (RD) model with control by augmented transmission capability that is provided either along nonlocal spatial coupling or by time-delayed feedback. We show that traveling pulses occur primarily as solutions to the RD equations, while augmented transmission changes excitability. For certain ranges of the parameter settings, defined as weak susceptibility and moderate control, respectively, the hybrid model can be mapped to the original RD model. This results in an effective change in RD parameters controlled by augmented transmission. Outside moderate control parameter settings new patterns are obtained, for example, stepwise propagation due to delay-induced oscillations. Augmented transmission constitutes a signaling system complementary to the classical RD mechanism of pattern formation. Our hybrid model combines the two major signaling systems in the brain, namely, volume transmission and synaptic transmission. Our results provide insights into the spread and control of pathological pulses in the brain.

In plants, the stem cells residing in shoot apical meristems (SAM) give rise to above-ground tissues (Aichinger et al., 2012). Hence, the maintenance of stem cell niches is of central importance to a plant's continued growth and development (Fletcher and Meyerowitz, 2000; Gordon et al., 2009). For the flowering plant Arabidopsis thaliana, the genetic determinants of stem cell growth, division, and localization have been identified, and negative feedback between a homeodomain transcription factor, WUSCHEL (WUS), and a receptor kinase, CLAVATA (CLV), is known to play a crucial role in controlling the reservoir of stem cells in the central domain of a SAM. The morphology of plant stems and floral organs is controlled in large part by the size and stability of SAMs, which is controlled, in turn, by spatiotemporal patterns of WUS and CLV expression in meristems. For example, loss of restrictive signals in clv mutants of Arabidopsis leads to enlargement of shoot and floral meristems, resulting in extra floral organs and club-shaped siliques (Jonsson et al., 2005). The size, localization and stability of stem cell domains should be determined, in principle, by the interactions of WUS and CLV proteins, especially by their propensities to diffuse through the domain and by the rates of the molecular reactions that control their activities. Within this paradigm, reaction-diffusion (RD) models of WUS-CLV interactions have been popular mathematical models of SAM dynamics (Jonsson et al., 2005; Hohm et al., 2010; Fujita et al., 2011). In RD models, the spontaneous generation of inhomogeneous distributions of WUS and CLV in SAM domains is usually attributed to mechanisms based on a “Turing” instability (Turing, 1952; Segel and Jackson, 1972). The generic RD equations for spatiotemporal changes in the concentrations, u(x,t) and v(x,t), of two interacting proteins are ∂u∂t=f(u,v)+Du∂2u∂x2, ∂v∂t=g(u,v)+Dv∂2v∂x2, where f(u,v) and g(u,v) are nonlinear functions describing their local chemical interactions. A unique, uniform, steady-state solution, u(x,t) = u0 = constant and v(x,t) = v0, of these equations can become unstable with respect to small, non-uniform perturbations, u(x,t) = u0 + eλt·δu·cos(qx) and v(x,t) = v0 + eλt·δv·cos(qx), δu > Du, generating standing waves of wavelength l ≈ 2π/qcrit in the simulations of the RD system (Gierer and Meinhardt, 1972; Murray, 2003). At present, the diffusive lengths of CLV and WUS in SAMs have not been determined, and there is no evidence to suggest that the Turing condition (diffusivity of CLV >> diffusivity of WUS) is satisfied in the central zone of a SAM.