Local Perturbation Analysis Research Articles

Identifying patterning behavior in a plant infestation of insect pests

In this study, we developed a mechanistic model formulated as a system of reaction–diffusion equations (RDE) to explore the spatiotemporal dynamics of a theoretical pest with a tillering host plant in a controlled rectangular plant field. Local perturbation analysis, a recently developed method of analysis for wave propagation, was utilized to determine patterning regimes resulting from the local and global behaviors of the slow and fast diffusing components of the RDE system, respectively. Turing analysis was done to show that the RDE system does not exhibit Turing patterns. With bug mortality as the bifurcation parameter, regions with oscillations and stable coexistence of the pest and tillers were identified. Numerical simulations illustrate the patterning regimes in 1D and 2D settings. The oscillations suggest that recurrences in pest infestation is possible. Moreover, simulations showed that patterns produced in the model are strongly influenced by the pests’ homogeneous dynamics inside the controlled environment.

Safety estimation for a new model of regenerative and frictional cutting dynamics

In our previous study on regenerative and frictional cutting dynamics, a new model considering both time-delayed regenerative effect and Stribeck effect as sources of cutting instability has been proposed [7], which has improved the prediction of linear stability in the zone of low cutting velocity. Based on the new model, this investigation explores complex nonlinear cutting dynamics. More specifically, the criticality of Hopf bifurcation on the cutting stability boundaries is studied by perturbation analysis showing the co-existence of cutting stationary and chatter in the linearly stable region, i.e., the unsafe zones (UZs). Then this analytical estimation is compared with numerical simulation revealing a possibility of underestimation due to the large-amplitude frictional chatter entering the stable region, which expands the UZs. Beside this local perturbation analysis, global bifurcation diagrams are constructed by numerical simulation, yielding various complex responses including multiple stability, regenerative chatter with loss of tool-workpiece contact and stick-slip frictional vibration. Finally, the cutting safety in the UZs is studied based on basin stability estimation, where the initial conditions are approximated by Fourier series and chatter occurrence is estimated via Monte Carlo simulation. It is found that from the statistical point of view, the large-amplitude frictional chatter hardly influences the UZs.

Local perturbation analysis of functional regression with applications

For a special functional regression model, i.e., both the dependent and independent variables are functions, we propose the concept of continuous-time perturbation analysis. For three perturbation schemes which correspond to the independent variable, dependent variable and weight function respectively,we obtain the closed form of the diagnosis function based on the functional optimization theory.As an application, we applied this approach to the system of ordinary differential equations (ODEs) and discovered the so-called boundary effect of two-step estimation ofODEs which leaves the possibility for future improvement of two-step estimation.

An elog-KdVB dynamics in non-thermal solar plasmas

This paper deals with a continued study on the basis of the gravito-electrostatic sheath (GES) model to explore the excitation of solitary and shock-like wave structures evolving in the non-thermal solar plasmas. The method applied here is based on a nonlinear local perturbation analysis over the GES structure equations designed in a thermostatistically modified form to arrive at an extended logatropic Korteweg-de Vries-Burgers (elog-KdVB) equation with a unique linear derivative source, which has in principle, a special set of multiparametric coefficients dependent on the diversified solar plasma parameters. A constructive numerical integration of the elog-KdVB equation yields the excitation of rarefactive shock-like wave patterns supported in the solar plasmas. Their noticeable unique characteristic feature is the naturalistic existence of distorted non-uniform tails. The shock-wave amplitude increases with the increase in the thermostatistical power (κ), and vice versa. In contrast, the shock-tail width decreases with the increase in the thermostatistical distribution power (κ), and vice versa. It implicates that the shock-tail width vanishes in the Boltzmann thermostatistical limit (). The corresponding gradients, phase portraits, and curvature dynamics associated with the fluctuations are illustratively depicted. The microphysical details behind the dynamics are analyzed. The elog-KdVB dynamical results explored are bolstered with the reinforcement of the earlier multispace satellitic observations and original probe measurements reported elsewhere. The non-trivial implications and applications are summarily highlighted in the real helioseismic contextual linkage.

A coupled bulk-surface model for cell polarisation

Several cellular activities, such as directed cell migration, are coordinated by an intricate network of biochemical reactions which lead to a polarised state of the cell, in which cellular symmetry is broken, causing the cell to have a well defined front and back. Recent work on balancing biological complexity with mathematical tractability resulted in the proposal and formulation of a famous minimal model for cell polarisation, known as the wave pinning model. In this study, we present a three-dimensional generalisation of this mathematical framework through the maturing theory of coupled bulk-surface semilinear partial differential equations in which protein compartmentalisation becomes natural. We show how a local perturbation over the surface can trigger propagating reactions, eventually stopped in a stable profile by the interplay with the bulk component. We describe the behavior of the model through asymptotic and local perturbation analysis, in which the role of the geometry is investigated. The bulk-surface finite element method is used to generate numerical simulations over simple and complex geometries, which confirm our analysis, showing pattern formation due to propagation and pinning dynamics. The generality of our mathematical and computational framework allows to study more complex biochemical reactions and biomechanical properties associated with cell polarisation in multi-dimensions.

Study of MRI in stratified viscous plasma configuration

We analyze the morphology of the magneto-rotational instability (MRI) for a stratified viscous plasma disk configuration in differential rotation, taking into account the so-called corotation theorem for the background profile. In order to select the intrinsic Alfvénic nature of MRI, we deal with an incompressible plasma and we adopt a formulation of the local perturbation analysis based on the use of the magnetic flux function as a dynamical variable. Our study outlines, as consequence of the corotation condition, a marked asymmetry of the MRI with respect to the equatorial plane, particularly evident in a complete damping of the instability over a positive critical height on the equatorial plane. We also emphasize how such a feature is already present (although less pronounced) even in the ideal case, restoring a dependence of the MRI on the stratified morphology of the gravitational field.

Analysis of a minimal Rho-GTPase circuit regulating cell shape

Networks of Rho-family GTPases regulate eukaryotic cell polarization and motility by controlling assembly and contraction of the cytoskeleton. The mutually inhibitory Rac–Rho circuit is emerging as a central, regulatory hub that can affect the shape and motility phenotype of eukaryotic cells. Recent experimental manipulation of the amounts of Rac and Rho or their regulators (guanine nucleotide-exchange factors, GTPase-activating proteins, guanine nucleotide dissociation inhibitors) have been shown to bias the prevalence of these different states and promote transitions between them. Here we show that part of this data can be understood in terms of inherent Rac–Rho mutually inhibitory dynamics. We analyze a spatio-temporal mathematical model of Rac–Rho dynamics to produce a detailed set of predictions of how parameters such as GTPase rates of activation and total amounts affect cell decisions (such as Rho-dominated contraction, Rac-dominated spreading, and spatially segregated Rac–Rho polarization). We find that in some parameter regimes, a cell can take on any of these three fates depending on its environment or stimuli. We also predict how experimental manipulations (corresponding to parameter variations) can affect cell shapes observed. Our methods are based on local perturbation analysis (a kind of nonlinear stability analysis), and an approximation of nonlinear feedback by sharp switches. We compare the Rac–Rho model to an even simpler single-GTPase (‘wave-pinning’) model and demonstrate that the overall behavior is inherent to GTPase properties, rather than stemming solely from network topology.

Morphology of two-dimensional MRI in axial symmetry

In this paper, we analyse the linear stability of a stellar accretion disk having a stratified morphology. The study is performed in the framework of ideal magneto-hydrodynamics and therefore results in a characterization of the linear unstable magneto-rotational modes. The peculiarity of the present scenario consists of adopting the magnetic flux function as the basic dynamical variable. Such a representation of the dynamics allows us to take account of the co-rotation theorem as a fundamental feature of the ideal plasma equilibrium and to evaluate its impact on the perturbation evolution. According to the Alfvenic nature of the magneto-rotational instability, we consider an incompressible plasma profile and perturbations propagating along the background magnetic field. Furthermore, we develop a local perturbation analysis around fiducial coordinates of the background configuration and deal with very small-scale linear dynamics in comparison to the background inhomogeneity size. The main issue of the present study is that the condition for the emergence of unstable modes is the same in the stratified plasma disk as in the case of a thin configuration. Such a feature is the result of the cancellation of the vertical derivative of the disk angular frequency from the dispersion relation, which implies that only the radial profile of the differential rotation is responsible for the trigger of the growing modes.

Local Perturbation Analysis: A Computational Tool for Biophysical Reaction-Diffusion Models

Diffusion and interaction of molecular regulators in cells is often modeled using reaction-diffusion partial differential equations. Analysis of such models and exploration of their parameter space is challenging, particularly for systems of high dimensionality. Here, we present a relatively simple and straightforward analysis, the local perturbation analysis, that reveals how parameter variations affect model behavior. This computational tool, which greatly aids exploration of the behavior of a model, exploits a structural feature common to many cellular regulatory systems: regulators are typically either bound to a membrane or freely diffusing in the interior of the cell. Using well-documented, readily available bifurcation software, the local perturbation analysis tracks the approximate early evolution of an arbitrarily large perturbation of a homogeneous steady state. In doing so, it provides a bifurcation diagram that concisely describes various regimes of the model’s behavior, reducing the need for exhaustive simulations to explore parameter space. We explain the method and provide detailed step-by-step guides to its use and application.

From simple to detailed models for cell polarization

Many mathematical models have been proposed for the process of cell polarization. Some of these are 'functional models' that capture a class of dynamical behaviour, whereas others are derived from features of signalling molecules. Some mechanistic models are detailed, and therefore complex, whereas others are simplified. Each type contributes to our understanding of cell polarization. However, the huge variety at different levels of detail makes comparisons challenging. Here, we provide examples of both elementary and more detailed models for polarization. We also display how a recent mathematical method, local perturbation analysis, can provide an appropriate tool for such comparisons. This technique simplifies and speeds up the model development process by revealing the effect of model extensions, parameter variations and in silico manipulations such as knock-out or over-expression of key molecules. Finally, simulations in both one dimension and two dimensions, and particularly in deforming two-dimensional 'cells', can highlight behaviour not captured by traditional simulation methods.

A model for intracellular actin waves explored by nonlinear local perturbation analysis

Waves and dynamic patterns in chemical and physical systems have long interested experimentalists and theoreticians alike. Here we investigate a recent example within the context of cell biology, where waves of actin (a major component of the cytoskeleton) and its regulators (nucleation promoting factors, NPFs) are observed experimentally. We describe and analyze a minimal reaction diffusion model depicting the feedback between signalling proteins and filamentous actin (F-actin). Using numerical simulation, we show that this model displays a rich variety of patterning regimes. A relatively recent nonlinear stability method, the Local Perturbation Analysis (LPA), is used to map the parameter space of this model and explain the genesis of patterns in various linear and nonlinear patterning regimes. We compare our model for actin waves to others in the literature, and focus on transitions between static polarization, transient waves, periodic wave trains, and reflecting waves. We show, using LPA, that the spatially distributed model gives rise to dynamics that are absent in the kinetics alone. Finally, we show that the width and speed of the waves depend counter-intuitively on parameters such as rates of NPF activation, negative feedback, and the F-actin time scale.

Modelling cell polarization driven by synthetic spatially graded Rac activation.

The small GTPase Rac is known to be an important regulator of cell polarization, cytoskeletal reorganization, and motility of mammalian cells. In recent microfluidic experiments, HeLa cells endowed with appropriate constructs were subjected to gradients of the small molecule rapamycin leading to synthetic membrane recruitment of a Rac activator and direct graded activation of membrane-associated Rac. Rac activation could thus be triggered independent of upstream signaling mechanisms otherwise responsible for transducing activating gradient signals. The response of the cells to such stimulation depended on exceeding a threshold of activated Rac. Here we develop a minimal reaction-diffusion model for the GTPase network alone and for GTPase-phosphoinositide crosstalk that is consistent with experimental observations for the polarization of the cells. The modeling suggests that mutual inhibition is a more likely mode of cell polarization than positive feedback of Rac onto its own activation. We use a new analytical tool, Local Perturbation Analysis, to approximate the partial differential equations by ordinary differential equations for local and global variables. This method helps to analyze the parameter space and behaviour of the proposed models. The models and experiments suggest that (1) spatially uniform stimulation serves to sensitize a cell to applied gradients. (2) Feedback between phosphoinositides and Rho GTPases sensitizes a cell. (3) Cell lengthening/flattening accompanying polarization can increase the sensitivity of a cell and stabilize an otherwise unstable polarization.

CONDITIONING OF STATE FEEDBACK POLE ASSIGNMENT PROBLEMS

In [26,27,35], condition numbers and perturbation bounds were produced for the state feedback pole assignment problem (SFPAP), for the single- and multi-input cases with simple closed-loop eigenvalues. In this paper, we consider the same problem in a different approach with weaker assumptions, producing simpler condition numbers and perturbation results. For the SFPAP, we shall show that the absolute condition number $\kappa \leq c_0 \|B^{\dagger}\| $ $ \left[ \kappa_X + \left(1 + \|F\|^2 \right)^{1/2} \right]$, where the closed-loop system matrix $A+BF = X \Lambda X^{-1}$, the closed-loop spectrum in $\Lambda$ is pre-determined, $\kappa_X \equiv\|X\| \|X^{-1}\|$, the operators $P_c (\cdot) \equiv (A+BF)(\cdot) - (\cdot) \Lambda$ and $\mathcal{N}(\cdot) \equiv (I - BB^{\dagger}) P_c(\cdot)$, and $c_0 \equiv \|I(\cdot) -P_c\left[ \mathcal{N}^{\dagger}(I - BB^{\dagger}) (\cdot) \right]\|$. With $c_B \!\equiv\!\|B\| \|B^{\dagger}\|$ and $c_1 \!\equiv\! (\|B\| \|F\|)^{-1}$, the relative condition number $\kappa_r \!\leq c_0 c_B \left[ c_1 \kappa_X \|\Lambda\| + \right. \left. \left( c_1^2 \|A\|^2 + 1 \right)^{1/2} \right]$. With $B$ well-conditioned and $\Lambda$ well chosen, $\kappa$ and $\kappa_r$ can be small even when $\Lambda$ (not necessary in Jordan form) possesses defective eigenvalues, depending on $c_0$. Consequently, the SFPAP is not intrinsically ill-conditioned. Similar results were obtained in [23], although differentiability was not established for its local perturbation analysis. Simple as well as general multiple closed-loop eigenvalues are treated.

Deterministic Versus Stochastic Cell Polarisation Through Wave-Pinning

Cell polarization is an important part of the response of eukaryotic cells to stimuli, and forms a primary step in cell motility, differentiation, and many cellular functions. Among the important biochemical players implicated in the onset of intracellular asymmetries that constitute the early phases of polarization are the Rho GTPases, such as Cdc42, Rac, and Rho, which present high active concentration levels in a spatially localized manner. Rho GTPases exhibit positive feedback-driven interconversion between distinct active and inactive forms, the former residing on the cell membrane, and the latter predominantly in the cytosol. A deterministic model of the dynamics of a single Rho GTPase described earlier by Mori et al. exhibits sustained polarization by a wave-pinning mechanism. It remained, however, unclear how such polarization behaves at typically low cellular concentrations, as stochasticity could significantly affect the dynamics. We therefore study the low copy number dynamics of this model, using a stochastic kinetics framework based on the Gillespie algorithm, and propose statistical and analytic techniques which help us analyse the equilibrium behaviour of our stochastic system. We use local perturbation analysis to predict parameter regimes for initiation of polarity and wave-pinning in our deterministic system, and compare these predictions with deterministic and stochastic spatial simulations. Comparing the behaviour of the stochastic with the deterministic system, we determine the threshold number of molecules required for robust polarization in a given effective reaction volume. We show that when the molecule number is lowered wave-pinning behaviour is lost due to an increasingly large transition zone as well as increasing fluctuations in the pinning position, due to which a broadness can be reached that is unsustainable, causing the collapse of the wave, while the variations in the high and low equilibrium levels are much less affected.

Local Perturbation Analysis of Linear Programming with Functional Relation Among Parameters

In this paper, the authors study the sensitivity analysis for a class of linear programming (LP) problems with a functional relation among the objective function parameters or those of the right-hand side (RHS). The classical methods and standard sensitivity analysis software packages fail to function when a functional relation among the LP parameters prevail. In order to overcome this deficiency, the authors derive a series of sensitivity analysis formulae and devise corresponding algorithms for different groups of homogenous LP parameters. The validity of the derived formulae and devised algorithms is corroborated by open literature examples having linear as well as nonlinear functional relations between their vector b or vector c components.

Contributions of demography and dispersal parameters to the spatial spread of a stage‐structured insect invasion

Stage-structured models that integrate demography and dispersal can be used to identify points in the life cycle with large effects on rates of population spatial spread, information that is vital in the development of containment strategies for invasive species. Current challenges in the application of these tools include: (1) accounting for large uncertainty in model parameters, which may violate assumptions of "local" perturbation metrics such as sensitivities and elasticities, and (2) forecasting not only asymptotic rates of spatial spread, as is usually done, but also transient spatial dynamics in the early stages of invasion. We developed an invasion model for the Diaprepes root weevil (DRW; Diaprepes abbreviatus [Coleoptera: Curculionidae]), a generalist herbivore that has invaded citrus-growing regions of the United States. We synthesized data on DRW demography and dispersal and generated predictions for asymptotic and transient peak invasion speeds, accounting for parameter uncertainty. We quantified the contributions of each parameter toward invasion speed using a "global" perturbation analysis, and we contrasted parameter contributions during the transient and asymptotic phases. We found that the asymptotic invasion speed was 0.02-0.028 km/week, although the transient peak invasion speed (0.03-0.045 km/week) was significantly greater. Both asymptotic and transient invasions speeds were most responsive to weevil dispersal distances. However, demographic parameters that had large effects on asymptotic speed (e.g., survival of early-instar larvae) had little effect on transient speed. Comparison of the global analysis with lower-level elasticities indicated that local perturbation analysis would have generated unreliable predictions for the responsiveness of invasion speed to underlying parameters. Observed range expansion in southern Florida (1992-2006) was significantly lower than the invasion speed predicted by the model. Possible causes of this mismatch include overestimation of dispersal distances, demographic rates, and spatiotemporal variation in parameter values. This study demonstrates that, when parameter uncertainty is large, as is often the case, global perturbation analyses are needed to identify which points in the life cycle should be targets of management. Our results also suggest that effective strategies for reducing spread during the asymptotic phase may have little effect during the transient phase.

Systematic reduction of complex tropospheric chemical mechanisms, Part I: sensitivity and time-scale analyses

Abstract. Explicit mechanisms describing the complex degradation pathways of atmospheric volatile organic compounds (VOCs) are important, since they allow the study of the contribution of individual VOCS to secondary pollutant formation. They are computationally expensive to solve however, since they contain large numbers of species and a wide range of time-scales causing stiffness in the resulting equation systems. This paper and the following companion paper describe the application of systematic and automated methods for reducing such complex mechanisms, whilst maintaining the accuracy of the model with respect to important species and features. The methods are demonstrated via application to version 2 of the Leeds Master Chemical Mechanism. The methods of Jacobian analysis and overall rate sensitivity analysis proved to be efficient and capable of removing the majority of redundant reactions and species in the scheme across a wide range of conditions relevant to the polluted troposphere. The application of principal component analysis of the rate sensitivity matrix was computationally expensive due to its use of the decomposition of very large matrices, and did not produce significant reduction over and above the other sensitivity methods. The use of the quasi-steady state approximation (QSSA) proved to be an extremely successful method of removing the fast time-scales within the system, as demonstrated by a local perturbation analysis at each stage of reduction. QSSA species were automatically selected via the calculation of instantaneous QSSA errors based on user-selected tolerances. The application of the QSSA led to the removal of a large number of alkoxy radicals and excited Criegee bi-radicals via reaction lumping. The resulting reduced mechanism was shown to reproduce the concentration profiles of the important species selected from the full mechanism over a wide range of conditions, including those outside of which the reduced mechanism was generated. As a result of a reduction in the number of species in the scheme of a factor of 2, and a reduction in stiffness, the computational time required for simulations was reduced by a factor of 4 when compared to the full scheme.

Effects of finite resistivity on magnetorotational instabilities in a realistic accretion flow

A local perturbation analysis is performed on a realistic background accretion flow in a global magnetic field. The adopted background model is an analytic solution to the resistive MHD equations and describes magnetically-controlled advection-dominated accretion flows (ADAFs) with an accuracy to the first order in the resistive corrections. The results show that there are three independent wave modes, which may be called the Rayleigh, Balbus–Hawley and resistive modes. Within our resistive-MHD corrections to the ideal-MHD limit, a Balbus–Hawley-like criterion for the instability of axisymmetric perturbations appears as a consequence of the competition between damping due to magnetic diffusion and excitation due to shear flow. As for non-axisymmetric perturbations, the former two modes are likely to be unstable in the presence of shears because the magnetic diffusion acts as a stabilizer only to axisymmetric perturbations within our approximation.

On the Conditioning of the Mixed H2/H∞-Optimisation Problem

Local perturbation analysis of the mixed H2/H∞-optimisation problem is presented. Linear (or asymptotic) perturbations bounds are obtained in terms of condition numbers.

CHAOTIC DYNAMICS OF THE PARTIALLY FOLLOWER-LOADED ELASTIC DOUBLE PENDULUM

The non-linear dynamics of the elastically restrained double pendulum, with non-conservative follower-type loading and linear damping, is re-examined with specific reference to the occurrence of chaotic motion. A local non-linear perturbation analysis is performed, showing that in three distinct regions of loading parameter space, small initial disturbances will result in, respectively, (1) static equilibrium solutions, (2) stable periodic motion, and (3) initially large changes in amplitude due to a destabilizing effect of both linear and non-linear forces. A global numerical analysis confirms the theoretical findings for regions (1) and (2), and shows that in region (3) almost all solutions are chaotic. It is suggested that chaos is triggered by a bifurcating cascade of large amplitude stable and unstable equilibrium points, which may be explored by orbits only when the zero-solution is destabilized by both linear and non-linear forces. Although heuristically based, this may be used as a practical and rather accurate predictive criterion for chaos to appear in the specific system.