Local Stability Criteria Research Articles

A PREDATOR–PREY INTERACTION MODEL WITH SELF- AND CROSS-DIFFUSION IN AQUATIC SYSTEMS

In this paper, the complex dynamics of a spatial aquatic system in the presence of self- and cross-diffusion are investigated. Criteria for local stability, instability and global stability are obtained. The effect of critical wavelength which can drive a system to instability is investigated. We noticed that cross-diffusion coefficient can be quite significant, even for small values of off-diagonal terms in the diffusion matrix. With the help of numerical simulation, we observed the Turing patterns (spots, strips, spot-strips mixture), regular spiral patterns and irregular patchy structures. The beauty and complexity of the Turing patterns are attributed to a large variety of symmetry properties realized by different values of predator's immunity, rate of fish predation and half saturation constant of predator population.

Strength and residual stress evaluation of stub columns fabricated from 800 MPa high-strength steel

In this study, stub columns subjected to concentric and eccentric loads were tested to check the applicability of the current local stability criteria to 800MPa high-strength steel (HSA800) recently developed in Korea. The key test variables in the concentrically loaded tests included the plate-edge restraints and the width-to-thickness ratio. Specimens made of ordinary steel (SM490) were also tested for comparison purposes. Eccentrically loaded stub column tests were conducted for a range of the P–M combinations by controlling the loading eccentricity. All the concentrically loaded specimens with non-compact and slender sections developed sufficient strengths according to the current local stability criteria. All the eccentrically loaded specimens with non-compact H-sections also exhibited a sufficient P–M interaction strength even higher than that of compact H-section counterparts. The experimental P–M interaction strength was very accurately predicted with the strain compatibility method by using the measured stress–strain curve, or by explicitly considering early strain-hardening property of high strength steel. Residual stresses were also measured by using the non-destructive indentation method to see their dependency or independency on the yield stress of steel material. The measured results of this study again indicated that the magnitude of residual stresses bears no strong relation to the yield stress of steel material, implying that the impact of residual stress on inelastic local buckling of high strength steels is less.

Unified approach of filament winding applied to complex shape mandrels

The filament winding process faces up limiting fabrication inconveniences when designing complex geometries of composite structures. Even the complete coverage of a cylindrical mandrel requires introducing deviation from geodesic trajectories. As a consequence, models for non-geodesic paths have been developed. The present research aims to establish, to solve and to validate a generic mathematical model that contributes either to wind complex shapes, or to solve common filament winding disadvantages, on the basis of an integrated strategy. This so-called unified approach leads to benefit of composite structures made by filament winding despite the limitations of the manufacturing process. Based on the mathematical description of the mandrel geometry, the theory of surfaces leads to express the local curvatures. Considering the slippage tendency of the fiber tow over the surface, a local stability criterion involving mathematical parameters of the mandrel surface is established, and a general fiber path equation can be formulated. A numerical tool is developed and applied to predict the evolution of the filament winding angle of the fiber tow placed over the surface of two axisymmetric geometries: a convex and a concave one. Experimental validation is carried out by manufacturing these geometries using a four axis filament winding machine.

Influence of prey reserve in two preys and one predator system

In this paper we have formulated and analyzed two preys one predator model with one prey dispersal in a two patch environment. Each patch is supposed to be homogeneous. Patch two constitutes a reserved area of prey and no predation is permitted in this zone whereas patch one is an open access predation zone. The growth of prey in each patch is assumed to be logistic. The transmission function from zone one due to predation is considered as a modified function of general nature. We have also studied the effect of team approach on this model. Criteria for local stability and instability of the non-negative equilibria are obtained. Using differential inequality, we obtain sufficient conditions that ensure the persistence of the system. The key results developed in this article have been illustrated using numerical simulation. These results can be interpreted in different contexts like resource conservation, pest management, bio-economics of a renewable resource etc. Keywords : Prey, Predator, Competition, Team Approach, Local Stability, Persistence.

The chaos and control of a food chain model supplying additional food to top-predator

The control and management of chaotic population is one of the main objectives for constructing mathematical model in ecology today. In this paper, we apply a technique of controlling chaotic predator–prey population dynamics by supplying additional food to top-predator. We formulate a three species predator–prey model supplying additional food to top-predator. Existence conditions and local stability criteria of equilibrium points are determined analytically. Persistence conditions for the system are derived. Global stability conditions of interior equilibrium point is calculated. Theoretical results are verified through numerical simulations. Phase diagram is presented for various quality and quantity of additional food. One parameter bifurcation analysis is done with respect to quality and quantity of additional food separately keeping one of them fixed. Using MATCONT package, we derive the bifurcation scenarios when both the parameters quality and quantity of additional food vary together. We predict the existence of Hopf point (H), limit point (LP) and branch point (BP) in the model for suitable supply of additional food. We have computed the regions of different dynamical behaviour in the quantity–quality parametric plane. From our study we conclude that chaotic population dynamics of predator prey system can be controlled to obtain regular population dynamics only by supplying additional food to top predator. This study is aimed to introduce a new non-chemical chaos control mechanism in a predator–prey system with the applications in fishery management and biological conservation of prey predator species.

Existence of spatial patterns in a predator–prey model with self- and cross-diffusion

In this work, we investigate the spatiotemporal dynamics of reaction–diffusion equations subject to cross-diffusion in the frame of a two-dimensional ratio-dependent predator–prey model. The conditions for diffusion-driven instability are obtained and the Turing space in the parameters space is achieved. Moreover, the criteria for local and global asymptotic stability of the unique positive homogeneous steady state without diffusion are discussed. Numerical simulations are carried out in order to validate the feasibility of the obtained analytical findings. Different types of spatial patterns through diffusion-driven instability of the proposed model are portrayed and analysed. Lastly, the paper finishes with an external discussion of biological relevance of the analysis regarding cross-diffusion and pattern issues.

Food web complexity analysis: effects of ecosystem changes

In this paper, we use a food web model to investigate the effects of habitat loss and species diversity on ecosystem spatial structure and its elasticity. Our model incorporates functional responses based on the Holling-type-II and the modified Lesie–Gower schemes. Criteria for local stability, global stability of the nonnegative equilibria are obtained. The permanent coexistence of the three species is also discussed. Finally, by computer simulations, we find that far-from equilibria, environmental quality is positively correlated patterns complexity communities. Knowledge of the distribution shape of interacting communities will allow us to make better predictions of environmental quality.

The Role of Viral Infection on a Bacterial Ecosystem with Nutrient Enrichment

A mathematical analysis of a virus–marine bacteria interaction model is proposed to study the role of virus infection on a bacterial community in the presence of a nutrient. Our model includes four state variables, namely, nutrient concentration, susceptible bacteria biomass, infected bacteria biomass and phage virus concentration. The phage virus infection induces viral lysis, which releases into the marine environment on average \(b\) (viral replication factor) viruses per cell. The dynamical behavior of the system is studied by deriving the existence and bounded conditions and criteria for local and global stability of solutions. The local stability for the interior positive equilibrium is studied by using Routh–Hurwitz conditions. The general bifurcation analysis is shown. The qualitative behaviour of the system is verified through numerical simulation. It is observed that inclusion of nutrients in the system changes the nature of stability of different equilibria.

Global Dynamics of a Tuberculosis Epidemic Model and the Influence of Backward Bifurcation

In this paper, we propose and analyze a tuberculosis (TB) model with exogenous re-infection. We assume that treated individual may be again infected by infectious individual. The model exhibits two bifurcations viz. transcritical bifurcation when the basic reproductive number R0 = 1 and backward bifurcation where the disease transmission rate β plays as control parameter. The persistent of the model and, the local and global stability criteria of disease-free and endemic equilibria are discussed. By carrying out bifurcation analysis, it is shown that the model exhibits the bistability and undergoes the Hopf bifurcation when immunological memory is everlasting i.e. when σ = 0. Lastly, some simulations are given to verify our analytical results.

Hall diffusion and the magnetorotational instability in protoplanetary discs

The destabilising effect of Hall diffusion in a Keplerian disc allows the MRI to occur for much lower ionisation levels than would otherwise be possible. However, simulations suggest that the consequences for the saturated state are not as significant as suggested by the linear instability. Close inspection reveals that that the simulations have not yet probed the Hall-dominated regime. Here we revisit the effect of Hall diffusion on the MRI and the implications for the extent of MHD turbulence in protoplanetary discs. We conduct a local stability analysis for a vertical, weak magnetic field subject to axisymmetric perturbations with a vertical wave vector. The diffusivity dependence is presented using analytic expressions and contours in the eta_H - eta_P plane for the maximum growth rate and corresponding wave number, the upper cut-off for unstable wave numbers, and the loci that divide the plane into regions of different characteristic behaviour. In the highly-diffusive limit the magnetic field decouples from the fluid perturbations and the diffusive MRI reduces to a diffusive plane-parallel shear instability with effective shear rate 1.5 Omega. We give analytic expressions for the growth rate and wave number of the most unstable mode. Finally, we illustrate the critical effect of Hall diffusion on the extent of dead zones in protoplanetary discs by applying a local stability criterion to a simple model of the minimum-mass solar nebula at 1 au, including x-ray and cosmic-ray ionisation and a population of 1 micron grains. Hall diffusion increases or decreases the MRI-active column density by an order of magnitude or more, depending on whether B is parallel or antiparallel to the rotation axis, respectively. Existing estimates of the depth of magnetically active layers in protoplanetary discs are likely to be wildly inaccurate. [Abridged]

고강도강재 단주의 압축강도 및 잔류응력 평가

본 연구에서는 인장강도 800MPa급 고강도강재(HSA800)의 단주 중심압축실험과 편심압축실험을 통해 균등압축과 휨-압축 부재의 강도를 평가하여 현행 강구조기준(KBC2009, AISC2005)의 적용성 여부를 검토하였다. 또한 잔류응력의 계측을 통하여 강재 항복강도와 잔류응력과의 상관성 여부도 검토하였다. 고강도강재와 일반강재의 국부좌굴 거동 차이의 여부를 확인하기 위하여 중심압축실험에 SM490 강재로 제작된 비교실험체도 포함시켰다. 강도로 무차원화한 판폭두께비와 판 단부의 지지조건을 주요변수로 하여 실험을 실시하였다. 편심압축실험은 HSA800 강재만을 대상으로 하였으며, 휨-압축의 조합력을 받는 부재의 P-M 상관관계를 알아보기 위해 가력 편심거리를 조정하여 다양한 P-M 조합에 대해 강도평가 실험을 수행하였다. 잔류응력은 중심압축실험에 사용된 H형단면 실험체를 대상으로 비파괴실험법인 압입법에 의해 가력 이전에 그 크기와 분포를 측정하였다. 실험결과 중심압축을 받는 모든 HSA800 단주는 판 단부의 지지조건 및 판폭두께비 조건에 따른 현행 강구조기준의 설계강도를 충분히 발휘하였다. 편심압축을 받는 실험체 역시 현행 설계기준의 P-M 상관관계를 충분히 안전측으로 충족하였다. 본 연구에서도 잔류응력의 크기는 강재의 항복강도와 무관하다는 선행연구결과와 합치하는 잔류응력 측정값이 얻어졌다. In this study, stub columns subjected to concentrical and eccentrical loads were tested to check the applicability of the current local stability criteria (KBC2009, AISC2005) to 800MPa high-strength steel (HSA800). The key test variables in the concentrically loaded tests included the plate-edge restraints and the width-to-thickness ratio normalized by the yield strength of steel. Specimens made of ordinary steel (SM490) were also tested for comparative purposes. Eccentrically loaded stub column tests were conducted for a range of the P-M combinations by controlling the loading eccentricity. All the concentrically loaded specimens with non-compact and slender sections developed sufficient strengths according to the current local stability criteria. All the eccentrically loaded specimens with non-compact H sections also exhibited a sufficient P-M interaction strength that was even higher than that of compact H- section counterparts. Residual stresses were also measured by using the non-destructive indentation method to demonstrate their dependency or independency on the steel material's yield strength. The measured results of this study also indicated that the magnitude of residual stresses bears no strong relation to the yield strength of the steel material.

MODELING EFFECTS OF TWO INTERACTING POPULATIONS ON A RENEWABLE RESOURCE

Abstract In this paper, a general mathematical model is presented to study the effect of two populations on a resource biomass. The interaction between two populations is assumed to be competition, predation, or cooperation. These two populations may depend on the resource biomass partially, wholly, or they may predate on the resource. In each case, criteria for local stability, instability, and global stability are obtained. Numerical simulation is presented to illustrate the theoretical results obtained in each case. It is shown that the depletion of resource biomass is maximum in the case of cooperation and is minimum in the case of competition.

DETERMINISTIC CHAOS VERSUS STOCHASTIC OSCILLATION IN A PREY-PREDATOR-TOP PREDATOR MODEL

The main objective of the present paper is to consider the dynamical analysis of a three dimensional prey-predator model within deterministic environment and the influence of environmental driving forces on the dynamics of the model system. For the deterministic model we have obtained the local asymptotic stability criteria of various equilibrium points and derived the condition for the existence of small amplitude periodic solution bifurcating from interior equilibrium point through Hopf bifurcation. We have obtained the parametric domain within which the model system exhibit chaotic oscillation and determined the route to chaos. Finally, we have shown that chaotic oscillation disappears in presence of environmental driving forces which actually affect the deterministic growth rates. These driving forces are unable to drive the system from a regime of deterministic chaos towards a stochastically stable situation. The stochastic stability results are discussed in terms of the stability of first and second order moments. Exhaustive numerical simulations are carried out to validate the analytical findings.

CFL Conditions for Runge–Kutta discontinuous Galerkin methods on triangular grids

We study time step restrictions due to linear stability constraints of Runge–Kutta Discontinuous Galerkin methods on triangular grids. The scalar advection equation is discretized in space by the Discontinuous Galerkin method with either the Lax–Friedrichs flux or the upwind flux, and integrated in time with various Runge–Kutta schemes designed for linear wave propagation problems or non-linear applications. Von–Neumann-like analyses are performed on structured periodic grids made up of congruent elements, to investigate the influence of element shape on the stability restrictions. We assess CFL conditions based on different element size measures, among which only the radius of the inscribed circle and the shortest height prove appropriate, although they are not totally independent of the triangle shape. We explain their general behaviour with respect to element quality, and report the corresponding Courant numbers with both types of flux and polynomial order p ranging from 1 to 10, for use as guidelines in practical simulations. We also compare the performance of the Lax–Friedrichs flux and the upwind flux, and we draw general conclusions about the relative computational efficiency of RK schemes. The application of CFL conditions to two examples involving respectively an unstructured and a hybrid grid confirms our results, although it shows that local stability criteria tend to yield too restrictive conditions.

A mathematical model for the effect of malicious object on computer network immune system

In this paper, a nonlinear mathematical model is proposed and analyzed to study the effect of malicious object on the immune response of the computer network. Criteria for local stability, instability and global stability are obtained. It is shown that the immune response of the system decreases as the concentration of malicious objects increases, and certain criteria’s are obtained under which it settles down at its equilibrium level. This paper shows that the malicious objects have a grave effect on cyber defense mechanism. The paper has two parts – (i) in the first part a mathematical model is proposed in which dynamics of pathogen, immune response and relative characteristic of the damaged node in the network is investigated, (ii) in second part the effect of malicious object on the immune response of the network has been examined. Finally how and where to use this modeling is discussed.

Stabilization of Discrete-time Nonlinear Systems subject to Input Saturations: a New Lyapunov Function Class

AbstractThis paper addresses the problem of stabilization of discrete-time systems including a cone-bounded nonlinearity and a saturating actuator. In the sense of Lyapunov stability, we introduce a new candidate Lyapunov function which takes nonlinearity behavior into account. The local stability criterion is formulated as a set of Bilinear Matrix Inequalities (BMI) conditions. We present an optimization problem in order to guarantee the closed-loop stability aiming the largest basin of attraction, which may be nonconvex, and/or, nonconnected. Furthermore, a simple iterative algorithm is proposed in order to solve our BMI problem. Some numerical examples are presented to highlight the relevance of the new Lyapunov function in regard to the classical quadratic function.

The minimal model of the hypothalamic–pituitary–adrenal axis

This paper concerns ODE modeling of the hypothalamic-pituitary- adrenal axis (HPA axis) using an analytical and numerical approach, combined with biological knowledge regarding physiological mechanisms and parameters. The three hormones, CRH, ACTH, and cortisol, which interact in the HPA axis are modeled as a system of three coupled, nonlinear differential equations. Experimental data shows the circadian as well as the ultradian rhythm. This paper focuses on the ultradian rhythm. The ultradian rhythm can mathematically be explained by oscillating solutions. Oscillating solutions to an ODE emerges from an unstable fixed point with complex eigenvalues with a positive real parts and a non-zero imaginary parts. The first part of the paper describes the general considerations to be obeyed for a mathematical model of the HPA axis. In this paper we only include the most widely accepted mechanisms that influence the dynamics of the HPA axis, i.e. a negative feedback from cortisol on CRH and ACTH. Therefore we term our model the minimal model. The minimal model, encompasses a wide class of different realizations, obeying only a few physiologically reasonable demands. The results include the existence of a trapping region guaranteeing that concentrations do not become negative or tend to infinity. Furthermore, this treatment guarantees the existence of a unique fixed point. A change in local stability of the fixed point, from stable to unstable, implies a Hopf bifurcation; thereby, oscillating solutions may emerge from the model. Sufficient criteria for local stability of the fixed point, and an easily applicable sufficient criteria guaranteeing global stability of the fixed point, is formulated. If the latter is fulfilled, ultradian rhythm is an impossible outcome of the minimal model and all realizations thereof. The second part of the paper concerns a specific realization of the minimal model in which feedback functions are built explicitly using receptor dynamics. Using physiologically reasonable parameter values, along with the results of the general case, it is demonstrated that un-physiological values of the parameters are needed in order to achieve local instability of the fixed point. Small changes in physiologically relevant parameters cause the system to be globally stable using the analytical criteria. All simulations show a globally stable fixed point, ruling out periodic solutions even when an investigation of the 'worst case parameters' is performed.

Hybrid, Multiresolution Wires with Massless Frictional Contacts

We describe a method for the visual interactive simulation of wires contacting with rigid multibodies. The physical model used is a hybrid combining lumped elements and massless quasistatic representations. The latter is based on a kinematic constraint preserving the total length of the wire along a segmented path which can involve multiple bodies simultaneously and dry frictional contact nodes used for roping, lassoing, and fastening. These nodes provide stick and slide friction along the edges of the contacting geometries. The lumped element resolution is adapted dynamically based on local stability criteria, becoming coarser as the tension increases, and up to the purely kinematic representation. Kinematic segments and contact nodes are added, deleted, and propagated based on contact geometries and dry friction configurations. The method gives a dramatic increase in both performance and robustness because it quickly decimates superfluous nodes without loosing stability, yet adapts to complex configurations with many contacts and high curvature, keeping a fixed, large integration time step. Numerical results demonstrating the performance and stability of the adaptive multiresolution scheme are presented along with an array of representative simulation examples illustrating the versatility of the frictional contact model.

The effect of environmental tax on the survival of biological species in a polluted environment: a mathematical model

In this paper, a nonlinear mathematical model is proposed and analyzed for the survival of biological species affected by a pollutant present in the environment. It is considered that the emission of the pollutant into the environment is dynamic in nature and depends on the environmental tax imposed on the emitters. It is also assumed that the environmental tax is imposed to control the emission of pollutants only when the concentration level of pollutants in the environment crosses a limit over which the pollutants starts causing harm to the population under consideration. Criteria for local stability, global stability and permanence are obtained using theory of ordinary differential equations. Numerical simulations are carried out to investigate the dynamics of the system using fourth order Runge–Kutta Method. It is found that, as the emission rate of pollutants in the environment increases, the density of biological species decreases. It may also be pointed out that the biological species may even become extinct if the rate of emission of pollutants increases continuously. However, if some environmental taxes are imposed to control the rate of emission of these pollutants into the environment, the density of biological species can be maintained at a desired level.

Hopf bifurcation in an eco-epidemic model

This paper deals with prey-predator system where the prey population is infected with a microparasite. Predator functional response is assumed to be Holling type I. All the feasible equilibrium points of the system are obtained and the condition for the existence of positive (interior) equilibrium point is also determined. The criteria for both local stability and instability involving system parameters are derived. The criterion for existence of Hopf-type small periodic oscillation is shown. The condition for survival of all the population is investigated. We have developed a necessary and sufficient criteria for uniform persistence. Finally, the numerical verifications of analytic results are done.