scholarly journals Stability and dynamic transition of vegetation model for flat arid terrains

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lan Jia ◽  
Liang Li
Keyword(s):  
Linear Part ◽  
Dissipative System ◽  
Rectangular Domain ◽  
Simple Eigenvalue ◽  
Real Eigenvalue ◽  
Continuous Transition ◽  
Center Manifold Reduction ◽  
Dynamic Transition ◽  
Dynamic Transition Theory ◽  
Exchange Of Stability

<p style='text-indent:20px;'>In this paper, we aim to investigate the dynamic transition of the Klausmeier-Gray-Scott (KGS) model in a rectangular domain or a square domain. Our research tool is the dynamic transition theory for the dissipative system. Firstly, we verify the principle of exchange of stability (PES) by analyzing the spectrum of the linear part of the model. Secondly, by utilizing the method of center manifold reduction, we show that the model undergoes a continuous transition or a jump transition. For the model in a rectangular domain, we discuss the transitions of the model from a real simple eigenvalue and a pair of simple complex eigenvalues. our results imply that the model bifurcates to exactly two new steady state solutions or a periodic solution, whose stability is determined by a non-dimensional coefficient. For the model in a square domain, we only focus on the transition from a real eigenvalue with algebraic multiplicity 2. The result shows that the model may bifurcate to an <inline-formula><tex-math id="M1">\begin{document}$ S^{1} $\end{document}</tex-math></inline-formula> attractor with 8 non-degenerate singular points. In addition, a saddle-node bifurcation is also possible. At the end of the article, some numerical results are performed to illustrate our conclusions.</p>

scholarly journals Composite Materials Behaviour. Study, Development and Implementation of the Hypoelastic Model

2021 ◽  
Vol 58 (2) ◽  
pp. 167-175
Author(s):  
Marcela Sava
Keyword(s):  
Numerical Simulation ◽  
Composite Materials ◽  
Linear Part ◽  
Plastic State ◽  
Powder Materials ◽  
Continuous Transition ◽  
Deformation Speed ◽  
Simple Compression ◽  
Hypoelastic Model ◽  
Behaviour Study

The purpose of this research is to study and develop the formulation of a rheological law for composite materials with elasto-plastic behaviour in cold compression. Starting from the generally known relationships in literature, the hypoelastic model proposed for the composite materials behaviour (as powder materials) has been developped/explained, ensuring the understanding of the research. The hypolastic theory has been used for modeling the continuous transition from elastic to plastic state for a powder material. The material behaviour is described through an isotropic tensor relationship between the deformation speed tensor, Cauchy�s stress tensor and its derivative in relation to time (the Jaumann�s derivative). Only the linear part has been used from the general form of the law which depends on scalar functions. The calculations lead to relationships depending on five parameters which are identified according to experimental data. A numerical simulation of the stress-strain evolution during the simple compression of a diepressed powder sample is made; the numerical simulation has been validated by the experimental results.

scholarly journals Dynamic transitions for the S-K-T competition system

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ruikuan Liu ◽  
Dongpei Zhang
Keyword(s):  
Boundary Condition ◽  
Neumann Boundary ◽  
Dynamical Behaviour ◽  
Critical Surface ◽  
Homogeneous State ◽  
Bifurcation Parameter ◽  
Continuous Transition ◽  
Dynamic Transition ◽  
Competition System ◽  
Jump Transition

<p style='text-indent:20px;'>This paper is concerned with dynamical transition for biological competition system modeled by the S-K-T equations. We study the dynamical behaviour of the S-K-T equations with two different boundary conditions. For the system under non-homogeneous Dirichlet boundary condition, we show that the system undergoes a mixed dynamic transition from the homogeneous state to steady state solutions when the bifurcation parameter cross the critical surface. For the system with Neumann boundary condition, we prove that the system undergoes a mixed dynamic transition, a jump transition and a continuous transition when the bifurcation parameter cross the critical number. Finally, two examples are provided to validate the effectiveness of the theoretical results.</p>

scholarly journals Dynamic transitions and bifurcations of 1D reaction-diffusion equations: The Self-adjoint case

2021 ◽  
Author(s):  
Taylan Sengul ◽  
Burhan Tiryakioglu
Keyword(s):  
Nonlinear Term ◽  
Linear Part ◽  
Dissipative System ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Second Order Reaction ◽  
Kpp Equation ◽  
Sivashinsky Equation

This paper deals with the classification of transition phenomena in the most basic dissipative system possible, namely the 1D reaction diffusion equation. The emphasis is on the relation between the linear and nonlinear terms and the effect of the boundaries which influence the first transitions. We consider the cases where the linear part is self-adjoint with 2nd order and 4th order derivatives which is the case which most often arises in applications. We assume that the nonlinear term depends on the function and its first derivative which is basically the semilinear case for the second order reaction-diffusion system. As for the boundary conditions, we consider the typical Dirichlet, Neumann and periodic boundary settings. In all the cases, the equations admit a trivial steady state which loses stability at a critical parameter. We aim to classify all possible transitions and bifurcations that take place. Our analysis shows that these systems display all three types of transitions: continuous, jump and mixed and display transcritical, supercritical bifurcations with bifurcated states such as finite equilibria, circle of equilibria, and slowly rotating limit cycle. Many applications found in the literature are basically corollaries of our main results. We apply our results to classify the first transitions of the Chaffee-Infante equation, the Fisher-KPP equation, the Kuramoto Sivashinsky equation and the Swift-Hohenberg equation.

In Vitro Incorporation of Sodium Acetate-l-C14 into Leukocyte Lipids of Normal and Leukaemic Subjects as a Function of Incubation Time

1962 ◽  
Vol 02 (02) ◽  
pp. 165-172
Author(s):  
C Miras ◽  
G Lewis ◽  
J Mantzos
Keyword(s):  
Sodium Acetate ◽  
Linear Part ◽  
Lipid Synthesis ◽  
Normal Subjects ◽  
Cytostatic Agents ◽  
Time Intervals ◽  
Total Blood ◽  
Effect Of Treatment ◽  
Product Relation

Summary1. Separated leukocytes or total blood from normal subjects, untreated leukaemic patients and from leukaemic patients treated with cytostatic agents were incubated with CH3COONa-l-C14. Radioactivity of mixed lipids was measured at standard time intervals.2. The time incorporation curve observed with leukocytes from treated leukaemic patients showed after an initial linear part, a more rapid levelling off than the curves observed with leukocytes from untreated and normal subjects.3. Therefore, an indirect effect of treatment on leukocyte lipid synthesis seems to be present.4. Phospholipid and neutral lipid synthesis by leukaemic leukocytes was also studied. The results give no evidence that these fractions as a whole have any precursor-product relation.

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